Journal of Statistical Planning and
Inference 136 (2006) 1911 – 1922
www.elsevier.com/locate/jspi
Asymptotically best response-adaptive
randomization procedures夡
Feifang Hua, b,∗ , William F. Rosenbergerc, d, e , Li-Xin Zhange, f
a Department of Statistics, University of Virginia, Halsey Hall, Charlottesville, VA 22904-4135, USA
b Division of Biostatistics and Epidemiology, Department of Health Evaluation Services, University of Virginia,
School of Medicine, USA
c Department of Applied and Engineering Statistics, George Mason University, 4400 University Drive, Fairfax,
VA 22030 USA
d Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle,
Baltimore, MD 21250, USA
e Department of Epidemiology and Preventive Medicine, School of Medicine, University of Maryland, USA
f Department of Mathematics, Zhejiang University, Xixi Campus, Zhejiang, Hangzhou 310028, PR China
Available online 31 August 2005
Abstract
We derive a lower bound on the asymptotic variance of the allocation proportions from responseadaptive randomization procedures when the allocation proportions are asymptotically normal. A
procedure that attains this lower bound is defined to be asymptotically best. We then compare the
asymptotic variances of five procedures, for which allocation proportions converge, to the lower
bound. We find that a procedure by Zelen and a procedure by Ivanova attain the lower bound and
a procedure by Eisele and its extension to K > 2 treatments can attain the lower bound but are, in
general, not asymptotically best. We discuss the tradeoffs among the benefits of randomization, the
benefits of attaining the lower bound, and the benefits of targeting an optimal allocation. We conclude
that none of these procedures possesses all of these benefits.
© 2005 Elsevier B.V. All rights reserved.
MSC: Primary 62G10
Keywords: Adaptive designs; Clinical trials; Doubly-adaptive biased coin design; Neyman allocation;
Rao–Cramér lower bound; Urn models
夡 This research was supported by NSF grant DMS-0204232 (Hu and Rosenberger) and grants from the National
Natural Science Foundation of China and the National Natural Science Foundation of Zhejiang Province (Zhang).
∗ Corresponding author. Department of Statistics, University of Virginia, Halsey Hall, Charlottesville,
VA 22904-4135, USA.
E-mail address: fh6e@virginia.edu (F. Hu).
0378-3758/$ - see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.jspi.2005.08.011
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F. Hu et al. / Journal of Statistical Planning and Inference 136 (2006) 1911 – 1922
1. Introduction
Response-adaptive randomization procedures for clinical trials incorporate sequentially
accruing response data into future randomization probabilities (Rosenberger and Lachin,
2002). Hu and Rosenberger (2003) have recently developed a template for a careful evaluation of a response-adaptive randomization procedure for binary responses. The template
considers the competing goals of desiring high power for treatment comparisons while assigning fewer patients to the inferior treatment. It requires (1) a single limiting allocation,
(2) a randomization procedure, and (3) the nondegenerate asymptotic distribution of the
allocation proportions to each treatment following that randomization procedure. In particular, if the observed allocation proportions are asymptotically normal, the asymptotic
power is an decreasing function of the asymptotic variance of the allocation proportions.
For requirement (2), Rosenberger and Lachin (2002) define three classes of responseadaptive randomization procedures: urn models, sequential estimation procedures, and treatment effect mappings. There is, as yet, little information on the asymptotic distribution of
allocation proportions from treatment effect mappings, and we will therefore focus in this
paper on specific randomization procedures from only the classes of urn models and sequential estimation procedures. For K = 2 treatments, many urn models have been proposed. In
this paper, we focus on two that have the same limiting allocation, for ease of comparison.
These are Wei and Durham’s (1978) procedure, often referred to as the “randomized playthe-winner rule”, and Ivanova’s (2003) procedure, which she refers to as the “drop-the-loser
rule”. Sequential estimation procedures for two treatments can be described in full generality for two treatments by Eisele’s (1994) procedure, which he called the “doubly-adaptive
biased coin design”. Each of these procedures yields asymptotically normal allocation proportions, and the asymptotic variance is known, thus satisfying requirement (3) above.
