A Competing Risks Model with Time-varying Heterogeneity and
Simultaneous Failure
Ruixuan Liu
Department of Economics
Emory University
Atlanta, GA 30322
This version: December 2015
Abstract
This paper proposes a new bivariate competing risks model that speci…es each marginal duration as
the …rst time a Lévy subordinator crosses a random threshold. Our speci…cation is a natural variant
of the competing risks version of the mixed proportional hazards model, but it allows time-varying
heterogeneity and simultaneous termination of both durations. When the structural multiplicative e¤ects
from covariates are of linear-index forms, we identify these e¤ects, the baseline hazard function, and the
characteristics of latent Lévy subordinators with competing risks data. Semiparametric estimation of
…nite dimensional parameter in our model is developed based on certain average derivative estimation.
Keywords: Competing Risks, Duration Analysis, First Passage Time, Lévy Copula, Lévy Processes.
JEL Code: C14, C24, C34.
I am deeply indebted to my mentor and supervisor Yanqin Fan for her continuing guidance and support. I also appreciate
detailed and inspiring comments from Jaap Abbring, Irene Botosaru, Emmanuel Guerre, Qi Li, Eric Renault, Aman Ullah, and
Jon Wellner.
1
Introduction
In empirical studies, a spell of duration could potentially end for various reasons, whereas only the realized
one is recorded in the dataset. This phenomenon is captured by the competing risks model, where two
or more types of durations may occur on a subject, but only the one occurring …rst is observed together
with its occurrence time, and other durations are censored. For two durations or failure times (Y1 ; Y2 ) of
interest, economists or econometricians only observe the minimum Y = min(Y1 ; Y2 ) and the cause of failure.
Applications of competing risks models in economics include Flinn and Heckman’s (1982), who investigated
the duration of unemployment where an individual can exit unemployment either by …nding a job or by
leaving the labor market; Katz and Meyer’s (1990), who studied the probability of leaving unemployment
through recalls or new jobs; Deng, Quigley, and Van Order’s (2000), who investigated mortgage termination
by prepayment or default; Honore and Lleras-Muney (2006) who studied changes in cancer and cardiovascular
mortality since 1970; and Brown and Dinc’s (2010), who examined banks’survival where the failure could
be triggered by bankrupcy or acquisition. The competing risks structure is generic for many other economic
models, including the classical Roy model in Heckman and Honore (1990), where workers self-select the
sector that gives them the highest expected earnings, and the …rst-price auction model in Athey and Haile
(2003) where only the winning price and winner’s identity are available in the data.
This paper proposes and studies a new bivariate competing risks model that speci…es each marginal
duration as the …rst time a stochastic process crosses a random threshold, and imposes dependence between
two durations via the unobservable dependent bivariate processes. Here each marginal stochastic process is
assumed to be a Lévy subordinator, i.e., a continuous-time process with stationary, independent, and nonnegative increments, and we resort to the Lévy copula (Kallsen and Tankov, 2006) to generate dependence
between two latent Lévy subordinators without restricting their marginal speci…cations. Meanwhile two
random thresholds are assumed to follow independent unit exponential distributions, and covariates enter
into the model in a multiplicative way, basically changing the magnitude of jumps of latent processes. Such
a model is of substantial interest because not only is it related to optimal stopping time models where
agents optimally time discrete actions (Heckman and Navarro, 2007; Abbring, 2012), but it also applies to
the scenario in which the termination of duration is caused by some gradual and irreversible accumulation
of damage (Singpurwalla, 1995,2005). The choice of latent processes, thresholds, covariates’ e¤ects and
dependence structure in our model could be seen as a natural variant of the competing risks version of the
mixed proportional hazards model (MPH) from a stochastic process point of view. In contrast with the
traditional hazard-based models including MPH, our model embeds time-varying heterogeneity represented
by the latent processes, and it allows simultaneous termination of both durations with positive probability.
Surprisingly, the conditional distribution function of two durations in our model is of the same form as that in
Marshall and Olkin (1967). Compared with the original and existing generalizations of Marshall-Olkin type
model (Klein, Keiding, and Kamby, 1989; An, Christensen, and Gupta, 2004), the durations in our model are
de…ned in a highly structural way and the concurrent termination of both durations is not necessarily driven
1
by some independent and common shock. Thus, I will name the present model the Extended Marshall-Olkin
(EMO) model. EMO exhibits tractable mathematical structures and interesting probablistic properties, and
has empirical content under a competing risks scheme where one does not observe both durations sequentially.
Our contributions are mainly three-fold. First of all, the current paper contributes to the literature on the
process-based duration models advanced by Abbring (2012) via proposing a bivariate durations model where
both durations are described by their threshold-crossing behavior. The combination of our latent processes
and threholds di¤er from the ones in Abbring (2012), hence resulting in di¤erent model setup, identi…cation
and inference procedures. In the sequel, we show that our construction resembles the bivariate version of
MPH, as the latter could be put into the same framework, with its duration de…ned by the …rst passage
time of a deterministically increasing function (multiplied by some static heterogeneity term) crossing a
unit exponential threshold, and the dependence between two durations is caused by unobservable static
heterogeneity terms.1 Our model complements Abbring’s (2012), as the sample path of the latent process
here is non-decreasing, therefore it is more suitable to capture the failure due to wearout or aging e¤ects.
Second, it is well-known that the competing risks scheme induces delicate (non-)identi…ability issues once
dependence between two durations is allowed, and it continues to attract attention from researchers, see
Heckman and Honore (1989,1990), Abbring and van den Berg (2003), Lee (2006), Lee and Lewbel (2012),
Khan and Tamer (2009), and Fan and Liu (2013). However, none of the aforementioned identi…cation
results apply to our present model. Instead, we make novel use of the analytical properties of characteristic
functions of Lévy processes and the special Marshall-Olkin structure arising from simultaneous jumping
behavior of the latent processes. Assuming multiplicative e¤ects from observable covariates take some
linear index forms, we could identify these e¤ects, the baseline hazard function, and the characteristics
of latent Lévy processes separately even in a competing risks setting. Last but not least, the identi…cation
results immediately suggest straightforward semiparametric estimators based on certain average derivative
estimation (Stoker, 1986; Powell, Stock, and Stoker, 1989). We establish root-n consistency and asymptotic
normality of proposed estimator for the …nite dimensional parameters in structural multiplicative e¤ects.
This is remarkable because although the competing risks version of MPH is known to be nonparametrically
identi…able (Heckman and Honore, 1989; Abbring and van den Berg, 2003), only partial answers have been
given regarding non-/semiparametric estimation, see Fermanian (2004) and Chen (2010). When it comes
to the existing process-based models, the literature is restricted to the univariate duration models and the
suggested estimation method is fully parametric with numerical inversion involved, see Abbring (2012),
Abbring and Salimans (2012).
We are now in a position to describe the agenda for this paper. Section 2 contains a brief review of related
literature. Section 3 presents the new univariate duration model in Liu (2015) after reexaming MPH from a
process point of view. In Section 4, we construct the extended Marshall-Olkin model for bivariate durations
1 Another close connection to MPH is that the distribution of non-negative heterogeneity term is often assumed to be in…nitely
divisible, such as gamma distribution in Lancaster (1979) and stable distribution in Hougaard (1986ab). Correspondingly in
the present model, it is well known that any Lévy subordinator has non-negative in…nitely divisible distribution (Sato, 1999).
2
and derive the conditional joint survivial function. Various probabilistic properties and its distinction with
the original Marshall-Olkin model are also discussed. Section 5 presents its empirical content under competing risks with the semiparametric estimation procedure for …nite dimensional parameters in Section 6. We
conclude in Section 7 discussing potential applications and possible extensions. The Appendices containing
all the proofs are further divided into three sections. Appendix A collects the proofs of main results. More
technical proofs of the linear representation of the semiparametric estimator are in Appendix B. Appendix
C lists various de…nitions and properties related to Lévy processes theory for easy reference.
2
Literature Review
In this section we review two branches of related literature on nonparametric identi…cation and estimation
of competing risks models and recent advance on process-based duration models.
The potential identi…cation failure of competing risks model has a long history dating back to Cox (1959).
Tsiatis (1975) comes up with an explicit construction to demonstrate the non-identi…ability of the marginal
distribution function of Y1 once the independence between (Y1 ; Y2 ) is dispensed with. Crowder (1991)
further ampli…es this identi…ability crisis by proving that even if the two marginal distribution functions
of (Y1 ; Y2 ) are given, their joint distribution still remains unidenti…ed generally. In response to the identi…ability crisis, three general approaches have been taken in the literature. First, covariate information
and speci…c model structures imposed on the marginal distributions may restore point identi…cation, see
e.g., the proportional hazards and accelerated failure time models in Heckman and Honore (1989), Abbring
and van den Berg (2003), Lee (2006), and Lee and Lewbel (2012); Second, in the spirit of Peterson (1976),
worst-case bounds have been adapted to accelerated failure time model with discrete regressors in Honore
and Lleras-Muney (2006), and censored quantile regression in Khan and Tamer (2009). Finally, assuming
a known copula for the individual risks, Zheng and Klein (1995) …rst extend point identi…cation results for
independent risks to dependent risks and propose a copula-graphic estimator of the marginal survival function. When the copula function is Archimedean and known, Rivest and Wells (2001) …rst derive an explicit
expression for the copula-graphic estimator of the survival function proposed in Zheng and Klein (1995). In
the context of a censored quantile regression where the independent variable is subject to certain dependent
censoring, Fan and Liu (2013) consider partial identi…cation where the copula function is allowed to vary
within a prespeci…ed class. Fan and Liu (2013) demonstrate the possibility of getting more informative
insight for the conditional quantile function over the approach taking worst-case bounds.
Even though MPH are known to be nonparametrically identi…able with competing risks data, the literature on non-/semiparametric estimation is still at its infancy. Fermanian (2004) has proved several
consistency results for di¤erent model primitives based on kernel estimators, but no asymptotic distribution
has been developed. Moreover, it is not clear what the optimal rates of convergence are. With a known
survival copula function between the heterogeneity terms, Chen (2010) has studied a semiparametric MLE
for …nite dimensional parameter in the model without restricting the baseline hazard functions. Burda,
3
Harding, and Hausman (2015) have proposed Bayesian estimation methods with a piece-wise linear hazard
function and heterogeneity terms having an in…nite mixture of generalized inverse Gaussian densities.
Parallel to the traditional hazard-based modeling strategy, many process-based models have been developed by making use of a speci…c parametric sub-class of Lévy processes. For example, the Brownian motion
appears in Lancaster’s (1972) strike model and Whitmore’s (1979) job tenure model, while the gamma process
has been utilized in Singpurwalla (1995) to model the degradation process. Abbring (2012) has pioneered
the study of identi…ability of process-based duration models, without making any parametric assumption.
The mixed hitting-time model in Abbring (2012) speci…es the duration formally as …rst passage time of
a random threshold by the sample path of a latent stochastic process. Abbring (2012) has demonstrated
that not only MHT incorporates the time-varying heterogeneity represented by the latent process, but also
it arises more naturally from the optimal stopping time problem where the solution could be described by
this threshold-crossing behavior. When the process is chosen to be the spectral negative Lévy process, the
identi…ability of all structural components could be shown, adapted from the strategies analyzing the identi…ability of MPH. So far the identi…cation of MHT is restricted to the univariate duration model, and the
suggested estimation is fully parametric. Botosaru (2013) also proposes another univariate duration model
making use of the Lévy subordinator, which could be either viewed as a hazard-based model with certain
random hazard density or a process-based model with the latent process being a double stochastic integral of
the Lévy subordinator. Identi…cation in Botosaru’s (2013) random hazard density model relates to solving
a nonlinear integral equation. For the identi…ed case, Botosaru (2013) presents a sieve type estimator and
proves its consistency. In a companion paper of the present work, Liu (2015) proposes a new univariate
duration model where the structural duration is the …rst passage time of a Lévy subordinator crossing a
unit exponential threshold. When the structural multiplicative e¤ect is parameterized as a linear index, the
model is equivalent as the single-index proportional hazards model where the unknown link function is the
Lévy-Laplace exponent function of the latent Lévy subordinator. Liu (2015) has studied a sieve maximum
partial likelihood estimator for the …nite dimensional parameter and proved its semiparametric e¢ ciency.
