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Working Paper Series
QUANTILE AND PROBABILITY CURVES
WITHOUT CROSSING
Victor
Chemozhukov
Ivan Fernandez- Val
Alfred Galichon
Working Paper 07-1
April 27, 2007
Room
E52-251
50 Memorial Drive
Cambridge,
MA 021 42
This paper can be downloaded without charge from the
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at
QUANTILE AND PROBABILITY CURVES WITHOUT CROSSING
VICTOR CHERNOZHUKOVt
ALFRED GALICHON*
IVAN FERNANDEZ- VAL§
Abstract. The most common approach
to estimating conditional quantile curves
is
to
fit
a curve,
typically linear, pointwise for each quantile. Linear functional forms, coupled with pointwise fitting, are
used
for
a number of reasons including parsimony of the resulting approximations and good computa-
tional properties.
The
resulting
fits,
however,
may
not respect a logical monotonicity requirement - that
the quantile curve be increasing as a function of probability. This paper studies the natural monotonization of these empirical curves induced by sampling from the estimated
non-monotone model, and then
taking the resulting conditional quantile curves that by construction are monotone in the probability.
This construction of monotone quantile curves
arrangement of the original
non-monotone
to the true curves in finite samples, for
conditional quantile curves have the
Under
may be
function.
seen as a bootstrap and also as a monotonic
It is
any sample
shown that the monotonized curves are
size.
Under
as the original
misspecification, however, the asymptotics of the rearranged curves
An
the estimates of conditional distribution functions.
(Hadamard)
may
analogous procedure
The
results are derived
is
non-monotone
developed to monotonize
by establishing the compact
the monotonized quantile and probability curves with respect to the
diflferentiability of
the compact differentiability of the rearrangement-related operators
Method, Hadamard
2000 subject
Date:
First version
of this paper
for
is
established.
Rearrangement Operators.
classification: Primary 62J02; Secondary 62E20,
is
of April
(partially)
6,
2005.
The
last
62P20
modification was done April 27, 2007.
borrowed from the work of Xuming He (1997), to
the inspiration and formulation of the problem.
Chesher, Phil Cross,
is
In doing so,
regression, Monotonicity, Rearrangement, Approximation, Functional Delta
Differentiability of
AMS
curves.
partially differ from the
original curves in discontinuous directions, tangentially to a set of continuous functions.
Keywords: Quantile
closer
correct specification, the rearranged
same asymptotic distribution
asymptotics of the original non-monotone curves.
re-
Raymond
Guiteras,
Xuming
We
would
like to
He, Roger Koenker,
whom we
The
title
are grateful
thank Josh Angrist, Andrew
Vadim Marmer,
Ilya
Molchanov,
Francesca Molinari, Whitney Newey, Steve Portnoy, Shinichi Sakata, Art Shneyerov, Alp Simsek, and
seminar participants at BU, Columbia, Cornell, Georgetown, Harvard-MIT, MIT, Northwestern,
and
UCL
for
very useful
comments that helped improve the paper.
1
UBC,
Introduction and Discussion
1.
The problem studied
sion as the prime
in this
paper can be best described using Hnear quantile regres-
example (Koenker, 2005). Suppose that
x'(3{u)
is
a linear approxima-
tion to the u-quantile Qo{u\x) of a real response variable Y, given a vector of regressors
X
=
u G
The
X.
(0, 1)
typical estimation
methods
producing an estimate x
fitting, are
fit
the conditional curve x'P{u) pointwise in
Linear functional forms, coupled with pointwise
(5{u).
used for a number of reasons including parsimony of the resulting approxi-
mations and good computational properties (Portnoy and Koenker, 1997). However, a
problem that might occur
that the
is
map
u
may not
be increasing
in u,
manifestation of this issue,
we
X P{u)
which violates the
known
the conditional quantile curves x
In the analysis
I—>
logical
monotonicity requirement. Another
as the "quantile crossing
f-^ x' (3[u)
may
problem" (He, 1997),
is
that
cross for different values of u.
shall distinguish the following
two
cases, each leading to the lack
of monotonicity or the crossing problem:
(1)
Monotonically correct case: The population curve u
and thus
u
(2)
H^-
satisfies the
x'P{u)
i—>
x'P{u)
is
increasing in u,
monotonicity requirement. However, the empirical curve
may be non-monotone due
to estimation error.