These procedures are generally analyzed according to a simple homogeneous parametric
structure, with pA and pB being the success probabilities for those patients assigned to
treatment A and B, respectively. We can also define qA = 1 − pA and qB = 1 − pB .
These procedures have analogs for K > 2 treatments, although the theoretical developments and practical consequences are far more difficult to ascertain, and are the subject of
current research. For example, Eisele’s procedure was generalized by Hu and Zhang (2004)
to K > 2 treatments; the extension of Wei and Durham’s procedure is the generalized Friedman’s urn model of Wei (1979); and the extension of Ivanova’s procedure is the birth and
death urn with immigration by Ivanova et al. (2000).
For requirement (1), various limiting allocations (pA , pB ) can be proposed, where is
the proportion of patients assigned to treatment A in the limit. In all practical cases, will
be a function of the unknown parameters, and hence the argument pA , pB . We define “urn
allocation”, which is the limiting allocation of many urn models, as (pA , pB ) = qB /(qA +
qB ). When the treatments generated by a procedure converge to a desired limiting allocation,
we say they target this allocation. Various target allocations have been proposed based
on formal optimality criteria (cf. Rosenberger et al., 2001). If one wishes to maximize
the power of the usual test comparing two binomial probabilities, it is well-known that
Neyman allocation, the ratio of the standard deviations, should be used, given by =
√
√
√
pA qA /( pA qA + pB qB ). However, Neyman allocation may be inappropriate for ethical
reasons because it assigns more patients to the inferior treatment if pA + pB > 1. An
F. Hu et al. / Journal of Statistical Planning and Inference 136 (2006) 1911 – 1922
1913
alternative allocation that minimizes the expected number of treatment failures for fixed
√
√
√
power is (pA , pB ) = pA /( pA + pB ) (Rosenberger et al., 2001), which we refer
to as “RSIHR allocation” as an acronym of the authors of the original paper. Other forms
of optimal allocation are given in that paper as well. An advantage of Eisele’s procedure
over the urn models is that we can select any desired allocation, including urn allocation,
Neyman allocation, and RSIHR allocation.
In this paper we derive a lower bound on the asymptotic variance of the allocation proportions for general response-adaptive randomization procedures. The technique used is an
extension of the Rao–Cramér lower bound to dependent sequences of random variables.
We show that Ivanova’s procedure has minimum asymptotic variance in the class of all
response-adaptive procedures with limiting urn allocation. While Eisele’s procedure does
not have minimum asymptotic variance, we can tune a parameter to obtain a nearly deterministic procedure that is minimum asymptotic variance. We also discuss several tradeoffs
that we encounter in practice: the desire for high power, the desire for a fully randomized
procedure, and the desire to target any allocation of interest. In Section 2, we present the
theorem for the special case of binary response and K = 2 treatments, and then demonstrate
the results for the response-adaptive randomization procedures we have discussed so far. In
Section 3, we generalize the results to any response-adaptive randomization procedure for
general K treatments under mild regularity conditions. In Section 4, we draw conclusions.
Proofs are relegated to an appendix.
2. Results for two treatments
In this section, we present the criterion for an asymptotically best response-adaptive
randomization procedure for K = 2 treatments. We first describe the response-adaptive
randomization procedures we will be examining and also give some important asymptotic
properties. Let NA (n) be a random variable indicating the number of patients assigned to
treatment A in n patients (NB (n) = n − NA (n) is then determined).
Wei and Durham’s procedure. Wei and Durham’s (1978) famous randomized play-thewinner rule uses an urn model to allocate treatments. The urn starts with a fixed number
of type A balls and type B balls in an urn. To randomize a patient, a ball is drawn, the
corresponding treatment assigned, and the ball replaced. An additional ball of the same
type is added if the patient’s response is success and an additional ball of the opposite type
is added if the patient’s response is a failure. The urn can only target one value, namely
(pA , pB ) = qB /(qA + qB ). Smythe and Rosenberger (1995) showed that if pA + pB < 23 ,
n1/2 (NA (n)/n − qB /(qA + qB )) → N(0, v),
in law, where
v=
qA qB (5 − 2(qA + qB ))
(2(qA + qB ) − 1)(qA + qB )2
.