The only existing process-based competing risks model is due to Ryu (1993), to the best of my knowledge.
Ryu (1993) studies two duration variables with one driven by the natural failure mechanism (characterized by
the threshold-crossing behavior of certain non-decreasing process) and the other one from economic agent’s
endogenous optimization behavior. The observed duration arises as a consequence of these two competing
forces. There is just one latent process in Ryu (1993) and it is assumed to be compound Poisson with
nonparametric increments, which actually belongs to Lévy subordinators. Apart from this process, all the
other components are parametric in Ryu (1993). The identi…ability of Ryu’s (1993) model remains to be
addressed.
4
3
A Uni…ed Perspective of Duration Analysis
We provide a uni…ed perspective on the structural duration model described by the threshold-crossing behavior of some latent process, including MPH in Lancaster (1979), MHT in Abbring (2012), and a random
hazard density model (RHD) in Botosaru (2013). Thereafter we introduce our new model in the univariate
case.
We start with a generic duration model where the duration outcome is formally de…ned as the …rst
passage time of a latent process
e¤ect
(t) crossing an unobservable threshold e, with structural proportional
(X) from covariates X acting on the process:
Y
inf ft :
In all models discussed in the paper, X,
(X)
(t)
eg :
(3.1)
(t) and e are assumed to be mutually independent. The rule
speci…ed in (3.1) resembles the classical binary choice model closely in a dynamic setup.2 But intead of
making a discrete choice based on static comparisons, it stems from the optimal stopping time model where
agent optimally chooses when to switch to another state at time Y , such as in …rms’ entry/exit choices
(Dixit, 1989), workers’ unemployment decisions (Mortensen and Pissarides, 1994), and real option type
investments (McDonold and Siegels, 1986). Speci…cally, the economic agent is facing a utility maximization
problem where the current state generates a payo¤ to the agent with e and switching to another state gives
(X)
(t),3 i.e., the duration in (3.1) is the optimal solution of
(Z
Z 1
Y
max
e exp ( s) ds +
(X) (t) exp (
Y
0
Y
s) ds
)
with the discount rate equal to .
In spite of the fact that MPH is constructed from a regression modelling on the hazard rate, it is interesting
to ask whether it could be put into the same framework where the duration is speci…ed as the …rst passage
time of a latent process. Indeed this is possible. This alternative point of view further reveals the restrictions
on the latent process and threshold imposed by MPH, and it provides the link of our speci…cation to MPH.
Example 3.1 (MPH) The duration outcome in Lancaster’s (1979) MPH could be de…ned as in (3.1) by
Y
where the latent process is equal to
cumulative baseline hazard function
inf ft :
(t) =
(X)
(t)
eg ;
(3.2)
(t) with an unobservable heterogeneity term
and the
(t). The threshold follows unit exponential distribution, i.e. e
Exp (1). To see this, note that
Pr fY > tjX; g = Pr fe >
(t) (X) jX; g = exp (
(t) (X)) :
2 In discrete-time setting, Heckman and Navarro (2007) have constructed a general mixture duration model based on a latent
process crossing thresholds.
3 Because only the comparison plays a role from the agent’s perspective, it is not restrictive to let one side to become time
invariant, see Honore and Paula (2010).
5
In MPH given the realization of , we just have a deterministic and nondecreasing trend
(t) approaching
the threshold from below. The individual heterogeneity does not evolve over time along the entire spell of
duration, hence it is certainly desirable to construct a model with time-varying heterogeneity. Apparently,
one has to make some assumptions on the underlying class for
(t), in order to maintain a tractable
structure. In the sequel, we shall restrict our attention to general Lévy processes.
Lévy processes constitute a very rich and attractive class of stochastic processes, including the commonly
encountered Brownian motion, gamma process and stable process as special cases. They have attracted
considerable attention due to the ‡exibility for a wide variety of modeling issues in …nance, insurance and
engineering, see Kyprianou (2006). Its adaption to duration models have been developed by making use
of a speci…c parametric sub-class of Lévy processes in Lancaster (1972), Whitmore (1979) and Singpurwalla (1995). Abbring (2012) has pioneered the study of nonparametric identi…cation of process-based
duration models. Before reviewing his contribution, it is necessary to introduce the formal de…nition of a
d dimensional Lévy process and two important subclasses. To avoid digressions, we refer the readers to
Sato for a authoritative treatment on the Lévy process theory.
De…nition 3.2 A d dimensional Lévy process L (t) is a right continuous stochastic process with left limits
such that for every t and r
0, the increment L (t + r)
L (t) is independent of fL (s) ; 0
s
tg and has
the same distribution as L (r). Two subclasses are of particular importance:
(i) L (t) is a Lévy subordinator if it has almost surely nondecreasing sample path, i.e., for t
L (t)
s one has
L (s).
(ii) L (t) is a spectral negative Lévy process if it has no positive jumps.
The most remarkable property of a Lévy process is that even it is purely de…ned in terms of descriptive
features about the sample path, it admits a very concrete analytic characterization via the well-known
Lévy-Khintchine representation, see Sato (1999). Speci…cally, its Laplace transform could be expressed as
E exp
where
zT L (t)
= exp [ t (z)] ;
( ) is known as the Lévy-Laplace exponent function, which is nonparametric and time-invariant.
This elegant analytic characterization plays a crucial role in identi…cation problems of those duration models
driven by Lévy processes, including Abbring (2012), Botosaru (2013), and Liu (2015).
Example 3.3 (MHT) Abbring (2012) starts with a model where the structural duration is de…ned to be
n
o
~ (t) e ;
Y
inf t : (X) L
(3.3)
~ (t) is a spectral negative Lévy process. The empirical content
where e is an arbitrary random threshold and L
of MHT is revealed through the conditional Laplace transform LY ( jX) of the duration outcome:
LY (sjX)
E fexp ( sY ) jXg = Le ~ (s) (X) ;
where Le is the Laplace transform of e and ~ (s) is the largest root satisfying
6
(3.4)
~ (s) = s.
Example 3.4 (RHD) The duratoin in Botosaru’s (2013) random hazard density model could be expressed
as
Y
inf ft :
(X)
(t)
eg ;
where the latent crossing process is a double stochastic integral of the Lévy subordinator, as
(t) =
RtRs
f (u) dL (u) ds with a transformation function f (u). When the process L (u) is taken to be a Lévy
0 0
subordinator, the conditional survival function could be written as
Z t
SY (tjx) = exp
( (x) f (u) (t
u)) du :
0
Inspired by Abbring (2012) and Botosaru (2013), I construct a new duration model under the framework
of (3.1), replacing the product of the cumulative baseline hazard function and the frailty term
(t) in
MPH all together with a latent stochastic process L ( (t)). Here L ( ) is a Lévy subordinator and
(t) is
a deterministically increasing time-change function. This time-change function could be seen as a suitable
transformation from certain abstract time scale that the process evolves to the rate of economic transactions
(Kyprianou, 2006), which simply turns out to be the cumulative baseline hazard function. Hence the duration
in our model becomes
Y
inf ft :
(X) L ( (t))
eg ;
(3.5)
where the random threshold is still assumed to be unit exponentially distributed as in MPH. Notice the
latent Lévy subordinator L ( ) has replaced the static frailty term
and its sample path property inherits
the key feature of a monotonic driving trend in MPH. Therefore, the latent Lévy subordinator can be viewed
as the process generalization of a standard frailty term, see Gjessing, Aalen, and Hjort (2003). Another
close connection to MPH is that the distribution of frailty term is often taken to be in…nitely divisible,
see Hougaard (1986ab). Correspondingly in our present model, it’s well known the Lévy subordinator has
non-negative in…nitely divisible distribution (Sato, 1999). The conditional survival function of Y
in (3.5)
could be derived in a straightforward way by the Lévy-Khintchine representation:
SY (tjx) = exp [
(t)
( (x))] :
(3.6)
Our new model complements the study in Abbring (2012) and Botosaru (2013), while the great advantage
of the current model is that the proportional hazard structure is maintained, which delivers very tractable
econometric or statistical properties, see Liu (2015).
4
An Extended Marshall-Olkin (EMO) Model
In this section, we will present the formal model setup of EMO and derive the conditional joint surivial
function of the bivariate structural durations (Y1 ; Y2 ), highlighting its connection with the bivariate MPH, a
continuous-time game model in Honore and Paula (2010). We further distinguish our modeling strategy with
the original Marshall-Olkin (1967) model and existing generalizations. Pertaining to the speci…c conditional
7
joint survival function, we discuss various probabilistic properties of EMO, especially its dependence structure
which has important implications for its empirical content under competing risks in the next section.
4.1
Construction of EMO
We propose a bivariate model where each individual duration is de…ned as the …rst passage time of a latent
Lévy subordinator crossing unit exponential threshold. And the dependence is generated via a bivariate
dependent Lévy subordinator whose two margins are denoted as L1 (t) and L2 (t). Formally those two
durations are de…ned by:
Y
where e
Exp (1) for
inf ft :
(X) L [ (t )]
e g,
(4.1)
= 1, 2. Notice that we assume the time change function
( ) is common for both
durations.
As already discussed in Section 3, not only is the class of Lévy subordinators ‡exible enough to incorporate
a variety of interesting parametric prcesses (see Appendix C), but it also admits a sharp characterization
via the well known Lévy-Khintchine representation. For each marginal Lévy subordinator L (t), its LévyLaplace exponent function
( ) is de…ned via the equation:
E fexp [ zL (t)]g = exp [ t
In fact
(z)] :
( ) could be expressed as the following integral with respect to the jump intensity measure
( ):
Z
(z) =
1 e zy
(dy) :
(4.2)
R+
Here the characterizing jump measure
( ) is known to be a Lévy measure (see the de…nition in Appendix
E) and we denote its corresponding Lévy density function as
( ), if
( ) is absolutely continuous with
respect to the Lebesgue measure.
Referring to a bivariate Brownian motion, it is obvious that covariance structure between those two
marginal Brownian motions captures all the dependence. The answer for pure jumping processes is not immediately clear, as the dependence comes from the relationship of their jump intensities. Such a dependence
characterization relies on the notion of the Lévy copula, introduced in Cont and Tankov (2003), Kallsen
and Tankov (2006). It is worth emphasizing that the Lévy copula is not a standard copula coupling two
marginal distribution functions, rather it is connected with the Lévy measure which measures the jump
intensity. There are mainly two advantages of the Lévy copula, compared with the standard distributional
copula. First, the laws of multivariate Lévy processes are conveniently speci…ed by their Lévy measures, and
only in a few cases the distribution or probability density function could be given in closed forms. In fact,
for given two marginal in…nitely divisible distributions, it is unclear which distributional copula generates a
joint in…nitely divisible distribution. Second, the distributional copula implied by the joint Lévy processes
is typically time-varying, whereas the Lévy copula is time-invariant. Nevertheless, it does share certain
similarities with the standard distributional or survival copula functoin, including the Sklar theorem and its
8
role in pinning down the joint Lévy-Laplace exponent function
E fexp [ z1 L1 (t)
Here we have
12
(u; v) =
ZZ
12
(z1 ; z2 ):
z2 L2 (t)]g = exp [ t
1
e
y1 u y 2 v
dCL
1
12
(z1 ; z2 )] :
(y1 ) ;
2
(y2 ) ;
(4.3)
R2+
(y) =
where CL ( ; ) is the Lévy copula coupling the two marginal Lévy tail integrals
R1
y
(y) d (y). We
refer the readers to Appendix C for a probablistic interpretation of the tail integral. Theorem 4.1 in Cont
and Tankov (2003) ensures that (4.3) could be marginalized into (4.2).
The next theorem contains the conditional joint survival function of (Y1 ; Y2 ). Notice that our model
exhibits a singularity, meaning there is a positive probability mass assigned on the diagonal, whose twodimensional Lebesgue measure is equal to zero.