Monotonically incorrect case: The population curve u
<-^
x'P{u)
is
non-monotone
due to imperfect approximation to the true conditional quantile function. Accordingly, the resulting empirical curve u
i-^ x' (5{u) is
also
non-monotone due
both non-monotonicity of the population curve and estimation
to
error.
Consider the random variable
Y^ :=
x%U)
where
U~
?7(0, 1).
This variable can be seen as a bootstrap draw from the estimated quantile regression
model, as in Koenker (1994), and has the distribution function
F{y\x)= I l{x'P{u)<y]du.
JQ
(1.1)
3
Moreover, inverting the distribution function, one obtains a proper quantile function
F-\u\x)
which
is
monotone
in u.
original curve x (3{u)
= mi{y:F{y\x)>u},
The rearranged
the original curve
if
(1.2)
quantile function F~^{u\x) coincides with the
is
increasing in u, but differs from the original
curve otherwise. Thus, starting with a possibly non-monotone original curve u
the rearrangement (1.1)-(1.2) produces a monotone quantile curve u
what
follows,
we
focus our attention on the interval
Indeed, any closed subinterval of
(0, 1)
tile
this
F^^{u\x).
name have a direct
erations research
In
loss of generality.
in Section 2.
regression bootstrap (Koenker, 1994), since the rearranged quantile curve
its
/3{u),
rearrangement mechanism has a direct relation to the quan-
by sampling from the estimated
and
x
can also be considered isomorphically to the
treatment of the unit interval case, as commented further
As mentioned above,
without
(0, 1)
t—»
h->
original quantile model. Moreover,
relation to rearrangement
maps
is
produced
both the mechanism
in variational analysis
and op-
Hardy, Littlewood, and Polya, 1952, and Villani, 2003). Further
(e.g..
important references on the rearrangement method are discussed below.
The purpose
of this paper
is
to establish the empirical properties of the rearranged
quantile curves and their distribution counterparts:
F
u
under scenarios
(1)
and
(2).
^{u\x)
The paper
and y
\-^
F{y\x),
also characterizes certain analytical
and approx-
imation properties of the corresponding population curves:
u
1-^
F~^{u\x)
=
inf{y
:
F{y\x)
>
u},
and y
i-^
F{y\x)
=
/
l{x'P{u)
<
y}du.
Jo
The
first
main
result of the
the rearranged curves.
We
paper establishes the improved estimation properties of
show that the rearranged curve F~^{u\x)
is
closer to the true
conditional quantile curve Qo{u\x) than the original curve. Formally, for each x,
that for
all
p G
[l,oo]
/ \Qo{u\x) - F-\u\x)\Pdu\
<(
f \Qo{u\x) -
x'p{u)\''du
we have
where the inequahty
a subset of
U
:=
strict for
is
increasing. This property
and
is
independent of the sample
x P{u)
<—>
u
i-^
is
decreasing on
Qo{u\x)
is
strictly
and thus continues to hold
size,
whether the linear quantile estimator x P{u)
also regardless of
estimates Qo{u\x) consistently or not,
x'P{u).
whenever u
(l,C)o)
of positive Lebesgue measure, while
(0, 1)
in the population,
p €
i.e.
=
whether Qq{u\x)
x'P{u) or Qo{u\x)
^
In other words, the rearranged quantile curves have smaller estimation error
than the original curves whenever the
latter are not
monotone. This
property that does not depend on the way the quantile model
not rely on any other specifics of the current context and
is
is
is
a very important
estimated.
It
also does
therefore applicable quite
generally.
Towards describing the essence
X
to x.
Suppose that P{u)
is
of the rest of results, let us fix the value of the regressor
an estimator
for
P{u) that converges weakly to a Gaussian
process G{u), so that
Vnx'0{u) as a stochastic process indexed
For sufficient conditions,
(1991),
by u
Piu))
=4>
x'G{u),
in the metric space of
(1.3)
bounded functions
example, Gutenbrunner and Jureckova (1992), Portnoy
see, for
and Angrist, Chernozhukov, and Fernandez- Val (2006).