The limit is presumably nonnormal for pA + pB > 23 .
Ivanova’s procedure. Ivanova (2003) considered an urn containing balls of three types,
type A, type B, and type 0. A ball is drawn at random. If it is type A or type B, the corre-
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F. Hu et al. / Journal of Statistical Planning and Inference 136 (2006) 1911 – 1922
sponding treatment is assigned and the patient’s response is observed. If it is a success, the
ball is replaced and the urn remains unchanged. If it is a failure, the ball is not replaced. If
a type 0 ball is drawn, no subject is treated, and the ball is returned to the urn together with
one ball of type A and one ball of type B. Ivanova (2003) shows that the limiting allocation
of the procedure is also (pA , pB ) = qB /(qA + qB ), and also that
n1/2 (NA (n)/n − qB /(qA + qB )) → N(0, v),
in law, where
v = qA qB (pA + pB )/(qA + qB )3 .
Eisele’s procedure. Eisele’s (1994) procedure is designed to target any desired allocation (pA , pB ). He defines a function g(x, ) that represents the closeness of the current
allocation proportion to the estimated target allocation, when replaced by the current values of maximum likelihood estimators of pA and pB . At the jth allocation, the procedure
allocates patient j to treatment A with probability g(NA (j − 1)/(j − 1), (p̂A , p̂B )). Hu
and Zhang (2004) define the following function g, evaluated at a generic value x, having
nice interpretive properties. For nonnegative integer ,
g(x, ) =
(/x)
,
(/x) + (1 − )((1 − )/(1 − x))
g(0, ) = 1,
g(1, ) = 0.
(1)
Hu and Zhang also show that
n1/2 (NA (n)/n − (pA , pB )) → N(0, v),
in law, where
v=
qA qB ((1 + 2)(pA + pB ) + 2)
(1 + 2)(qA + qB )3
.
Note that v is a decreasing function of .
We now discuss how to find a lower bound on the asymptotic variance of a responseadaptive randomization procedure. The lower bound depends on the unknown parameters as
well as a limiting allocation . Any procedure that has the same limiting allocation can then
be compared to this lower bound. The general theorem establishing a lower bound on the
asymptotic variance of the allocation proportions for K treatments and general parameter
vector will be given in Section 3 as Theorem 1 and proved in the appendix. To illustrate
the method, we now state the theorem for binary responses and K = 2 treatments.
Let I (pA , pB , NA (n)) be the Fisher’s information, where the expectation is taken conditional on NA (n), for estimating pA and pB . Suppose the following regularity conditions
hold:
1. pA , pB ∈ (0, 1);
F. Hu et al. / Journal of Statistical Planning and Inference 136 (2006) 1911 – 1922
1915
2. NA (n)/n converges to (pA , pB ) ∈ (0, 1) almost surely for the particular responseadaptive
randomization procedure;
√
3. n(NA (n)/n − (pA , pB )) converges in law to N(0, v) for the particular procedure.
Then a lower bound on the asymptotic variance of NA (n)/n is given by
j(pA , pB ) j(pA , pB ) −1
I (pA , pB , n(n, pA , pB ))
jpA
jpB
j(pA , pB ) j(pA , pB ) .
(2)
×
jpA
jpB
We can compute
⎡
p A qA
n(p
A , pB )
I −1 (pA , pB , n(pA , pB )) = ⎣
0
⎤
0
⎦.
p B qB
n(1 − (pA , pB ))
We refer to a response-adaptive allocation procedure that attains the lower bound as asymptotically best for that particular allocation (pA , pB ). It is best in the sense that, for a fixed
allocation (pA , pB ), it maximizes an asymptotic approximation to the power of chi-square
test for the difference of proportions
(p̂A − p̂B )2
pA qA /NA (n) + pB qB /NB (n)
as described in Hu and Rosenberger (2003), where p̂A and p̂B are the maximum likelihood
estimators of pA and pB .