Theorem 4.1 Assume that the bivariate durations are de…ned by (4.1) with mutually independent covariates X, Lévy subordinators (L1 (t) ; L2 (t)), and unit exponential thresholds (e1 ; e2 ). If two thresholds are
independent of each other, then we have the conditional joint survival function as
Pr fY1 > t1 ; Y2 > t2 jX = xg
=
exp [
1
(x)
(t1 )
2
(x)
(4.4)
(t2 )
3
(x)
[max (t1 ; t2 )]] ;
where
moreover all
j
(x)
1
(x)
=
12
(
1
(x) ;
2
(x))
2
(
2
(x)) ;
2
(x)
=
12
(
1
(x) ;
2
(x))
1
(
1
(x)) ;
3
(x)
=
1
(
1
(x)) +
2
(
2
(x))
12
(
1
(4.5)
(x) ;
2
(x)) ;
0 for j = 1; 2; 3:
The following table summaries the comparsion made for three models. Clearly, our EMO embeds timevarying heterogeneity and inherits nice properties from MPH in terms of identi…ability and ‡exiable semiparametric estimation.
Remark 4.2 The bivariate MPH model in Heckman and Honore (1989), Abbring and van der Berg (2003)
starts with a speci…cation on the marginal conditional hazard density function
Y
(tjX; ) for
= 1; 2, in
terms of the proportional e¤ ect from covariate, the frailty term, and the baseline hazard function:
Y1
(tjX; )
=
1
(X)
1 1
(t) ;
Y2
(tjX; )
=
2
(X)
2 2
(t) :
And the underlying dependence between (Y1 ; Y2 ) is produced via the dependent frailty pair ( 1 ;
tically in our construction, L ( (t)) plays a similar role as
9
2 ).
Heuris-
(t). Analogously our model speci…es the
Structural Component
Threshold
Latent Process
Sample Path
Heterogeneity
Univariate Case
Identi…cation
Semiparametric Est.
Competing Risks
Identi…cation
Semiparametric Est.
Table 1: Comparison of Three Models
MHT
MPH
EMO
Unrestricted
Spectral Negative Lévy
No Positive Jump
Time-varying
Unit Exponential
Baseline Hazard Function
Deterministic & Increasing
Static
Unit Exponential
Lévy Subordinator
Increasing
Time-varying
Yes
N.A.
Yes
Yes
Yes
Yes
N.A.
N.A.
Yes
N.A.
Yes
Yes
structural duration individually as the …rst passage time via (4.1), and the dependence is generated by the
bivariate Lévy subordinators (L1 (t) ; L2 (t)), essentially through the Lévy copula.
Remark 4.3 Honore and Paula (2010) have studied the identi…cation of a simultaneous equation model
where bivariate durations are determined by the optimal choice of two strategic agents exhibiting the thresholdcrossing feature:
Y1
inf ft1 :
1
(X)
(t1 ) exp ( 1 fY2
t1 g)
e1 g ;
Y2
inf ft2 :
2
(X)
(t2 ) exp ( 1 fY1
t2 g)
e2 g :
In this model, thresholds (e1 ; e2 ) are allowed to be arbitrary, but the leader-follower e¤ ect parameter
the deterministic trend function
and
( ) are assumed to be common for both players. The system resembles a
classical simultaneous equations model in which endogenous variables interact with observable and unobservable components. Despite the presence of multiple equilibria, all model primitives are shown to be identi…able
by observing bivariate durations (not with competing risks) in Honore and Paula (2010). In extension to
incorporate simultaneous stopping phenomenon, Honore and Paula (2014) consider the Nash bargaining solution in a cooperative game. In comparison, our model could be viewed as a seemingly unrelated regression
as the dependence is produced via the unobservable dependent processes.
Remark 4.4 Assuming (L1 (t) ; L2 (t)) is a compound Poisson process with bivariate non-negative shocks,
our model boils down to the bivariate random shock model studied by Marshall and Shaked (1979), in the
absence of covariates. Under this circumstance, the joint survival function of (Y1 ; Y2 ) could be determined
explicitly without referring to the Lévy-Khintchine representation. Let marginal Lévy subordinator be L (t) =
PN (t)
i=1 Wi; for = 1, 2, and the arrival of shocks is governed by a common Poisson process N (t) with rate
10
.4 The joint survival function of (Y1 ; Y2 ) is
Pr fY1 > t1 ; Y2 > t2 g
1 X
1
k
X
exp ( t1 ) ( t1 ) exp (
=
k!
(t2
t1 )) ( (t2
l!
t1 ))
Pr fY1 > t1 ; Y2 > t2 g
1 X
1
k
X
exp ( t2 ) ( t2 ) exp (
=
k!
(t1
t2 )) ( (t1
l!
t2 ))
k=0 l=0
for t1
h
i
E F [k;k+l] (e1 ; e2 ) ;
k
h
i
E F [k+l;k] (e1 ; e2 ) ;
t2 , or
k=0 l=0
for t1
t2 . In the above expression,
F [k1 ;k2 ] (w1 ; w2 )
Pr
8
k1
<X
:
Wj;1
w1 ;
k2
X
Wj;2
j=1
j=1
and the expectation is computed with respect to the thresholds (e1 ; e2 ).
4.2
k
w2
9
=
;
;
The Original Marshall-Olkin Model and Existing Generalizations
The prominent Marshall-Olkin (1967) model is widely adopted in biostatistics, engineering, and life testing
where simultaneous failures of human organs, components, or engines are present. The original model does
not involve any Lévy subordinator and is constructed with three types of independent shocks: one leading
to joint spell termination and two leading to individual spell termination. The occurrence times of three
shocks Z1 , Z2 , and Z3 are assumed to be independent exponential random variables with parameters
and
3,
1,
2,
respectively. So Y1 = min (Z1 ; Z3 ) and Y2 = min (Z2 ; Z3 ) follow the joint distribution given below:
Pr fY1 > t1 ; Y2 > t2 g = exp [
In comparison, our construction speci…es Y
1 t1
2 t2
= inf ft : L (t)
3
max (t1 ; t2 )] :
e g for
= 1; 2, and those parameters could
equivalently be expressed as
1
=
12
(1; 1)
2
(1) ;
2
=
12
(1; 1)
1
(1) ;
3
=
1
(1) +
2
(1)
12
(1; 1) :
Maintaining the presence of a common shock, Klein, Keiding and Kamby (1989), An, Christensen and
Gupta (2004) have generalized the Marshall-Olkin model with covariates and frailty terms. The observed
bivariate durations are still assumed to be Y1 = min (Z1 ; Z3 ) and Y2 = min (Z2 ; Z3 ), then a conditional
marginal hazard density model like MPH is speci…ed for each Zj for j = 1; 2; 3. Although estimation
4 The single Poisson process is not restrictive because several Poisson processes can always be superimposed to yield a new
Poisson process and the several sequences of damage distributions can be replaced by a sequence of appropriate mixture, see
Marshall and Shaked (1979).
11
procedures have been suggested in both papers, neither of them presents any large sample theory for the
proposed estimators. Honore and Paula (2010, 2014) also criticize this type of generalization of MarshallOlkin model due to a lack of structural interpretation and the fact that the existence of a common shock
to both durations seems too restrictive for economic applications. In sharp contrast with the existing
generalization of Marshall-Olkin (1967) model, our construction has circumvanted those shortcomings.
4.3
Dependence Properties
In EMO, dendence between the two duration variables is generated by dependence between the unobserved
characteristics of latent stochastic processes. We are also interested how the dependence between latent
processes transfers into the dependence between structural durations. Both the Pearson’s correlation coe¢ cient and two popular rank correlation coe¢ cients, Kendall’s tau and Spearman’s rho for the bivariate
durations are given in closed forms in EMO. In this context, the advantage of EMO is borne out, compared
with the calculation and bounds analysis for bivariate MPH in van den Berg (1997).
With the help of the Lévy copula, our goal is to understand the dependence structure imposed on the
resulting …rst passage times, or the structural durations. A straightforward algebra shows the conditional
survival copula function of two durations is:
CS (u; vjx) = SY1 ;Y2
SY 1 (ujx) ; SY 1 (vjx) jx = uv min u
1
! 1 (x)
2
;v
! 2 (x)
;
(4.6)
where
! 1 (x) =
To see this, note that SY 1 (ujx) =
CS (u; vjx) =
3 (x)
and ! 2 (x) =
(x)
+ 3 (x)
1
1
(
log u
(x)+ 3 (x)
uv 1 !2 (x) ;
u1 !1 (x) v;
3 (x)
:
(x)
+ 3 (x)
2
for = 1; 2, then by monotonicity of
if
if
log u
1 (x)+
( ) we get:
log v
3 (x)
log u
1 (x)+ 3 (x)
2 (x)+
3 (x)
log v
2 (x)+ 3 (x)
:
This is known as the Marshall-Olkin copula or the generalized Cuadras-Auge copula. Although this copula
is singular without a copula density function, it is an extreme copula with its Pickands’function (see Section
3.3.4 in Nelsen, 2006):
A (tjx) = 1
min (! 2 (x) t; ! 1 (x) (1
t)) :
Notice that the conditional survival copula function (4.6) changes with x as well, in contrast with the
commonly considered situations with an invariant copula in Heckman and Honore (1989), Abbring and van
der Berg (2003). The conditional survival copula reveals that our subsequent analysis will not …t into the
framework in Fan and Liu (2013), because it is not Archimedean, neither it admits a copula density function
with respect to the two dimensional Lebesgue measure. Nevertheless, the dependence structure still delivers
very informative empirical content.
Next theorem summarizes several dependence measures which are directly linked to this (conditional)
Marshall-Olkin copula, see Nelson (2006). In comparison with the bivariate MPH in van den Berg (1997),
only certain bounds for Kendall’s tau are available there.
12
Theorem 4.5 Let
K
(x) stand for (conditional) Kendall’s tau and
S
(x) for (conditional) Spearman’s rho.
Moreover, let tU (x), tL (x) denote the (conditional) upper and lower tail dependence parameters. Then they
are of closed forms in EMO:
K
(x)
=
S
(x)
=
tU (x)
=
! 1 (x) ! 2 (x)
;
! 1 (x) + ! 2 (x) ! 1 (x) ! 2 (x)
3! 1 (x) ! 2 (x)
;
2 [! 1 (x) + ! 2 (x)] ! 1 (x) ! 2 (x)
min (! 2 (x) ; ! 1 (x)) ; tL (x) = 0:
For Lévy subordinators, any Lévy copula is between the two extreme cases, the independent Lévy copula
CL;? (u; v) and the completely dependent Levy copula CL;k (u; v) with the following expressions (Cont and
Tankov, 2003):
CL;? (u; v)
= u I fv = 1g + v I fu = 1g ;
CL;k (u; v)
=
min fu; vg :
A convenient parametric Lévy copula which lies between those two extremes is the Lévy-Clayton copula
with a single parameter
> 0:
CL; (u; v) = u
1=
+v
:
(4.7)
One nice property related to the Lévy-Clayton copula is that the joint Lévy exponent function (4.3) can be
futher simpli…ed as:
12
(u; v) =
where
(u; v; ) = uv
ZZ
1
(u) +
y1 u y2 v
e
R2+
2
(v)
CL;
(u; v; ) ,
1
(y1 ) ;
2
(y2 ) dy1 dy2 ,
(4.8)
see Proposition 1 in Epifani and Lijoi (2010). Note that the function (u; v; ) summarizes all the dependence
structure when the Lévy-Clayton copula is adopted. Given the representation (4.8), we have the following
expressions for the coe¢ cients in the extended Marshall-Olkin model:
1
(x)
=
1
(
1
(x))
(
1
(x) ;
2
(x) ; ) ;
2
(x)
=
2
(
2
(x))
(
1
(x) ;
2
(x) ; ) ;
3
(x)
=
(
1
(x) ;
2
(4.9)
(x) ; ) :
Consequently, the expressions for those exponents in (4.6) simplify to
! 1 (x) =
(
1
(x) ; 2 (x) ; )
and ! 2 (x) =
1 ( 1 (x))
(
1
(x) ; 2 (x) ; )
:
2 ( 2 (x))
Moreover, the dependence of two Lévy processes could be stated in terms of the (conditional) Pearson’s
correlation coe¢ cient between Y1 and Y2 . For bivariate MPH with Weibull baseline hazard functions, van
den Berg (1997) obtains a closed form expression for Pearson’s correlation coe¢ cient, see equation (6) in
13
van den Berg (1997). In comparison, our general expression (4.10) in the next proposition does not restrict
the baseline hazard function.