The second main
result of the
paper
MF{y\x) -
is
that in the monotonically correct case
F{y\x)) =^ F'{y\x)[x'G{F{y\x))],
'
(1),
(1.4)
where
as a stochastic process indexed by y in the metric space i°°{y),
of K-;
XI
£°°(0, 1).
3^ is
the support
and
^'(y|^)
=
d(3{u)
1
-T^TTT
x'/3'(u)
,,.'
^ith p'{u):=
du
u=F{y\x)
Moreover, we show that
Vn{F-\u\x) - F-\u\x))
as a stochastic process indexed
first
by
u, in i°°{0, 1);
^ x'G{u),
(1.5)
which, remarkably, coincides with the
order asymptotics (1.3) of the original curve. This result has a convenient practical
implication:
if
the population curve
is
monotone, then the empirical non-monotone
curve can be re-arranged to be monotonic without affecting
properties.
To
derive the above results
we
its (first
find the functional
order) asymptotic
Hadamard
derivatives
5
of F{y\x)
and F~^{u\x) with respect to perturbations of the underlying curve x'P{u)
continuous functions, and then
in discontinuous directions, tangentially to the set of
use the functional delta method.
Establishing the
rearranged distribution and quantile curves
main
in
Hadamard
differentiability of the
discontinuous directions
is
the second
theoretical result of the paper.
The
third
main
result
that in the monotonically incorrect case
is
^™.) -.(„.,). g'||M^.
as a stochastic process indexed by y
defined in the next section.
=
equation y
E K,
in i°°{K),
<
Here Ui{y\x)
x'P{u), assuming that K{y\x)
is
...
<
where
K
(1.)
is
an appropriate
set
UK{y\x){y\x) are solutions to the
bounded.
Similarly, for the rearranged
quantile curve,
(1.7)
as a stochastic process indexed
by u G
A'', in
i°°{K'),
where K'
is
an appropriate
set
defined in the next section.
Analogously to quantiles, most estimation methods
tions
for conditional distribution func-
do not impose monotonicity, and therefore can give
rise to
non-monotonic empirical
conditional distribution curves; see, for example. Hall, Wolff, and
Yao
(1999).
A
similar
monotone rearrangement can be applied to these distribution curves by exchanging the
roles plaj^ed
by the quantile and the probability spaces. Thus, suppose that P{y\x)
candidate estimate of a conditional distribution function, which
is
The rearranged monotone
is
quantile curve associated with P{y\x)
/oo
Q{u\x)=
curve,
not monotone in
a
y.
^0
l{P{y\x)<u]dyJo
The rearranged
is
\{P
{y\x)
>
u]dy.
J-oo
probability curve can then be obtained as the inverse of this quantile
i.e.,
F{y\x)
=
inf
ju
:
Q {u\x) >
yj
,
which
is
monotone by construction. Section 3 shows
in
more
detail that similar
estimation properties and an asymptotic distribution theory goes through for
The
improved
Q
and F.
distributional results in the paper do not rely on the sampling properties of the
method
particular estimation
used, because they are expressed in terms of the differ-
entiability of the operator with respect to the basic estimated process.
Moreover, the
results that follow are derived without imposing linearity of the functional forms.
only conditions required are that
The
a central limit theorem like (1.3) applies to the
(1)
estimator of the curve, and (2) the population curves have some smoothness properties.
The
exact nature of these population curves does not affect the validity of the results.
For example, the results hold regardless of whether the underlying model
is
an ordinary
or an instrumental quantile regression model.
There
He
exist other
methods
monotonic
to obtain
fits
based on quantile regression.