We now apply this result for the three allocation rules mentioned in Section 1: urn
allocation, Neyman allocation, and RSIHR allocation. For urn allocation, given by qB /(qA +
qB ), from (2), the lower bound is given by
qA qB (pA + pB )
n(qA + qB )3
.
Since this is the asymptotic variance of Ivanova’s procedure, the procedure attains the
lower bound. Hence, Ivanova’s procedure is the asymptotically best procedure among all
procedures with limiting allocation qB /(qA + qB ).
It is interesting to note that the Zelen’s (1969) deterministic procedure, termed “play-thewinner rule”, which assigns the opposite treatment to the next patient if the previous patient
was a treatment failure, and the same treatment if the previous patient was a treatment
success, is also an asymptotically best procedure. However, unlike Ivanova’s procedure,
Zelen’s procedure is not randomized. That Zelen’s procedure satisfies regularity condition
2 was known in the original paper. To our knowledge, condition 3 along with v had not
previously been known. This leads to the following proposition:
Proposition 1. For Zelen’s (1969) procedure,
n1/2 (NA (n)/n − qB /(qA + qB )) → N(0, v),
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F. Hu et al. / Journal of Statistical Planning and Inference 136 (2006) 1911 – 1922
in law, where
v = qA qB (pA + pB )/(qA + qB )3 .
For Eisele’s design with the g function given in (1), it can be seen that v qA qB (pA +
pB )/(qA + qB )3 , with equality holding when → ∞. Thus the lower bound can be
attained, but the result is a mostly deterministic procedure that assigns the next patient to A
with probability 1 if the current allocation is less than (p̂A , p̂B ) and to B with probability
1 if the current allocation is greater than (p̂A , p̂B ). If the current allocation is equal to
(p̂A , p̂B ), then allocation is according to a biased coin.
Wei and Durham’s procedure is not an asymptotically best procedure (except trivially
when pA =pB ), as their asymptotic variance v is always larger than for Ivanova’s procedure,
a fact first pointed out by Ivanova (2003).
Complete randomization, with limiting allocation (pA , pB ) = 21 , satisfies regularity
conditions 2 and 3 with v = 41 , and this is not asymptotically optimal except in the trivial
case where pA =pB = 21 . The asymptotic variance of Ivanova’s procedure is often less than 41 ,
except for very large values of pA and pB . So Ivanova’s procedure is fully randomized, and
assigns more patients to the better treatment (asymptotically) often with smaller variability
than complete randomization.
When the limiting allocation is Neyman allocation, the lower bound is computed from
(2) as
1
pB qB (qA − pA )2
pA qA (qB − pB )2
.
+
√
√
3
√
√
pA q A
pB qB
4n pA qA + pB qB
Eisele’s procedure does not attain the lower bound (except as → ∞). When = 0,
we have a procedure explored by Melfi and Page (1998, 2000) with limiting Neyman
allocation, and they discovered that this procedure was highly variable. The high variability
can reduce power substantially, which is undesirable when the desired allocation is supposed
to maximize power. Essentially this procedure allocates to A with probability (p̂A , p̂B ) at
each stage.
Finally, when the limiting allocation is RSIHR allocation, the lower bound is computed
from (2) as
p B qA
1
pA qB
.
√
√ 3 √p + √p
A
B
4n pA + pB
The use of Eisele’s procedure targeting RSIHR with = 2 was strongly recommended in
Hu and Rosenberger (2003), although we see that this is not a best asymptotic procedure.
We discuss this further in Section 4.