Theorem 4.6 (Epifani and Lijoi, 2010) Let
when the Lévy copula is (4.7), then
RR
R2+
(Y1 ; Y2 jx) =
In the special case when
Q
(t)
e
j=1;2
r
(Y1 ; Y2 jx) stand for the conditional correlation coe¢ cient
1 ( 1 (x))
2
R1
0
te
(t)
2 ( 2 (x))
(t)
j
(
j (x))
e
(s^t) (
dt
R1
0
1 (x); 2 (x);
e
(t)
j
(
)
1 dsdt
j (x))
2
:
(4.10)
dt
(t) = t, we generalize the classical result as in Section 3.2 of Marshall and
Olkin (1967):
(
(Y1 ; Y2 jx) =
5
1(
1 (x)) +
2(
1
(x) ; 2 (x) ; )
( 1 (x) ;
2 (x))
2
(x) ; )
:
Semiparametric Identi…cation with Competing Risks
In this section we consider the empirical content of EMO under competing risks. In a competing risks model,
we only observe Y = min (Y1 ; Y2 ) and the cause of failure D de…ned slightly di¤erent from the one in a
standard competing risks model without simultaneous failure. Here D = 1 if Y1 < Y2 , D = 2 if Y2 < Y1 ,
and D = 3 if Y1 = Y2 . Apparently, the sampling information is summarized by the sub-distribution
functions:
FY;D=j (tjX = x) = Pr fY
t; D = jjX = xg for j = 1; 2; 3:
From the discussion in Section 4, (Y1 ; Y2 ) are still dependent conditional on covariates, hence the identi…cation analysis calls for extra attention. The following lemma collects the expressions for the sub-density
functions and already appears in Marshall and Olkin (1967). This convenient form is also utilized by Basu
and Ghosh (1978) to show the identi…ability of standard MO distribution under competing risks. A short
proof is given in the Appendix A for completeness.
Lemma 5.1 (Basu and Ghosh, 1978) Assume
( ) is absolutely continuous with its density function
( ),
then conditional sub-density functions fY;D=j (tjx) = @FY;D=j (tjx) =@t are equal to
fY;D=j (tjx) =
(t)
j
(x) exp [
(t)
12
(
1
(x) ;
2
(x))] ; for j = 1; 2; 3:
It is worthwhile highlighting the novelty in our identi…cation strategy as existing methods do not apply
here. In our model, the conditional survival copula function of two durations changes with covariates’value,
unlike the models in Heckman and Honore (1989), Abbring and van den Berg (2003), Lee (2006), and Lee
and Lewbel (2013). Furthermore, this conditional survival copula is not Archimedean and does not even
admit a density function, which goes beyond the framework in Fan and Liu (2013). But thanks to the
Marshall-Olkin structure, we can …rst identify certain reduced-form functions which are determined jointly
by the structural multiplicative e¤ects from observable covariates and the characteristics of latent Lévy
14
subordinators. When covariates enter into the model via linear index, those reduced-form functions generate
two single-index models (Ichimura, 1993) from which we could identify …nite dimensional parameter in the
indices and marginal Lévy measure. Thereafter we proceed to identify the Lévy copula.
We …rst list all the regularity assumptions, and then present the main theorem establishing the identi…ability of all model primitives.
Assumption (A)
(1) The time-change function
denoted by
( ), moreover
: R+ ! R+ , is nondecreasing and absolutely continuous with its density
(0) = 0 and
(0) = 1.
R
(2) The two marginal Lévy measures satisfy R+ y
(dy ) < 1 for = 1; 2 and the joiny Lévy measure
R
satis…es R2 y1 y2 12 (dy1 ; dy2 ) < 1. Moreover, two marginal tail integrals
are continuous for = 1; 2.
+
(3) The support of (
1
(X) ;
of R , moreover
2
d 1
S+
d
S+
1
(X)) contains an open rectangle in R2+ .
(x) = exp xT
(4) For = 1; 2 assume
d
2
, and the support of X is not contained in any linear subspace
where
f : k k2 = 1; and ’s …rst non-zero coordinate is positiveg :
Theorem 5.1 Let Assumptions (A1)-(A4) hold, then all the structural components ( ;
; CL ) with = 1; 2
are point identi…ed.
The normalization assumption in (A1) by restricting
(0) = 0,
(0) = 1 is innocuous since we do not
observe the magnitude of the latent process anyway. Besides the obvious example with (t) = t, one could
P
also consider higher order polynomials (t) =
cm tm or the following logarithm function which leads to
the Pareto type distribution:
(t) =
log 1 +
t
with
> 0:
Assumption (A2) requires the marginal Lévy measure has …nite …rst moment, which turns out to be equivalent
as the corresponding moment restriction on L1 (1) or L2 (1), see Wolfe (1971). Correspondingly, assuming
the …niteness of the …rst moment of a frailty term is widely adopted in the identi…cation results of MPH,
see Elbers and Ridder (1982), Heckman and Honore (1989), and Abbring and van den Berg (2003). In fact,
such an assumption is almost necessary to identify MPH, see Ridder (1990). Again this con…rms that a Lévy
subordinator is a natural generalization of frailty term from a process point of view. Here we also need the
assumption E [L1 (1) L2 (1)] < 1, restricting the joint Lévy measure. The variation restriction in (A3) on
structural regressions is rather mild in view of Abbring and van den Berg (2003).
With the linear index restrictions on covariates, the reduced-form conditional sub-density functions
fY;D (tjx) belong to the multiple indices models, whose identi…cation typically rely on exclusion restrictions as in Ichimura and Lee (1991). But in our context, such an assumption could be circumvented since
we could …rst identify certain single-index model because
0
exp x
=
12
(
1
(x) ;
2
15
(x)) E [D jX = x] , for = 1; 2;
(5.1)
where auxiliary binary variables are
Di = I fDi = or 3g for = 1; 2;
which prove to be useful in the estimation part as well. The function
(5.2)
12
(
1
(x) ;
2
(x)) is identi…ed from
the sampling information in Y :
Pr [Y > tjX = x] = exp [
(t)
In those compositions, the unknown link function
12
(
1
(x) ;
2
(x))] :
(marginal Lévy-Laplace exponent function) could be
identi…ed without extra regularity assumptions, as those Lévy-Laplace exponent functions are automatically
very smooth. In the end, after all the marginal components are identi…ed, the Lévy copula is also uniquely
determined given the Sklar’s theorem in Kallsen and Tankov (2006).
6
Semiparametric Estimation with Competing Risks
Considering semiparametric estimation, we will focus on estimating
for
base-line hazard function. We denote the cumulative hazard function as
function as
(t; ), with its univariate true parameter
= 1; 2 with a parameterized
(t; ) and the hazard density
0.
Notice for the random variable Y , its conditional hazard function is shown to be of proportional structure
in Section 5:
Y
(tjX = x) =
(t) exp [ (Xi )] ;
where
(Xi ) = log
Therefore we could estimate
12
(
1
(Xi ) ;
2
(Xi )) :
( ) and its derivatives using Fan, Gijbels and King (1997) even when Y is
subject to some addtional random censoring C, due to drop-out or early termination of the study. Although
independent censoring can be handled by traditional hazard-based models in duration or survival analysis,
its incorporation into the recent structural duration model calls for extra attention. For example, even
though Abbring’s (2012) MHT is naturally expressed in terms of the conditional Laplace transform, one has
to invert the Laplace transform back to the probability density function to handle additional independent
censoring. In comparison, such a complication could be easily embedded in our EMO. Let V = min (Y; C)
and
= I [Y < C]. From now on we shall explicity work with i:i:d:obervations Zi = (Vi ;
i;
i Di ; Xi ).
We
can only distinguish Di if Yi is not subject to addtional independent censoring from Ci :
It is well known that the …nite-dimensional parameter in the single-index model is only identi…able upto a
scale-normalization (Ichimura, 1993). Referring to (5.1), we see the following parameter
derivatives are scale-equivalent to
based on average
(Stoker, 1986; Powell, Stock, and Stoker, 1989):
= E [aD (Xi ) bY (Xi ) exp (aY (Xi )) + exp (aY (Xi )) bD (Xi )] ; for = 1; 2;
16
(6.1)
where aD (x) = E [D jX = x], bD (x) =
natural estimator for
1
n
b
@
@x E
[D jX = x], aY (x) =
(x), and bY (x) =
@
@x
(x). Hence, the
are
o
b
aD (Xi ) bbY (Xi ) exp (b
aY (Xi )) + exp (b
aY (Xi )) bbD (Xi ) ; for = 1; 2;
n n
X
i=1
(6.2)
with b
a ( ) and bb ( ) being estimated intercept term and slope coe¢ cients obtained from local polynomial
…tting. Starting from Fan and Gijbels (1996), the local polynomial estimator has become a default choice
if one plans to estimate either the function value or its various partial derivatives. Chaudhuri, Doksum,
and Samarov (1997), Li, Lu, and Ullah (2003) have studied the average derivative estimators from a general
multivariate local polynomial …tting, and both demonstrated its better …nite sample performance, compared
with the local constant based approaches in Hardle and Stoker (1989), Powell, Stock, and Stoker (1989).
Here we follow this route, since our "average derivative" involves both the function values and its …rst order
derivatives all together.
Let us pin down some notations for the local polynomial estimator with order p (adapted from Tsybakov,
2008), with d dimensional covariates. Let v = (v1 ;
number of v with jvj
T
d
Q
i=1
+ vd , and let P be the total
p: The following vector U (z) is in RP :
U (z) =
where v! =
; vd ) with jvj = v1 +
zv
; jvj
v!
p ;
(6.3)
vi !, the vectors of v 2 N d being ordered lexicographically, and z v =
d
Q
i=1
zivi . Furthermore,
we de…ne two types of selection vectors e 2 RP to pick up the estimators for the function value and its …rst
order derivatives respectively:
eT0;P
=
eTs;P
= @0; 0; :::; 1::::; 0; 0:::; 0A ;
| {z }
(1; 0; 0; :::; 0) ;
0
1
d
where 1
s
d.
We need to carry out the local polynomial estimation for the dependent variables D and Y separately.
For notational brevity, we shall use the same kernel density function K (u), the bandwidth hn and p th
polynomial in both estimation. We write Khn ( ) = hn d K ( =hn ) and Uhn ( ) = U ( =hn ). The local polynomial …tting associated with D is standard, which minimizes the following weighted least squares problem,
see Fan and Gijbels (1996)
Xh
bD (x) = arg min 1
Di
n i=1
n
T
Uhn (Xi
i2
x)
i Khn
(Xi
x) :
(6.4)
Whenever we evaluate the estimate at a sample point Xi , we shall use its leave-one-out version as in Li, Lu,
and Ullah (2003):
Xh
bD (Xi ) = arg min 1
Dj
n
T
Uhn (Xj
j6=i
17
i2
Xi )
i Khn
(Xj
Xi ) :
The weigths (
n
i )i=1
indicate that the estimation is only carried out for the uncensored subsample. Let the
estimators for the intercept and slope coe¢ cients be denoted as:
and bbD ( ) = bbD
;1
b
aD (Xi ) = eT0;P bD (Xi ) ; and bbD
( ) ; :::; bbD
;s
T
;d
()
.
(x) = hn 1 eTs;P bD (Xi ) ;
As for the second local polynomial estimation, we consider the minimizer of the local likelihood function
in Fan, Gijbels, and King (1997), where bY (x) ; b (x) = arg min
ln (x;
Y; )=n
1
n h
X
i=1
i
n
log (Vi ; ) +
T
o
x)
Uhn (Xi
;
ln (x;
(Vi ; ) exp
Y
; ) with:
T
Uhn (Xi
x)
i
Khn (Xi
x) :
Therefore b
aY (Xi ) and bbY (Xi ) are de…ned in a similar way based on the leave-one-out version.