(1997), for example, proposes to impose a location-scale regression model, which
naturally satisfies monotonicity. This approach
is fruitful
for location-scale situations,
but in numerous cases data do not satisfy the location-scale model, as discussed,
example,
in
Lehmann
(1974),
Doksum
(2005) develop a computational
(1974),
method
and Koenker (2005). Koenker and Ng
for quantile regression that
crossing constraints in simultaneous fitting of quantile curves.
fruitful in
Clearly,
The
delta
many
situations, but the statistical properties of the
Koenker and Ng's proposal
is
different
imposes the non-
This approach
may be
method remain unknown.
from the rearrangement method.
distributional results obtained in the paper can also be viewed as a functional
method
for the rearrangement-related operators (1.1)
inverse (quantile) operators as a special case.
results
for
by
Gill
and Johansen (1990), Doss and
is
(1.2) that include the
In this sense, they extend the previous
Gill (1992),
(1999) on compact differentiability of the quantile operator.
here, as well as in the quantile case,
and
and Dudley and Norvaisa
The main
technical difficulty
that differentiability needs to be established in
discontinuous directions (that converge to continuous directions,
i.e.,
tangentially to
the set of continuous functions), because the empirical perturbations of the quantile
processes are typically step functions.
Both the
work
statistical
of Dette,
and mathematical
Neumeyer, and
results of this
Pilz (2006),
paper complement the important
which applies the rearrangement operators to
kernel
mean
in the regressors.
•are
mean
regressions, in order to obtain
Our
results
on Hadamard
regression functions that are monotonic
differentiability in discontinuous directions
new. They complement the local expansions in smooth directions subsumed
proofs in Dette
et.
for the case called here the
al.
monotonically correct case.
addition, our results cover monotonically incorrect cases.
The mathematical
differs quite substantially.
results
on directional
by Mossino and
in the
The
results of this
statistical
In
problem
also
paper also complement the
differentiability of Li- functionals of rearranged functions like (1.2)
Temam
(1981).
The
results for Li -functionals
results of this paper, such as (1.4)-(1.7), but the converse
discussion after Proposition 4 in Section 2 for
more
do not imply the main
shown
is
to be true.
(See
details.)
There are many potential applications of the estimation and
differentiability results of
the paper to objects other than probability or quantile curves. For example, in a com-
panion work we present applications to economic demand and production functions and
to biometric growth curves, where monotonization
is
or logical restrictions (Chernozhukov, Fernandez- Val,
We
used to impose useful theoretical
and Galichon, 2006a).
organize the rest of the paper as follows. In Section 2.1
we
describe
analytical properties of the rearranged population curves. In Section 2.2
functional differentiability results.
In Section 2.3
we present estimation
the rearranged curves and establish their limit distributions.
illustrate the
main
2.
In Section 5
basic
derive the
properties of
we extend
In Section 4.1
rearrangement procedure with an empirical application, and
we provide a Monte-Carlo example.
we
In Section 3
the previous results to monotonize estimates of distribution curves.
some
we
in Section 4.2
we conclude with a summary
of the
results.
Rearranged Quantile Curves: Analytical and Empirical Properties
In this section the treatment of the problem
troduction. In particular,
y^ := Q{U\x), where
we
U ~
somewhat more general than
in the in-
replace the linear functional form x'/3{u) by Q{u\x). Define
Uniform(W) with
y]du be the distribution function
quantile function oiYx.
is
of Y^^
U =
(0,1).
Let F{y\x) := J^ l{Q{u\x)
and F~^{u\x) := inf{y
:
F{y\x)
>
<
u} be the
Remark. We
we
consider the interval
(0, 1)
without
loss of generality.
+
are interested in a particular subinterval {a, a
Indeed, suppose
we may
For example,
b) of (0,1).
wish to focus estimation on a particular range of quantiles or to avoid estimation of
we
quantiles. For this purpose,
= Q{a +
Y^ := Q{U\x)
F~^{u\x) := inf{y
functions
Q
:
>
G {Q{u\x)
:
ii} for
U~
u G
u E {a,a
=
The
(0, 1).
U
+
(a,
a+b):
analysis of the paper applies to the
u G
for
€
< y}du, and
F{y\x) := /^ l{Q{u\x)
C/(0, 1),
Q{{u — a)/b\x)
we can use
(a,
a
+ b)
and F{y\x)
= a + bF{y\x)
b)}.
We
Basic Analytical Properties.