3. Main results for K treatments
We now establish a data structure which allows for simplification of the proof of the main
results for K treatments with general outcomes. The main results involve the asymptotic
F. Hu et al. / Journal of Statistical Planning and Inference 136 (2006) 1911 – 1922
1917
properties of the maximum likelihood estimators from an exponential family and then the
Rao–Cramér lower bound based on the Fisher information matrix. For the generalized
Friedman’s urn model with K treatments, Rosenberger et al. (1997) proved these results,
but the required conditions are not satisfied for many other response-adaptive randomization
procedures.
Let Tij , i =1, . . . , n, j =1, . . . , K be the n treatment assignments, where Tij =1 if the ith
patient is assigned to treatment j, 0 otherwise. Let N(n)=(N1 (n), . . . , NK (n)) be the sample sizes on each treatment. Now define T1 = (T11 , . . . , T1K ), . . . , Tn = (Tn1 , . . . , TnK ).
Note that each vector is a vector of K − 1 0s and one 1. For the responses, let X1 =
(X11 , X12 , . . . , X1K ), . . . , Xn = (Xn1 , Xn2 , . . . , XnK ), where Xi represents the sequence
of responses that would be observed if each treatment were assigned to the ith patient independently. However, only one element of Xi will be observable. We assume that X1 , . . . , Xn
are independent and identically distributed, with
X1j ∼ fj (·, j ),
j = 1, . . . , K,
where j ∈ j . We thus assume that Xm is independent of X1 , . . . , Xm−1 , T1 , . . . , Tm ,
but that Tm depends on X1 , . . . , Xm−1 , T1 , . . . , Tm−1 , m = 1, . . . , n. The value of this
data structure for theoretical purposes is obscured by the complexity of the notation. We
therefore give a simple example to illustrate the notation before moving to our main result.
Example. If K =2 and we use Wei and Durham’s procedure, then f1 (·, 1 ) is Bernoulli(pA )
and f2 (·, 2 ) is Bernoulli(pB ). Also Tm = (1, 0) if treatment A was assigned to the mth
patient, and Tm = (0, 1) if treatment B was assigned. Then Xm = (Xm1 , Xm2 ), where Xm1 ∼
f1 and Xm2 ∼ f2 . Clearly Xm is independent of X1 , . . . , Xm−1 and also of T1 , . . . , Tm , but
we only observe the element of Xm corresponding to the element of Tm that is 1. However,
Tm depends on all previous treatment assignments and responses according to the urn model.
We now state the regularity conditions for the main result:
1. The parameter space j is an open subset of Rd , d 1, for j = 1, . . . , K.
2. The distributions f1 (·, 1 ), . . . , fK (·, K ) follow an exponential family.
3. For limiting allocation () = (1 (), . . . , K ()) ∈ (0, 1)K ,
Nj (n)
→ j ()
n
almost surely for j = 1, . . . , K.
4. For positive definite matrix V(),
N(n)
− () → N(0, V())
(n)
n
in law.
We require the following lemma, which gives the asymptotic distribution of the maximum
likelihood estimator ˆ of :
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F. Hu et al. / Journal of Statistical Planning and Inference 136 (2006) 1911 – 1922
Lemma 1. Under regularity conditions 1–3, we have
√
n(ˆ − ) → N(0, I−1 ()),
where I() = diag{1 ()I1 (1 ), . . . , K ()IK (K )} and Ij (j ) is the Fisher’s information
for a single observation on treatment j = 1, . . . , K.
Remark. We can generalize to nonexponential families by imposing further regularity
conditions, such as those given in Rosenberger et al. (1997). However, they replace condition
3 essentially with a requirement that
E(Tij |Fi−1 ) → j (),
j = 1, . . . , K
almost surely, where Fn is the sigma algebra generated by the first n assignments and
responses. This condition is not satisfied for Ivanova’s procedure and Zelen’s procedure,
for example. However, our condition 3 is satisfied for all the procedures we examine.
Now we state the main result:
Theorem 1. Under regularity conditions 1–4, there exists a 0 ⊂ = 1 ⊗ · · · ⊗ K
with Lebesgue measure 0 such that for every ∈ − 0 ,
V()
j() −1 j() I ()
.
j
j
We can now rigorously define an asymptotically best response-adaptive procedure as one
in which V() attains the lower bound
B() =
j() −1 j() I ()
j
j
for a particular target allocation ().