The following theorem shows the desired asymptotic normality of our estimator, and we spell out the
form of its asymptotic covariance in Appendix B.
Theorem 6.1 Under Assumptions (B1)-(B5) and (C1)-(C5) in Appendix B, we have the following asymptotic normality holds for = 1; 2:
p
7
Conclusion
n b
=) N (0;
):
This is the …rst paper on a competing risks model where the structural durations are driven by dependent
stochastic processes, without parameterizing the latent processes. The latent Lévy subordinators could be
seen as the natural variant of the static heterogeneity terms in MPH from a process point of view, and it
resembles the notion of usage and wear-out e¤ect (Singpurwalla, 1995) closely. Hence, besides those optimal
stopping time problems in economics, our model also applies to the scenario concerning the life/failure time
in biology, medical science, and engineering, where the traditional hazard-based models dominate. In this
Extended Marshall-Olkin (EMO) model, it is transparent how the deep structural parameters of theoretical
models can be related to reduced-form functions, and all model primitives could be point identi…ed for
various semiparametric speci…cations. The identi…cation strength further leads to root-n consistent and
asymptotically normal estimators for parameters in the structural proportional e¤ects.
Interestingly, the model generates equal durations with positive probability, motivated from an empirical
application on the joint retirement decisions of married couples, as a signi…cant portion of couples choose
to retire at the same time documented across di¤erent datasets (Honore and Paula, 2014). This key feature
is ruled out from the very beginning in the traditional duration or survival analysis. In this application,
the Lévy subordinator with its increasing sample path is more suitable to represent the latent aging process
or the degradation of health condition of the elderly, as opposed to the spectrally negative Lévy process in
Abbring (2012) which ‡uctuates up and down. There is no positive jump in the spectrally negative Lévy
process by de…nition, but unexpected health shocks, such as heart attacks or new cancer diagnoses, are quite
18
common for people near retirement (Coile, 2004). The machinery developed in this paper provides a nice
complementary tool compared with existing methods that have been used.
Numerious extensions are possible. First, one could incorporate endogenous and time-varying covariates
by modeling additional stochastic processes explicitly. Renault, van den Heijden, and Werker (2014) present
a structural model for transaction time (duration) and asset prices (associated marks) driven by multivariate
Brownian motion. In their model, successive passage times of one latent Gaussian component relative to
random boundaries de…ne durations and the other correlated components generate the marks. It is both
interesting and challenging to search for other processes which lead to tractable structure. Second, it is
desirable to relax the parametric assumption on the threshold variable and to consider a more general Lévy
process, i.e., the spectral positive Lévy process (Boyarchenko and Levendorskii, 2014). Last but not least,
it is worthwhile to study the empirical content of stochastic game models driven by a Lévy process; see
Chapter 11 in Kyprianou (2006).
8
Appendix A. Proofs of Main Results
In this section, we prove the main theoretical results in the paper, including driving the expressions of
the joint survival function, the sub-density functions, proving the identi…ability of all model primitives and
showing asymptotic normality of the semiparametric estimator.
Proof. (of Theorem 4.1) We consider two cases separately in the proof. When t1
t2 , we get
Pr fY1 > t1 ; Y2 > t2 jX = xg
=
Pr fe1 >
1
= E fexp (
1
= E fexp [ (
=
(x) L1 [ (t1 )] ; e2 >
(x) L1 [ (t1 )]
1
2
(x) L1 [ (t1 )] +
exp [ ( (t2 )
(t1 ))
2
(
2
2
(x) L2 [ (t2 )]g
(x) L2 [ (t2 )])g
2
(x) L2 [ (t1 )])
(x))
(t1 )
12
(
2
1
(x) (L2 [ (t2 )]
(x) ;
2
L2 [ (t1 )])]g
(x))] :
In the string of above equalities, the …rst one comes from the way we de…ne Y1 , Y2 as …rst passage times,
while the second one utilizes the distributional assumption on the random thresholds. The third equality is
prepared for using the independence between increments of Lévy processes, while the …nal one follows the
Lévy-Khintchine representation.
Analogously for the other case t1
t2 , we have:
Pr fY1 > t1 ; Y2 > t2 jX = xg
=
exp [ ( (t1 )
(t2 ))
1
(
1
(x))
(t2 )
12
(
1
(x) ;
2
(x))] :
Now the uni…ed expression in (4.4) just summarizes those two cases and those functions
j
(x) arise from a
simple reparameterization.
When it comes to the non-negativeness of
max f
1
(u) ;
2
j
(v)g
(x), we have the following inequalities hold
12
19
(u; v)
1
(u) +
2
(v) ,
for the marginal and joint Lévy-Laplace exponent functions for non-negative (u; v). For an arbitrary Lévy
copula CL de…ned on R2+ , we have the Frechet-Hoe¤ding type inequality (Cont and Tankov, 2004) hold
CL;?
CL
CL;k
where
CL;? (u; v)
= u I fv = +1g + v I fu = +1g ;
CL;k (u; v)
=
min fu; vg :
CL;? (u; v) is the independence copula and CL;k (u; v) = min fu; vg is the complete dependence copula funcy 1 u y2 v
tion, respectively. Notice that 1 e
is the distribution function of a bivariate exponential distribution
function with parameters (u; v), which we denote by Fu;v ( ; ). It is clear that
12
=
ZZ
(u; v)
1
y1 u y 2 v
e
dCL
1
(y1 ) ;
2
(y2 )
R2+
ZZ
=
R2+
ZZ
R2+
=
1
CL
1
(y1 ) ;
CL;?
(u) +
2
1
(y2 ) dFu;v (y1 ; y2 )
2
(y1 ) ;
2
(y2 ) dFu;v (y1 ; y2 )
(v) ;
where integration-by-parts is used in the second equality. Similarly, we have
(u; v)
ZZ
CL
12
=
(y1 ) ;
1
2
(y2 ) dFu;v (y1 ; y2 )
R2+
ZZ
=
ZZ
R2+
CL;k
1
1
(y1 ) ;
y 1 u y2 v
e
1
(u) ;
2
(y2 ) dFu;v (y1 ; y2 )
dCL;k
R2+
max f
2
1
(y1 ) ;
2
(y2 )
(v)g ,
where the …nal inequality follows from the fact that
1
e
y 1 u y2 v
max 1
y1 u
e
and the integration is carried out on the line (y1 ; y2 ) :
;1
1
e
(y1 ) =
y2 v
2
,
(y2 ) .
Proof. (of Lemma 5.1) Recall that conditional joint survival function is
S (t1 ; t2 jx) = exp [
1
(x)
(t1 )
2
20
(x)
(t2 )
3
(x)
[max (t1 ; t2 )]] ;
whence for t1 < t2 , it simpli…es to exp [
1
(x)
(t1 )
[
2
(x) +
3
(x)]
(t2 )]. Upon di¤erentiation, it has
a density of the following form:
1
(x) [
2
(x) +
(x)] (t1 ) (t2 ) exp [
3
Integrating the above density over the range [t; +1]
1
(x)
(t1 )
[
(x) +
2
3
(x)]
(t2 )] :
(t1 ; +1] delivers the conditional sub-survival function
for D = 1:
SV;D=1 (tjx)
Z Z
=
1 (x) [ 2 (x) + 3 (x)] (t1 ) (t2 ) exp [
1 (x) (t1 )
t t1
Z
=
(x)
exp [ [ 1 (x) + 2 (x) + 3 (x)] (t1 )] d (t1 )
1
[
2
(x) +
3
(x)]
(t2 )] dt2 dt1
t
=
(x)
1 (x) + 2 (x) +
1
3
(x)
exp [ [
1
(x) +
2
(x) +
3
(x)]
(t)] :
A similar derivation gives one the conditional sub-survival function for D = 2:
2
SV;D=2 (tjx) =
1
(x) +
2
(x)
(x) +
3
(x)
exp [ [
1
(x) +
2
(x) +
3
(x)]
(t)] :
Hence subtracting the sum of those two sub-survival functions from the joint survival function evaluated at
t1 = t2 = t, we get
3
SV;D=3 (tjx) =
1
(x) +
2
(x)
(x) +
3
(x)
exp [ [
1
(x) +
1
(x) ;
2
(x) +
3
(x)]
(t)] :
Notice that the following equation holds
1
(x) +
2
(x) +
3
(x) =
(
12
2
(x)) ;
given the reparameterization in (4.5). Finally the claimed expressions of those conditional sub-density
functions follow by di¤erentiating the corresponding sub-survival function.
Proof. (of Theorem 5.1) (Step 1) First, let us …rst summarize the sampling information in the competing risks model that is crucial for subsequent analysis. Referring to the auxilary random variables
Di = I fDi = or 3g with = 1; 2, we have
(x) + 3 (x)
(x)
+
1
2 (x) + 3 (x)
Pr [D = 1jX = x] =
(A.1)
from Lemma 5.1. The numerator on the right hand side of (A.1) delivers two single-index models as
0
exp x
=
(x) +
(x) ; for = 1; 2;
3
(A.2)
which leads to
0
exp x
Pr [D = 1jX = x] =
12
(
1
(x) ;
2
(x))
for = 1; 2:
Moreover, we have
Pr [Y > tjX = x] = exp [
(t)
21
12
(
1
(x) ;
2
(x))] ;
(A.3)
which asserts the condtional hazard function of Y is of proportional structure:
Y
(tjX = x) =
(t)
(
12
(Step 2) Notice that we would be able to identify
1
(x) ;
and
2
(x)) :
(A.4)
from these two single-index models in (A.2)
if the denominator in (A.3) is identi…able. This is indeed the case as
12
(
1
(x) ;
2
(x)) = fV (0jx) given
the multiplicative structure in (A.4) and the normalization assumption (A1) imposed on
time-change function
. Thereafter, the
(t) could be identi…ed from the conditional survival function of Y . The identi…ability
follows from Theorem 1 in Lin and Kulasekera (2007) given our Assumption (A4) and the smoothness of
Lévy-Laplace exponent functions
0
. As link functions,
are identi…ed within the support of exp x
as shown by Lin and Kulasekera (2007). The real analytical property of
allows a unique extension on the
whole positive real line, by Proposition 1 in Abbring and van den Berg (2003).
(Step 3) Given the identi…cation of
way. Recall that
(u) =
, we can identify the marginal Lévy measure
Z
1
e
yu
in the following
(dy) , for = 1; 2:
R+
1
( );
2
( ) belong to the class of so-called Bernstein functions, meaning their derivatives are completely
monotone as in Theorem 3.2 in Schilling, Song, and Vondracek (2012). Let the operator L denote the Laplace
transform of a positive measure and de…ne two positive measures e (dy) = y
(dy) for
= 1; 2. Then we
get
@
@u
1
@
@v
2
(u)
Z
=
e
yu
y
1
e
yv
y
2
R+
(v)
Z
=
R+
(dy) = L e 1 ;
(dy) = L e 2 ;
by di¤erentiating w.r.t u and v respectively. Assumption (A2) guarantees that e ( ) are …nite measures for
= 1; 2. Hence, by Theorem 12b in Widder (1947) and Proposition 1 in Abbring and van den Berg (2003)
we could identify e 1 ; e 2 from L e 1 ; L e 2 , as long as we get enough variation on some non-empty open
sets as in Assumption (A3). Thereafter,
are also identi…ed for = 1; 2.
(Step 4) It su¤cies to identify the Lévy copula function to fully pin down the characteristics of the Lévy
subordinators. So far, we have already identi…ed
Hence
12
(u; v) is also identi…ed. Notice that
Z
@2
e
12 (u; v) =
@u@v
R2+
Whence, we could identify e 12 (dy) = y1 y2
12
12
(
y 1 u y2 v
(dy1
1
(x) ;
y 1 y2
12
2
(x)) and
(dy1
1
(x) = exp(xT
) for
= 1; 2.
dy2 ) = L e 12 :
dy2 ) by the bivariate Laplace inversion theorem in
Widder (1947). Given the joint Lévy measure, its copula function CL is unique by the Sklar theorem in
Kallsen and Tankov (2006) given the absolutely continuity of the tail integrals associated with the marginal
Lévy measure..