2.1.
objects conditionally on the event
In order to go back to the unconditional quantities,
the transformations Q{u\x)
for y
all
bU\x), where
F{y\x)
and F"^.
define
tail
by developing some basic properties
start
for
F{y\x) and F~^{u\x), the population counterparts of the rearranged distribution curve
and
We
its inverse.
need these properties to derive various empirical properties stated
in the next section.
Recall
the following definitions from Milnor (1965):
first
A
continuously differentiable function.
derivative of g at this point does not vanish,
regular point
if
is
A
called a critical point.
^"^ ({y}) contains only regular points,
which
is
not a regular value
Denote by y^ the support
We
assume throughout that
compact subset
x, or
of R''.
we might be
be a singleton
i.e.,
is
if
g~^
point u which
X
{x}.
X
of Y^,
y^;
C
yX
:= {{y,x)
y, which
is
:
y
R
({y}),
compact subset
the
not a
9' (^)
A
7^ 0.
value y
of R,
UX =14
x X.
and that x E X,
a.
of interest are not functions of
X
is
taken to
the following assumptions about Q{u\x):
a continuously differentiable function in both arguments,
Q{u\x)
(b)
For each x E X, the number of elements of {u E
is
is
if
called a regular value of g
is
Ey^^x E X}, and
some applications the curves
We make
-^
he a
M.
called a critical value.
(a)
:
A
g' (u) 7^ 0.
Vu G
^
U C R
called a regular point of g
value y E g {U)
i.e.,
:
interested in a particular value x. In this case, the set
X=
U
In
is
eU
point u
Let g
U
Q'{u\x)
\
=
0}
is
finite
and
uniformly bounded on x E X.
Assumption
U
(b) implies that, for each
x E X, Q'{u\x)
and can only switch sign a bounded number
values oiu>-^ Q{u\x) in
3^^,
and
yX*
is
not zero almost everywhere on
of times. Let
:= {{y,x)
:
y
y* be the subset
Ey*,x E X}.
of regular
9
Proposition
(b), the
1 (Basic Properties of F{y\x)
and F~^{u\x)). Under assumptions
(a)
-
functions F{y\x) and F~^{u\x) satisfy the following properties:
1.
The
2.
For any y e y*
set of critical values,
yx
\ yi, is finite,
and Jyxy. dF{y\x)
=
0.
K{y\x)
F{y\x)
= Y^
sign{Q'{ukiy\x)\x)}uk{y\x)
+
<
l{Q'{uK{y\x){y\x)\x)
0},
fc=i
where
{ufc(y|x),
=
fork
l,...,K{y\x)
<
oo} are the roots of Q{u\x)
=
y in increasing
order.
3.
For any y E y*,
=
the ordinary derivative f{y\x)
dF{y\x)/dy
exists
and takes
the
form
^^^1^^
=
IJ
which
is
continuous at each- y €
absolutely continuous
Nikodym
4-
and
For any y &
3^*.
strictly increasing in
\ y^, set f{y\x) := 0.
y E yx-
Moreover, f{y\x)
is
F{y\x)
a
is
Radon-
The quantile function F~^{u\x) partially coincides with Q{u\x); namely
provided that Q{u\x)
5.
y
derivative of F{y\x) with respect to the Lebesgue measure.
F-\u\x)
for y
\Q'{u,{y\x)\x)y
=
=
Q{u\x),
increasing at u, and the equation Q{u\x)
is
=
y has unique solution
F~^{u\x).
The quantile function F~^{u\x)
is
equivariant to location and scale transformations
ofQ{u\x).
6.
The quantile function F~^{u\x) has an ordinary continuous derivative
l/f{F-\u\x)\x),
when F~^{u\x) G
3^*.
This function
is
also a
Radon-Nikodym
derivative with respect to
the the Lebesgue measure.
7.
The map {y,x)
continuous on lAX
i—>
F{y\x)
is
continuous on
yX
and the map {u,x)
i-^
F''^{u\x)
is
10
The
some
following simple example illustrates
where the
initial
of these basic properties in a situation
population pseudo-quantile curve
is
highly non-monotone.