We now show that an extension of Eisele’s procedure for K > 2 treatments (Hu and
Zhang, 2004) is not an asymptotically best procedure, in general, but can attain the lower
bound.
Hu and Zhang’s procedure. In the generalization of Eisele’s procedure to K > 2 treatments, g is a function mapping [0, 1]K ⊗ [0, 1]K to [0, 1]K . We allocate to treatment j with
probability gj (N(n)/n, ()), where g(x, y) = (g1 (x, y), . . . , gK (x, y)) and
yj (yj /xj ) ∧ c
gj (x, y) = K
j =1 {yj (yj /xj ) ∧ c}
for some c > 1, where a ∧ b is the minimum of a and b. The constant c is introduced for
technical reasons, and if c is chosen large it will have little influence on the allocation.
F. Hu et al. / Journal of Statistical Planning and Inference 136 (2006) 1911 – 1922
1919
Regularity conditions 1–4 hold with
V() =
1
2(1 + )
1 () +
B(),
1 + 2
1 + 2
where 1 () = diag{()} − ()() (see Hu and Zhang, 2004).
Note that, for any (), the procedure is asymptotically best when → ∞, as in Eisele’s
procedure for K = 2 treatments. The form of the asymptotic variance is particularly interesting because it demonstrates the relationship between the asymptotic variance and the
lower bound for all choices of target allocations ().
4. Conclusions
We have established a lower bound on the asymptotic variability of response-adaptive
randomization procedures which provides a baseline for comparison of existing procedures and further guidance in developing new procedures. As such, this paper represents
an evolving body of knowledge about these complex procedures; however, it also presents
numerous dilemmas regarding the tradeoffs among randomization, variability, and optimality. On the surface, Ivanova’s procedure would seem to give us everything we want in a
response-adaptive randomization procedure: it is fully randomized and it attains the lower
bound. However, it can only target qB /(qA + qB ), which is not optimal in any formal sense,
and previously reported simulations have shown that it can be slow to converge for large
values of pA and pB . It also becomes more deterministic for small values of pA and pB
(see Hu and Rosenberger, 2003). Eisele’s procedure and Hu and Zhang’s extension solve
some of these deficiencies, in that they can target any desired allocation, and can approach
the lower bound for large values of . However, the procedure becomes more deterministic
as becomes larger, and hence careful tuning of must be done in order to counter the
tradeoff. This leads to a challenge for researchers in the area: can a procedure be found
that preserves randomization, attains the lower bound, and targets any allocation? We also
note that Eisele’s and Hu and Zhang’s procedures are the only procedures mentioned in this
paper that can be used for more general response types.
Acknowledgements
The authors thank an anonymous referee for helpful comments. This paper was revised
while the authors were visiting the Department of Statistics and Applied Probability, National University of Singapore. They thank the Department for its hospitality.
Appendix
Proof of Proposition 1. Zelen’s rule induces a Markov chain {Tn , n 1} with states {0, 1}
and transition matrix
pA qA
P=
.
qB p B
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F. Hu et al. / Journal of Statistical Planning and Inference 136 (2006) 1911 – 1922
The stationary probabilities are =(, 1−), where =qB /(qA +qB ). The n-step transition
matrix can then be computed as
⎡ q + q (1 − q − q )n q − q (1 − q − q )n ⎤
B
n
P
⎢
(n)
= ((pij )) = ⎣
qB
A
A
B
A
A
A
B
⎥
qA + q B
.
qA + qB (1 − qA − qB )n ⎦
qA + q B
qA + qB
− qB (1 − qA − qB )n
qA + q B
By the central limit theorem for Markov chains (cf. Theorem 17.0.1 of Meyn and Tweedie,
1993), it follows that
n
√
1 NA
L
(Tm − E (T1 )) → N(0, v),
− = √
n
n
n
m=1
where
v = E (T1 − E (T1 ))2 + 2
∞
E {(T1 − E (T1 ))(Tm+1 − E (T1 ))}.
m=1
We compute
E (T1 − E (T1 ))2 = 1 2 =
q A qB
(qA + qB )2
and
E {(T1 − E (T1 ))(Tm+1 − E (T1 ))} = P (T1 = 1, Tm+1 = 1) − (P (T1 = 1))2
(m)
= p11 P (T1 = 1) − (P (T1 = 1))2
qA qB (1 − qA − qB )m
=
.