22
Proof. (of Theorem 6.1) The proof of main theorem is patterned after Hardle and Stoker (1989), Li, Lu,
and Ullah (2003), but the detail is slightly more involved, as we are dealing with various intercept and slope
estimates all together. We decompose the statistic into three categories: the …rst is the centered i:i:d average
of the infeasible estimator while the second part consists of several U-statistics. The third remainder term
is shown to be negligible by applying supremum bounds on various local polynomial estimators. The second
U-statistic would further be linearized via Hoe¤ding decomposition. Thereafter the desired asymptotic
normality follows from the behavior of those centered linear terms.
(Step 1) We prove the uniform consistency of the local polynomial estimators and show the rates of
convergence are
for D or Y .
b
a (x)
s
a (x) = Op
log n
nhdn
!
; and bb (x)
b (x) = Op
s
log n
nhd+2
n
!
;
(Step 2) We have the following decomposition of
b
where
tn0
= n
1
1
1
= tn0 +
4
X
tnl + op n
1=2
;
l=1
n n
o
X
aD (Xi ) bY (Xi ) eaY (Xi ) + eaY (Xi ) bD (Xi ) ;
i=1
tn1
tn2
= n
= n
1
1
n
X
i=1
n
X
i=1
tn3
tn4
= n
= n
1
1
n
X
i=1
n
X
i=1
.
bY (Xi ) eaY (Xi ) [b
aD (Xi )
h
eaY (Xi ) bbD (Xi )
aD (Xi )] ;
i
bD (Xi ) ;
eaY (Xi ) (aD (Xi ) bY (Xi ) + bD (Xi )) [b
aY (Xi )
h
aD (Xi ) eaY (Xi ) bbY (Xi )
aY (Xi )] ;
i
bY (Xi ) :
(Step 3) The statistics tnl all belong to U-statistics for l = 1; :::; 4. We obtain the linear representation
for
tnl = n
1
n
X
'l (Zi ) + op n
1=2
; for l = 1; :::; 4:
i=1
We refer readers to Lemma B.5 and Lemma B.8 for explicit expressions of 'l ( ), l = 1; :::; 4:
(Step 4) The terms associated with '3 ( ) and '4 ( ) can be written as local martingales attached to
the counting process Ni (t) = I [Vi
(
1 ; :::;
n)
t;
i
= 1], i = 1; :::; n. Let the associated random intensity process
speci…ed as
i
(t) = I (Vi > t) (t) exp
23
(Xi ) :
Thus
Mi (t) = Ni (t)
Z
t
i
(u) du;
0
are local martingales. Because the regressor is time-invariant, the local martingale has zero mean conditional
n
on all fXi gi=1 . Hence the asymptotic normality of tn0 ; tn1 ; tn2 could be checked apart from tn3 ; tn4 . Those
two parts are orthogonal in fact. Then the asymptotic normality follows by verifying the conditions in
Rebollebo’s CLT for local martingales and standard Lindeberg CLT applies to the sample average of i:i:d:
terms.
9
Appendix B. Proofs of Linear Representations
Our analysis of average derivative type estimators follow the same scheme as in Powell, Stock, and Stoker
(1989), Hardle and Stoker (1989), Chaudhuri, Doksum, and Samarov (1997), and Li, Lu, and Ullah (2003).
Two key components are the uniform linear representation of local polynomial estimators and linearization
of U-statistics. For the …rst purpose, we rely on some recent results by Einmahl and Mason (2005), while for
the second we resort to Gine and Mason (2007). We let M be a generic …nite positive constant whose value
might change from line to line, and occasionally we use subscript when more than one constant appear in
the same display.
First recall some standard de…nitions in Einmahl and Mason (2005). Let F collects some functions
f : Rd ! R. We say the class F is of VC type with respect to an envelope F if the covering number
N (F; L2 (Q) ; "), the smallest number of L2 (Q) open balls of radius " required to cover F, statis…es
!v
M kF kL2 (Q)
; for 0 < " 2 kF kL2 (Q) ,
N (F; L2 (Q) ; ")
"
for some universal positive constant M; v for every probability measure Q on the underlying space. Also by
pointwise measurable class F, we mean there exists a countable subclass F0 of F s.t. we can …nd for any
f 2 F a sequence of functions ffm g in F0 for which fm (x) ! f (x) for x 2 Rd . We shall apply the following
lemma when f is the identity map and the generalized kernel function K
x
hn
is K
x
hn
U
x
hn
.
Lemma B.1 For i:i:d: observations, we consider the following generalized kernel type estimator
gnhn (x) =
n
where the classes F = ff g and K = K
type. Moreover, C
n
1 X
cf (x) K
nhdn i=1
x
hn
x
Xi
hn
: hn > 0; x 2 Rdx
o
f (Zi ) ;
are both pointwisely measurable and of VC
fcf : f 2 Fg is relatively compact w.r.t the supnorm topology and either
9M > 0 : F (Z) 1 fx 2 J g
M < 1,
(B.1)
sup E F 2+ (Z) jX = x = M < 1;
(B.2)
or
x2J
24
with
> 2 and the envelope function F for class F. we have
p
nhdn kgnhn (x) Egnhn (x)k
p
lim sup sup sup
< 1;
d log hn _ log log n
n!1 h2Hn f 2F
(B.3)
where
Hn =
with
log n
n
hn : C
= 1 in the bounded case (B.1), or
=1
hn
h0 ;
2= (2 + ) under the moment restriction in (B.2).
To handle the U-processes which are related to tnj , we need to introduce addtional terminologies and
notations, see Gine and Mason (2007). For a kernel function f of k variables, we denote
Un(k) (f ) =
where Inm = f(i1 ;
; im ) : 1
ij
(n
n!
k)! X
f (Xi1 ;
; X ik ) ;
k
i2In
n; ij 6= ik if j 6= kg. Now suppose f is symmetric in its entries, we have
the well-known Hoe¤ding decomposition:
Un(m) (f )
Ef =
m
X
Un(k) (
kf ) ;
k=1
where
kf
2
Moreover let
=(
x1
P)
(
P)
xk
Pm
k
f:
(which we call maximal variance) be any number satisfying
P mf 2
2
F
M 2:
Lemma B.2 (Gine and Mason, 2007, Theorem 8) Let F be a collection of measurable symmetric functions
f : S m ! R, bounded up by M in absolute values, and let P be any probability measure on (S; S) : Assume
F is of VC type with envelope function F
and A
m
e ;v
1; there exist constants C1; C2 , s.t. for any k = 1; :::; m;
nk E Un(k) (
assuming n
2
M and with characteristics A and v. Then for every m 2 N ,
C2 log
A
2
kf )
F
C1 2k
2
log
A
k
;
(B.4)
.
When we simply have k = 1, the bound in (B.4) is equivalent as Proposition 1 in Einmahl and Mason
(2005).
It’s clear from Section 5 that our estimators for the structural regression parameter would be based on
various conditional moments and corresponding …rst order derivatives. We let m ( ) be the conditional mean
function of D :
Di = m (Xi ) + "i ;
where E ["i jXi ] = 0 holds almost surely. The local polynomial estimator takes a closed form as:
25
bD
where
"
#
n
1X
(x) = Gn 1 (x)
n i=1
i Di Khn
(Xi
x) Uhn (Xi
x) ;
(B.5)
n
Gn (x) =
Let q (x) = Pr f
i
1X
n i=1
i Khn
(Xi
x) UhTn (Xi
x) Uhn (Xi
x) :
= 1jXi = xg and fX (x) denote the marginal density function of covariates X: Now we
list the regularity assumptions needed.
Assumption (B)
(1) We assume the conditional independence censoring mechanism (Yi ? Ci ) jXi . In addition, q (x)
M0 > 0 uniformly in x 2 X .
(2) The conditional mean function is p + 1 order di¤erentiable with bounded (p + 1)-th order derivative
uniformly in x 2 X , thus
0
jm(v) (x)
m(v) x
for all v 2 N d such that jvj = p, one has m(v) (x) =
j
v
@x11
0
Ljjx
@v
v
@xdd
x jj;
m (x), stands for partial derivative and jj jj
for the Euclidean norm.
(3) The marginal density function fX (x) exist and is second order continuously di¤erentiable. Moreover
it is bounded up and below uniformly, i.e. 8x 2 X , fX (x)
M0 > 0 and fX (x)
M1 < 1 hold.
d
(4) The kernel density function K is a compactly supported (w.l.o.g let it be [ 1; 1] ), bounded kernel
n
o
(up by Kmax ). Furthermore, K = K xhn : hn > 0; x 2 Rd is pointwisely measurable and of VC type.
R
Moreover, p (K) = K (u) U (u) U T (u) du is a positive de…nite P P matrix.
(5) The bandwidth sequence satis…es: hn ! 0,
log n
p d+1
nhn
! 0; and nh2p
n ! 0:
It calls for a uniform asymptotic analysis of the derivative estimators from local polynomial …tting. We
shall focus on bbD
bbD
;t
;t
(x), as the analysis of b
aD (x) could be carried out upon change of notation. Recall
(x) = hn 1 eTs;PbD (x), where eTs;P is the particular selection vector picking up s-th component of …rst
order partial derivatives. Write it as a linear smoother
bbD
with
s
Wni
(x) =
;s
(x) =
X
s
Di Wni
(x) ;
1 T
e G 1 (x) D Khn (Xi
nhn s;P n
x) Uhn (Xi
(B.6)
x) :
(B.7)
Consider the p-th order Taylor expansion for a generic smooth function m ( ) locally around x, where the
window size is controlled by hn .
m (Xi ) =
X
0 jsj p
m(s) (x)
(Xi
s!
26
s
x) + R (Xi ; x)
where R (Xi ; x) =
P
jsj=p
p+d 1
d 1
jrs (Xi ; x) j, the total number of the sum equals to Np =
. Thus we could
decompose it into two parts by Proposition 1.12 in Tsybakov (2008):
bbD
;s
(x)
We …rst bound the bias term
bD
P
;s
(x) =
X
s
R (Xi ; x) Wni
(x) +
X
s
"i Wni
(x) :
s
R (Xi ; x) Wni
(x) uniformly in covariates’value.
P
s
(x)j = Op (hpn ) :
Lemma B.3 Under Assumptions (B1)-(B5), we get supx j R (Xi ; x) Wni
Proof. Recall R (Xi ; x) =
one could de…ne Ns ; 0
s
P
jsj=p
x
, similarly
p:
jrs (Xi ; x) j = j
with
p+d 1
d 1
jrs (Xi ; x) j, the total number of the sum equals to Np =
m(s) (
p!
x)
m(s) (x)
(Xi
p!
p
(Xi
x)
L p+1
h ;
p! n
p
x) j
being some intermediate value in neighborhood around x:Therefore, the reminder term is controlled
LNp p+1
p! hn ;
(uniformly in x) of the following order jR (Xi ; x) j
s
jWni
(x) j
1
n
hn 1 jjGn 1 (x) Khn (Xi
and with probability 1 we have
x) Uhn (Xi
C
;
nhn
x) jj
where we use the fact that Gn (x) is positive semide…nite and symmetric with probability approaching 1,
thus jjGn 1 (x) vjj
Cjjvjj. Finally the bias term could be bounded up uniformly by
sup
x
X
CNp p+1 X
s
h
jWni
(x) j
p! n
s
R (Xi ; x) Wni
(x)
Chpn :
In the current context, we need to expand the inverse Gn 1 (x) into two terms similarly as Lemma 4.2 in
Chaudhuri, Doksum, and Samarov (1997), or Lemma A.1 in Li, Lu, and Ullah (2003). For a kernel function
K, we de…ne
p (K) =
Z
U (u) U T (u) K (u) du;
T
p (K) =
Z
U (u) K (u) du:
Furthermore let uT = (u1 ; :::; ud ) , and for 1 s d,
Z
Z
#p;s (K) = U (u) U T (u) K (u) us du; {p;s (K) = U (u) K (u) us du:
Lemma B.4 Under Assumptions (B1)-(B5), we have
1
sup Gn (x)
x
1
fX (x) q
1
(x)
1
p
where
G (x)
2
fX (x) q
2
(x)
1
p
(K)
(K)
d
X
s=1
27
hn G (x) = Op
(1)
[fX (x) q (x)]s
h2n
+
s
!