Consider
the following pseudo-quantile function:
Q{u)
The
left
panel of Figure
ticular, the slope of
1
=
(u + - sm{2TTu)]
5
shows that
this function
Q{u) changes sign twice
curve F~^{u), also plotted in this panel,
The
1,
results
1, 2,
4
is
at 1/3
(2.1)
.
non-monotone
is
In par-
and 2/3. The rearranged quantile
illustrated in the right panel of Figure
Here we can see that
which plots the original and rearranged distribution curves.
is
[0, 1].
continuous and monotonically increasing.
and 7 of the proposition are
the rearranged distribution function
in
continuous, does not have mass points, and co-
incide with the original curve for values of y where the original curve
is
one to one and
increasing.
Figure 2 illustrates the third and sixth results of the proposition by plotting the
sparsity function for F~^{u)
in the right panel is
and the density function
The
of F{y).
continuous at the regular values of Q{u).
function for F~^{u) in the
left
panel
is
derivative of F{y)
Similarly, the sparsity
continuous at the corresponding image values
(under F{y)).
Functional Derivatives. Next, we
2.2.
Hadamard
differentiability of F{y\x)
main
results of the
paper on
and F~^{u\x) with respect to Q{u\x), tangentially
to the space of continuous functions on
for deriving the
establish the
UX.
This differentiability property
is
important
asymptotic distributions of the rearranged estimates. In particular, the
property allows us to establish generic convergence results for rearranged curves based on
any
initial
quantile estimator, provided the initial estimator satisfies a functional central
limit theorem.
The property
also
imphes that the bootstrap
inference on the rearranged estimates, provided the bootstrap
estimates. This result follows from the functional delta
Theorem
In
>
is
valid for performing
valid for the initial
bootstrap
for the
(e.g.,
13.9 in van der Vaart, 1998).
what
UX —
method
is
R,
follows,
C{UX)
£°°{UX) denotes the
set of
bounded and measurable functions h
denotes the set of continuous functions mapping h
:
UX
:
-^ R, and
11
^^^
Q'ly)
^^--'^''''^^
F(y)
y/"T"""'^
I
f
\
/
/\
/
Figure
1.
Left:
The pseudo-quantile
quantile function F^^{u). Right':
The
function Q{u) and the rearranged
pseudo-distribution function Q~^{y)
and the rearranged distribution function F{y).
,(y,
;
-
w
Figure
tile
2.
Left:
The
density (sparsity) function of the rearranged quan-
function F~^{u). Right:
tribution function F{y).
The density function
of the rearranged dis-
12
i^{UX) denotes the
<
oo,
^
and
measurable functions h
set of
UX -^ M such that /^ /^ \h{u\x)\dudx
where du and dx denote the integration with respect to the Lebesgue measure on
A", respectively.
Proposition 2 (Hadamard Derivative
:= /q \{Q{u\x)
+
<
tht{u\x)
X), for every
\ht
-
of F{y\x) with respect to Q{u\x)). Define F{y\x, ht)
y}du. Under assumptions (a)-(h), as
(2.3)
^
0,
- Y,
I'r^r'r'li
\Q'{uk{y\x)\x)\
:=
^
in
where
^
any compact subset
G
ht
^°°
The convergence holds uniformly
- hU
\ht
The convergence
^
0,
Moreover,
:= {(y,a:)
:
j/
of F~^{u\x) with respect to Q{u\x)).
^"''"l''''">,-^"''"l"'
in
G 3^*,x 6
Under
in the
^i).M.).
as-
where
ht
E
i°°
=
(2.5)
{{u,x)
:
{F~^ {u\x) x) 6
,
{UX), and h G C{UX).
on regions that exclude the
critical values,
(2.4)
^P.(F-'M..)W.
any compact subset ofUX*
results hold uniformly
mapping u h^ Q{u\x). At the
to decreasing.
ofyX*
{UX), and h E C{UA:).