(qA + qB )2
It follows that
v=
qA qB
1+2
(qA + qB )
2
∞
(1 − qA − qB )m =
m=1
qA qB (pA + pB )
(qA + qB )3
.
Proof of Lemma 1. Let
L() =
K
n (fj (Xi,j , j ))Ti,j =
i=1 j =1
K
Lj (j )
j =1
be the likelihood function based on the observed sample. It is sufficient to show that
1 j log Lj (j )
+ op (n−1/2 )
jj
Nj (n)
n
j log fj (Xi,j , j )
1
= −
Ti,j
+ op (n−1/2 ),
nj (j )
jj
(
j − j )Ij (j ) = −
i=1
(3)
F. Hu et al. / Journal of Statistical Planning and Inference 136 (2006) 1911 – 1922
1921
j = 1, . . . , K, as
n
Ti,j
i=1
j log fj (Xi,j , j )
= Op (n1/2 ).
jj
Let
Yi,j = −Ti,j
j log fj (Xi,j , j )
jj
and An = (X1 , . . . , Xn , T1 , . . . , Tn+1 ). Then {Yi = (Yi,1 , . . . , Yi,K ), Ai ; i = 1, 2, . . .} is
a martingale sequence with
1
1
E(Yi Yi |Ai−1 ] =
diag(Ti,1 I1 (1 ), . . . , Ti,K IK (K ))
n
n
i=1
i=1
NK (n)
N1 (n)
p
IK (K ) → I().
I1 (1 ), . . . ,
= diag
n
n
n
n
Conditions of the martingale central limit theorem (e.g., Billingsley, 1961) hold by regularity
conditions 2 and 3, and it follows that
1 L
Yi → N(0, I()).
√
n
n
i=1
It remains to show (3). For fixed j = 1, 2, . . . , K, define i (j ) = min{k : Nj (k) = i} =
min{k > i−1 (j ) : Tk,j = 1}, where min{∅} = +∞. Let {i,j } be an independent copy of
i,j =X (j ),j I {i (j ) < ∞}+i,j I {i (j )=
{Xi,j }, which is also independent of {Ti }. Define X
i
∞}, i = 1, 2, . . . . Then {Xi,j ; i = 1, 2, . . .} is a sequence of i.i.d. random variables, with
the same distribution as X1,j . Let
j (j ) = L
j (j ; Nj (n)) =
L
Nj (n)
i,j , j )
fj (X
i=1
j ). Then on the event {Nj (n) → ∞},
and let Nj (n),j maximize L(
j (j ; Nj (n)) and j = Nj (n),j .
Lj (j ) = L
(4)
i,j : i = 1, 2, . . . , n} are independent and identically distributed. Under
Notice that {X
regularity condition 2, we have
j (j ; Nj (n))
1 j log L
(
j − j )Ij (j ) = −
+ op (n−1/2 ).
n
jj
By (4) and regularity condition 3, (3) follows.
1922
F. Hu et al. / Journal of Statistical Planning and Inference 136 (2006) 1911 – 1922
i,j : i=1, 2, . . . , m} are
Proof of Theorem 1. From the proof of Lemma 1, we know that {X
independent and identically distributed. Therefore the Rao–Cramer lower bound
of () is
j() −1 j() I ()
.
j
j
Because N(n)/n is an asymptotically unbiased estimator of () and satisfies regularity conditions 2 and 4, by Theorem 4.16 of Shao (1999, p. 249) the theorem follows
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