#p;s (K)
log n
nhdn
1
p
!
(K) :
;
Proof. Note that
Z
Xi x
1
Xi x
Xi x
K
EGn (x) =
U
UT
dFX (Xi )
hdn
hn
hn
hn
Z
=
fX (x + uhn ) q (x + uhn ) K (u) U (u) U T (u) du
= fX (x) q (x)
p (K) + hn
d
X
(1)
[fX (x) q (x)]s #p;s (K) + O h2n :
s=1
And by the results in Doney, Einmahl, and Mason (2006), we get
s
sup jGn (x)
EGn (x) j = Op
x
log n
nhdn
!
:
Hence we use von Neumann series expansion for the inverse matrix:
Gn 1 (x)
=
1
fX (x) q (x)
1
p
hn
2
fX (x) q 2 (x)
(K)
1
p
(K)
d
X
(1)
[fX (x) q (x)]s
!
#p;s (K)
s=1
1
p
(K) + Op
h2n
+
s
log n
nhdn
!
:
Thus the linear representations we are seeking are
b
aD (x)
aD (x) =
and
bbD
where 1
;s
(x)
bD
X
1
eT0;P
nfX (x) q (x)
;s
(x)
1
p
(K) "i
i Khn
(Xi
x) Uhn (Xi
x) + op n
X
1
eTs;P p 1 (K) "i i Khn (Xi x) Uhn (Xi
nhn fX (x) q (x)
1X T
es;P G (x) "i i Khn (Xi x) Uhn (Xi x) + op n
n
=
s
d. From the previous two lemmas we immediately obtain
s
!
log n
; sup bbD (x) bD (x) = Op
sup jb
aD (x) aD (x)j = Op
nhdn
x2X
x2X
We de…ne the matrix A via AT = e1;P ; :::; ed;P
T
1
hn
s
Lemma B.5 Under Assumptions (B1)-(B5), we have for l = 1; 2:
n
1X
' (Zi ) + op n
n i=1 l
1=2
;
where the in‡uence functions are de…ned by
'1 (Zi ) = "i
and
'2 (Zi ) = "i
i
aY (Xi )
bY
ie
(Xi ) ;
n
eaY (Xi ) bY (Xi ) + eaY (Xi ) fX (Xi ) [AG (Xi )
28
; (B.8)
x)
(B.9)
1=2
!
;
:
. The following lemma gives the expression for the
in‡uence function in those terms tn;1 and tn;2 .
tnl =
log n
nhdn
1=2
p
o
(K)] :
Proof. The goal is to collect the dominating linear terms from various U-statistics. We only carry out a
detailed proof tn2 as the analysis for tn1 follows the same type of argument somewhat simpler. Let tn2;s
denote the s th element of vector tn2 . Given (B.9) and the restriction on the bandwidth hn , we have
1
tn2;s = n
n
X
i=1
1
=
where
Un(2) (f1 )
=
Un(2) (f2 )
=
hd+1
n
Un(2) (f1 )
h
eaY (Xi ) bbD
;s
(Xi )
bD
1 (2)
U (f2 ) + op n
hdn n
;s
1=2
i
(Xi )
:
0
n
X
1 X
eaY (Xj )
Xi Xj
@
eTs;P p 1 (K) "i i K
Uhn (Xi
2
n j=1 fX (Xj ) q (Xj )
hn
i6=j
0
1
n
Xi Xj
1 X aY (Xj ) @X T
e
es;P G (Xj ) "i i K
U (Xi Xj )A :
n2 j=1
hn
1
Xj )A ;
i6=j
These three terms could be analyzed via U-statistics by a simple symmetrization, say for the …rst one
0
1
n
aY (Xj )
1 X@ 1 X T
e
"
X
X
i
i
j
i
es;P p 1 (K)
K
Uhn (Xi Xj )A
n j=1 nhd+1
fX (Xj )
hn
n
i6=j
"
X
eaY (Xi ) "j j
1
eaY (Xj ) "i i
2
Xi Xj
T
1
es;P p (K) d+1 K
Uhn (Xj Xi ) +
Uh (Xi
=
n (n 1)
hn
fX (Xi ) q (Xi )
fX (Xj ) q (Xj ) n
hn
1 i<j n
The classes of functions that come into play in the U-processes are
F1
=
F2
=
Xi Xj
eaY (Xj ) "i
K
Uhn (Xi Xj ) : hn 2 H ;
fX (Xj ) q (Xj )
hn
Xi Xj
f2 : f2 = eTs;P G (Xi ) "i eaY (Xj ) K
U (Xi Xj ) : hn 2 H :
hn
f1 : f1 = eTs;P
1
p
(K)
the second order tem of the Hoe¤ding decomposition is of smaller order
hn d
1
Un(2) (
2 f1 )
= Op
log n
nhdn
;
by the maximal moment bound in Lemma B.4 as we could bound the maximal variance by an order of hdn .
The …rst order term in the Hoe¤ding decomposition for f1 is
=
=
1
U (1)
d+1 n
hn
n
X
1
n
(
1 f1 )
Ej eTs;P
1
p
(K)
i=1
n
1X
" eT
n i=1 i s;P
1
p
(K)
Z
1
hd+1
n
eaY (xj )
K
hd+1
n
n
=
=
1 X eaY (Xi ) "i T
es;P
n i=1
hn
K
Xi
eaY (Xj ) "i
U
fX (Xj )
Xj
hn
xj
Xi
hn
U
Xi
Xi
Xj
hn
xj
dxj
hn
n
1
p
1 X aY (Xi ) T
(K) p (K) +
"e
es;P
n i=1 i
n
1 X aY (Xi )
"e
bY;s (Xi ) + op n
n i=1 i
1=2
29
1
p
(K)
"
d
X
s=1
#
bY;s (Xi ) #p;s (K) + op n
1=2
#
Xj ) :
where we have used eTs;P
p
1
R
(K) K (u) U (u) du =
p
1
(K)
p
(K)
s+1;1
= 0 for 1
s
d as in Lemma
A.3 of Li, Lu, and Ullah (2003) and the equality (4.27) in Chaudhuri, Doksum, and Samarov (1997).
When it comes to f2 ;we get
1 (1)
U (
hdn n
n
1X
" fX (Xi ) eaY (Xi ) eTs;P G (Xi )
1 f2 ) =
n i=1 i
(2)
Similarly, for the other second order terms, we have hn d Un (
2 f2 )
p
(K) + op n
log n
nhd
n
= Op
1=2
;
:
The proof of obtaining the in‡uence function from local likelihood …tting falls into the same framework as
the analysis above. The complication is due to the presence of this extra parameter
in the hazard function.
Following Fan, Gijbels, and King (1997), it will prove to be convenient to introduce some additional notations
(t; ) = log (t; ).
(x)
E [ (V ; 0 ) jX = x] ;
h 0
i
(x) = e (x) E
(V ; 0 ) jX = x ;
h
00
00
(x) = E
(V ; 0 ) + e (x) (V ;
& (x)
= e
i
)
jX
=
x
:
0
Moreover, let
S0 (x) =
and
& (x)
(x)
fX (x)
p
T
p
(K)
(K)
(x)
p (K)
;
(x)
!
Pd
Pd
(1)
(1)
[fX (x) & (x)]s #p;s (K)
[fX (x) (x)]s {p;s (K)
s=1
s=1
;
Pd
(1) T
0
s=1 [fX (x) (x)]s {p;s (K)
S1 (x) =
It is straightforward to verify that the matrix S0 (x) is invertible:
S0 1 (x) =
where
(x) =
&
1
(x) p 1 (K) 0
+
0T
0
(x)
2
e0;P eT0;P 2 (x) &
eT0;P (x) (x) &
1
(x)
2
1
(x)
(x)
1
e0;P (x) (x) &
1
(x)
(B.10)
(x) =& (x) :
Assumption (C)
(1) The function
(x) is (p + 1) th order continuously di¤erentiable with bounded (p + 1)-th order
derivative uniformly in x 2 X for all x 2 X .
(2) The following conditional mean functions
E ( jX) , E ( (V ;
0
E
(V ;
0 ) jX
0 ) jX) ,
0
E
0
, and E
(V ;
(V ;
0 ) jX
00
,E
(V ;
0 ) jX
0 ) jX
are all uniformly continuous for x 2 X .
(3) There exists some
E
2+
> 0, such that
(V ;
0 ) jX
,E
0
2+
(V ;
0)
are …nite and uniformly continuous for X = x.
30
jX
and E
0
2+
(V ;
0)
jX
,
(4)
0
is an interior point of the parameter space. In addition, there exists a function M (v), with
EM (V ) < 1, such that
@3
@3
<
M
(v)
;
and
(v;
)
@ 3
@ 3
for all v and all
in a neighborhood of
(v; ) < M (v) ;
0.
(5) We assume
00
flog
(s)
Let
(V ; )g
00
0, and
(V ; )
0.
T
(x) ; 1
jsj
where H = diag (hvn ; 1
p
jvj
collect all the partial derivatives of the function
T
Y
(x), and
(x) =
(s)
T
(x) ; 1
jsj
p
H,
p). Strengthening the result in Theorem 1 in Fan, Gijbels and King (1997),
we get the uniform consistency results:
sup
x
n
bY (x)
Y
o
j
= op (1) ;
0
(x) + jb (x)
moreover, the following expansion holds
bY (x)
b (x)
Y
(x)
= Sn 1 (x) l_n (
Y
(x) ;
0)
0
The score function is l_n (
(x) ;
Y
@ln ( ; )
@
= n
@ln ( ; )
@
= n
1
1
0)
n h
X
i
T
(Vi ; ) exp
i
i=1
n h
X
0
T
@ln ( ; ) @ln ( ; )
; @
@ T
=
0
(Vi ; )
2
bY (x)
+ Op
Y
(x)
2
0j
+ jb (x)
:
where
Uhn (Xi
(Vi ; ) exp
x)
T
i
Uhn (Xi
Uhn (Xi
x)
i=1
x) Khn (Xi
i
Khn (Xi
x) ;
x) :
and the Hessian matrix Sn (x) is de…ned as
= n
Sn (x)
n
X
1
Khn (Xi
x)
(Vi ;
0) e
0
i=1
T
Uhn (Xi x)
(Vi ;
0) e
T
x) UhTn (Xi
Uhn (Xi
Uhn (Xi x)
UhTn (Xi
0
x)
00
x)
(Vi ;
0) e
(Vi ;
0) e
T
T
Uhn (Xi x)
Uhn (Xi
Uhn (Xi x)
+
Assumption (C5) ensures Sn (x) to be negative de…nite.
Let
rn (x) = n
1
n
X
(Vi ;
0
i=1
(Xi
(Vi ; 0 )
We shall decompose the score function l_n (
l_n (
Y
(x) ;
x) h
0 ) Uhn
0) = n
1
Y
(x) ;
0)
e
e(
(Xi )
T
Uhn (Xi x))
x)
Khn (Xi
where
=
i
vi
=
i
(Vi ;
0
(Vi ;
Khn (Xi
into two parts
n
X
"i Uhn (Xi
vi
i=1
"i
i
0)
31
(Xi )
0) e
0
; and
(Vi ;
0) e
(Xi )
x) + rn (x) ;
x) :
i
(Vi ;
x)
!