P>W:=- ^(^_.'„|^)|^)
every
-
0,
^>,Mx.O:^
yX*}, for
0,
DMr)
Proposition 3 (Hadamard Derivative
t
^^
(22)
h\oo
surnpitions (a)-(h), as
t
DM.,t)^!M^hLzim^O,(yM,
The convergence holds uniformly
the
:
critical values of
Q{u\x) possibly changes from increasing
monotonically correct case
(1),
the following result
is
worth emphasizing;
Corollary
1 (Monotonically correct case). Suppose
each {u,x) G
UX,
then
yX* =
yX
and
UX* = UX.
Propositions 2 and 3 holds uniformly over the entire
over,
Dh{u\x)
to the original
=
h, i.e., the
curve
is
Hadamard
u k^ Q{u\x) has Q'{u\x) >
yX
0,
for
Therefore, the convergence in
and
UX,
respectively.
More-
derivative of the rearranged quantile with respect
the identity operator.
13
The convergence
uniform over the entire domain
is
in the
monotonically correct case.
This result raises naturally the question of whether uniform convergence can be achieved
by some operation of smoothing
either over y (or over u).
The
in the
The answer
monotonically incorrect case - namely integrating
is
indeed yes.
Hadamard
following proposition calculates the
tionals obtained
{y',x)^
by
derivative of the following func-
integration;
/ l{y <y'}g{y\x)F{y\x)dy,
{u',x)^
Jy
/ \{u
<u']g{u\x)F~\u\x)du,
JU
with the restrictions on g specified below.
These elementary functionals are useful
building blocks for various statistics, as briefly mentioned in the next section.
Proposition
specified
4.
results are true with the limits being continuous
uniformly in
f
Jy
\{y
<
y'}g{y\x)Dh,{y\^,t)dy-^_
[
l{y
<
y'}g{y\x)Dh{y\x)dy
Jy
(y',
x)
/
l{u
€ yX, for any g G
i°°{yPt!) such that
x
i—>
g{y\x)
is
continuous for
y.
2.
<
u'}g{u\x)Dfn{u\x,t)du -^
Ju
/
l{u
<
u'}g{u\x)Dh{u\x)du
JU
uniformly in {u',x) 6 UA!, for any g E d-^iJAX) such that x
a.e.
on the
domains:
1.
a.e.
The following
(-+
g[u\x)
is
continuous for
u.
This proposition essentially
(l)-(2) follow
from the
is
a corollary of Propositions 2 and
fact that the pointwise
coupled with the uniform integrability shown in
interchange of limits and integrals.
An
3.
Indeed, the results
convergence of Propositions 2 and
Lemma
alternative
way
3 in the
3,
Appendix, permits the
of proving result (2), but not
any
other result in the paper, can be based on exploiting the convexity of the functional in
(2)
with respect to the underlying curve, following the approach of Mossino and
(1981),
and Alvino, Lions, and Trombetti (1989).
pursue this approach
in
in this paper.
Due
to this limitation,
Temam
we do not
However, details of this approach are described
Chernozhukov, Fernandez- Val, and Galichon (2006b) with an application to some
nonparametric estimation problems.
14
It is also
worth emphasizing the properties of the following smoothed functionals. For
a measurable function /
:
R
h->-
R
smoothing operator as
define the
Sf{y):= jh{y-y')f{y')dy\
where
ks{v)
=
<
l{\v\
5}/25 and 5
>
is
(2.6)
a fixed bandwidth. Accordingly, the smoothed
curves SF{y\x) and 5F~^(u|x) are given by
SF{y\x) := f ks{y - y')F{y'\x)dy\
SF-\u\x) :=
- i^)F-\u'\x)dv!
j ks{u
Since these curves are merely formed as differences of the elementary functionals in
Proposition
4,
Corollary
2.
followed by a division by
We
5,
have that SDhi{y\x,t)
the following corollary
—
>
is
immediate.
SDh{y\x) uniformly in {y,x) G
SDht{u\x,t) -^ SDh{u\x) uniformly in {u,x) e
J^^Y,
and
UX.
Note that smoothing accomplishes uniform convergence over the entire domain, which
is
a good property to have from the perspective of data analysis.
2.3.
Empirical Properties of F{y\x) and F~^{u\x).
main
We
are
now ready
to state the
results for this section.
Proposition 5 (Improvement
Suppose that
curve Qo{-\-).