0)
:
both have conditional zero mean by Proposition 1 in Fan, Gijbels, and King (1997). Apply (B.2) to the
centered part in l_n (
(x) ;
0 ),
sup n
1
Y
x
it holds that
n
X
"i Uhn (Xi
vi
i=1
x)
p
log n
nhdn
x) = Op
Khn (Xi
:
(B.11)
Lemma B.6 Under Assumptions (C1)-(C5), we have supx jrn (x)j = Op hp+1
.
n
Proof. First of all, the expected value of rn (x) is equal to
Ern (x) = f (x) e
E ( jX = x)
(x) p+1
hn
0
E
(V ;
RP
(v)
jvj=p+1
0 ) jX = x
RP
(x) v
u U
v!
(u) K (u) du
(v) (x)
jvj=p+1
v!
uv K (u) du
;
+ op hp+1
n
(B.12)
extending the proof of Theorem 2 in Fan, Gijbels and King (1997) into the multivariate case. and similarly
p
we get Ern2 (x) = O hp+1
. Thus by the uniform bound in (B.4), we have
n
s
!
log
n
sup jrn (x) Ern (x)j = Op hp+1
;
n
nhdn
x
as desired.
Lemma B.7 Under Assumptions (C1)-(C5), we have
1
1
sup Sn (x)
h2n
hn S (x) = Op
fX (x) S0 (x)
x
where S (x)
1
+
s
log n
nhdn
!
;
(B.13)
fX 2 (x) S0 1 (x) S1 (x) S0 1 (x) :
The proof of Lemma B.7 follows the same line as in Lemma B.4, mutatis mutandis, hence we omit it.
We de…ne the matrix B via BT = e1;P +1 ; :::; ed;P +1
T
. Now we are ready to get the in‡uence functions for
tn3 and tn4 .
Lemma B.8 Under Assumptions (C1)-(C5), we have for l = 3; 4:
n
tnl =
1X
' (Zi ) + op n
n i=1 l
1=2
;
where the in‡uence functions are de…ned by
'3 (Zi ) = eaY (Xi ) [aD (Xi ) bY (Xi ) + bD (Xi )] "i
(Xi ) & (Xi ) + 2 (Xi )
(Xi ) & 2 (Xi )
vi
(Xi )
;
& (Xi )
and
'4 (Zi )
= eaY (Xi ) BS (Xi )
0 (
B "i
B
+A B
@
p
1
"i
p
(K)
vi
(K)
Pd
s=1
h
ea Y ( X i )
&(Xi )
i(1)
s
vi e0;P
{p;s (K) +
Pd
s=1
32
e0;P eT0;P
(Xi )eaY (Xi )
(Xi )&(Xi )
Pd
s=1
2
(Xi )eaY (Xi )
(Xi )& 2 (Xi )
(1)
{p;s (K)
s
(1)
) 1
{p;s (K)
s
C
C
C:
A
Proof. Here we only highlight the di¤erence apart from the proof of Lemma B.5 regarding tn4 . Combining
(B.11)(B.12)(B.13), it follows
tn4;s =
1
hd+1
n
Un(2) (g1 ) +
1 (2)
U (g2 ) + op n
hdn n
1=2
:
where
Un(2)
(g1 )
=
n
1 XX 1
"i Uhn (Xi
fX (Xj ) eaY (Xj ) eTs;P +1 S0 1 (Xj )
n2 j=1
vi
Xj )
Xi
K
i6=j
Un(2) (g2 )
=
n
1 X X aY (Xj ) T
"i Uhn (Xi
e
es;P +1 S (Xj )
n2 j=1
vi
Xj )
K
Xi
i6=j
Xj
hn
Xj
hn
;
:
It follows the same argument that the second order terms in the Hoe¤ding decomposition are of smaller
order, so we focus on the expressions of the …rst order term:
1
hd+1
n
=
=
=
Un(1) (
1 g1 )
Z
n
Xi xj
"i Uhn (Xi xj )
1 X T
e
S0 1 (xj ) fX 1 (xj ) eaY (xj ) K
dFX (xj )
s;P +1
h
vi
nhd+1
n
n
i=1
Z
n
"i U (u)
1 X T
e
S0 1 (Xi uhn ) eaY (Xi uhn ) K (u)
du
nhn i=1 s;P +1
vi
)
0 (
(1)
2
Pd h eaY (Xi ) i(1)
Pd
(Xi )eaY (Xi )
1
T
"i
{p;s (K) + e0;P e0;P s=1
{p;s (K)
n
p (K)
s=1
&(Xi )
(Xi )& 2 (Xi )
1X T B
s
B
s
es;P B
(1)
n i=1
@
a (Xi )
Pd
i )e Y
vi e0;P s=1 (X(X
{p;s (K)
i )&(Xi )
s
+op n
1=2
:
where we have taken a …rst-order Taylor expansion for S0 1 (Xi
uhn ) eaY (Xi
uhn )
1
C
C
C
A
and the zero-order term
vanishes similarly as in Lemma A.3 of Li, Lu, and Ullah (2003).
For g2 ; the linear term via Hoe¤ding decomposition is
1 (1)
U (
hdn n
10
n
1 g2 )
=
"i p (K)
1 X aY (Xi ) T
e
es;P +1 S (Xi )
+ op n
n i=1
vi
1=2
:
Appendix C. A Review of Lévy Processes
Lévy process, a process with independent and stationary increments, is the natural generalization of random
walk in continuous time and it encompasses many well studied ones, such as Brownian motion, (compound)
Poisson process, gamma process, and inverse Gaussian process. The distribution of Lévy process at a …xed
time coincides with the in…nitely divisible law, which also arises as the most general limit law in the classical
limit theorems for null arrays, see Sato (1999). Two sub-families of the general Lévy process are of particular
interest. One is the so-called spectral negative Lévy process used in Abbring (2012), which cannot have any
33
positive jump. While the other is the Lévy subordinator, which has non-decreasing sample path. The
Gaussian component would necessarily be ruled out in a Lévy subordinator. In this Appendix, I collect some
characterization and terminology about Levy process, and refer the interested readers to Sato (1999) and
Cont and Tankov (2003) for further information.
Despite the richness of Lévy class, it admits a concise characterization via the celebrated Lévy-Khintchine
representation. A d dimensional Lévy process L (t) is completely characterized by the triplet ( ; ;
)
through it characteristic function
E fexp [izL (t)]g = exp [t
with
where i =
p
(z) = h ; zi
1,
1 0
z z+
2
is the Lévy measure.
Z
Rd
h
eihy;zi
2
Rd
1
i hy; zi 1fjyj
1g
i
(dy) ;
on Rd n f0g is called Lévy measure if it satis…es the following
De…nition 10.1 A positive Radon measure
condition:
Z
(z)] ; z 2 Rd ;
1 ^ jyj
(dy) < 1:
Recall that a Lévy subordinator L (t) has almost surely nondecreasing sample path, i.e., for t
has L (t)
s one
L (s). The following lemma gives an equivalent characterization of the subordinator in terms of
the triplet ( ; ;
).
Proposition 10.2 L (t) is a Lévy subordinator i¤ .
R
positive quadrant with Rd (1 ^ y) (dy) < 1.
= 0;and
0;
assigns zero measure outside the
+
The restriction
R
Rd
+
(1 ^ y)
(dy) < 1 arises because the nondecreasing process has bounded variation.
We could simplify its Lévy-Laplace exponent function in this case a bit:
D 0 E Z h
i
(z) = i
;z +
eihy;zi 1
(dy) ;
Rd
where
0
=
Z
y
(dy) :
jyj 1
In this paper we restrict our attention to the case where
0
= 0 without any deterministic linear trend.
Otherwise, we need to take second order derivatives to identify
and
. By Theorem 4.1 in Cont and
Tankov (2003), there is no linear trend for each marginal Lévy process either.
The next two lemmas are about the equivalence between moment conditions for L (t) and moment
conditions for the corresponding Lévy measure. For the absolute moment conditions, see Corollary 25.8 in
Sato (1999).
34
Proposition 10.3 (Theorem 25.17 Sato, 1999) 8 non-negative u, the exponential moment E fexp [ uL (t)]g
is …nite for some t; if and only if it is …nite for all t, if and only if
Z
e hy;ui (dy) < 1.
jyj 1
In this situation, we have
E fexp [ uL (t)]g = exp [t
(iu)] .
Proposition 10.4 (Wolfe, 1971) The existence of moment for marginal distribution L (t) is equivalent as
the existence of moment for the Lévy measure
.
For most Lévy processes, the jump intensity would explode to in…nity as the jump size converges to
zero. The resulting implication is that the survival function version of Lévy measure is more tractable than
its distribution function version. Such survival function attached to the Lévy measure is the so-called tail
integral which we de…ne next.
De…nition 10.5 (Tail Integral) A two-dimensional tail integral is a function
(1)
is a 2-increasing function;
(2)
is equal to zero if one of its arguments is equal to +1;
(3)
is …nite everywhere except possibly at zero.
The probabilistic interpretation of the tail integral measure
: R2+ ! R+ s.t.
(x) is the following. Let N (t; x) denote
the number of jumps of magnitude larger than the positive constant x during the time interval [0; t]. Then
N (t; x) is a Poisson process with parameter
(x) t. With the tail integral, we could de…ne the notion of
Lévy copula and its role in pinning down the joint Lévy measure.
De…nition 10.6 (Lévy Copula) A two dimensional Lévy copula for Lévy subordinators is a 2 increasing
2
grounded function CL (u; v) : [0; 1] ! [0; 1] with uniform margins, i.e.
CL (u; 1) = CL (1; u) = u:
Theorem 10.7 (Sklar Theorem) Let
be a two-dimensional tail integral with margins
1;
2,
then there
exists a positive Levy copula CL s.t.
(u; v) = CL
if
1;
2
1
(u) ;
2
(v) ;
(C.1)
are continuous, the copula function CL is unique. Conversely for a given Lévy copula CL and two
marginal tail integrals
1;
2,
(C.1) de…nes a two-dimensional tail integral.
Recall the Clayton-Lévy copula takes the form CL; (u; v) = (u
+v
)
1=
, with a single parameter
2 (0; 1). A straightforward yet slightly tedious computation leads to the following expressions of its
35
partial derivative w.r.t the arguments u or v:
@
CL; (u; v)
@u
@
CL; (u; v)
@v
1
= u
1
u
+v
= v
1
u
+v
1
1
1
;
;
and its derivative w.r.t he parameter :
@
CL; (u; v)
@
1
=
u
log (u
1
+v
+v
log u1 + v
(u + v
u
)
2
When the marginal Lévy measure has a density function
log
)
1
v
:
( ) with respect to the Lebesgue measure and
the Lévy copula function is di¤erentiable, the joint Lévy-Laplace exponent function admits the following
alternative expression:
12 (z1 ; z2 ) =
ZZ
1
e
@2
CL ( ; ) ju=
@u@v
z1 y1 z2 y2
R2+
1 (y1 )
v=
1
(y1 )
2
(y2 ) dy1 dy2 ;
2 (y2 )
Next we list some commonly used univariate Lévy densities, from Cont and Tankov (2003), Gjessing, Aalen
and Hjort (2003).
Example 10.8 The stable process has a power parameter
2 (0; 1) and stable index
2 (0; 2], the marginal
Lévy density is
S
(y) =
) y 1+
(1
;
and the Lévy exponent function
S
(u) = u :
Example 10.9 The gamma process has a shape parameter
and scale parameter
, the marginal Lévy
density is
G
(y) =
y
e
;
y
and the Lévy exponent function
G
(u) = log 1 +
u
:
Example 10.10 The power variance function process is a general process nesting the above gamma process
as a special case, the marginal Lévy density is
S
with
> 0, n >
(y) =
n n
yn
(n + 1)
1
y
e
;
1 and n > 0 and the Lévy exponent function is
PV F
(u) =
n
1
1+
The gamma process arises as a borderline case when n = 0:
36
u
no
:
Finally we would like to comment on the compound Poissson process L (t) =
P
i N (t)
Wi , because it is
the only one having piece-wise constant sample path (almost surely) and its correponding Lévy measure is
R
…nite, i.e. Rd (dy) < 1. In addition, when Wi has a density function f (w) and the arrival rate of N (t)
is , the Lévy measure has a density function
CP
(z) =
(w) = f (w) and the Lévy exponent function simpli…es to
Z h
i
eihw;zi 1 f (w) dw.
Rd
References
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