Q{'\-) is
in
Estimation Property Provided by Rearrangement).
an estimator (not necessarily consistent) for some true quantile
Then, the rearranged curve F^^{u\x)
Q{u\x) in the sense
\Qo{u\x)
that,
for each x ^
- F-\u\x)\Pduj
where the inequality
is strict
the curve
is
obtained,
is
closer to the true curve than
X
<U\QQ{u\x)-Q{u\x)\Pdu]
for p G (l,oo) whenever Q{u\x)
oflA of positive Lebesgue measure, while Qo{u\x)
The above property
is
is
is
,pe[l,oo],
decreasing on a subset
increasing on
U
.
independent of the sample size and of the way the estimate of
and thus continues
to hold in the population.
This proposition establishes that the rearranged quantile curves have smaller estimation error than the original curves whenever the latter are not monotone. This
is
a very
15
important property that does not depend on the way the quantile model
It
also does not rely
The
on any other
and
specifics
is
estimated.
thus applicable quite generally.
is
following proposition investigates the asymptotic distributions of the rearranged
curves.
Proposition 6 (Empirical Properties
Suppose that
Q{-\-) is
UX, and
—
^/n{Q[u\x)
and
(b).
Then
Assume
in i°°{K), where
K
is
that, in
Q{u\x))
UX,
G
as a stochastic process indexed by {u,x)
process with continuous paths.
F{y\x) and {u,x)
i-^
F"^(u|x)).
Q{-\-) that takes its values in the space of
an estimator for
measurable functions defined on
^
of {y,x)
^°°{UX),
^ G{u\x),
where {u,x)
Q{u\x)
also that
bounded
i-^
G{u\x)
is
a Gaussian
satisfies the basic conditions (a)
any compact subset ofyX*,
^{F{y\x) -
F{y\x))
G
as a stochastic process indexed by (y, x)
yX*
;
^ DG{y\x)
and in 1°°1JAX k), with
UXk =
{(^i x)
:
{F-\u\x),x)^K],
^/n{F-\u\x)- F-^{u\x))^Dg{u\x),
as a stochastic process indexed by {u,x)
Corollary 3 (Monotonically correct
G
for each (u,x)
UX,
then
G
UXk-
case).
yX* =
yX
Suppose u
and
\-^
UX* = UX.
vergence in Proposition 5 holds uniformly over the entire
Dg{u\x)
=
G{u\x),
i.e.,
Q{u\x) has Q'{u\x)
Accordingly,
yX
the rearranged quantile curves have the
and
UX.
same
>
the con-
Moreover,
order as-
first
ymptotic distribution as the original quantile curves.
Thus,
and
in the
initial
monotonically correct case, the
quantile estimates coincide.
first
Hence,
all
order properties of the rearranged
the inference tools that apply to
original quantile estimates also apply to the rearranged quantile estimates. In particular,
if
the bootstrap
is
valid for the original estimate,
estimate, by the functional delta
of Section 4,
we
method
it
is
also valid for the rearranged
for the bootstrap.
In the empirical
exploit this useful property to construct uniform confidence
example
bands
for
the conditional quantile functions based on the rearranged quantile function estimates.
16
In addition to
tlie results
on quantile function estimates, Proposition 6 provides the
asymptotic properties of the distribution function estimates.
The preceding remark
about the vahdity of bootstrap applies also to these estimates.
In the monotonically incorrect case, the large sample properties of the rearranged
quantile estimates differ from those of the initial quantile estimates.
Proposition 6
enables us to perform inferences for rearranged curves in this case, including by the
bootstrap, but only after excluding certain nonregular neighborhoods (for the distribution estimates, the neighborhood of the critical values of the
map u
t—>
Q{u\x), and, for
the rearranged quantile estimates, the image of the latter neighborhood under F{y\x)).
However,
we consider
if
the following linear functionals of the rearranged quantile and
distribution estimates:
{y',x)^
/
l{y
<
y'}g{y\x)F{y\x)dy,
{u,x)^
Jy
< u'}g{u\x)F~\u\x)du,
l{u
/
Ju
then we no longer need to exclude the nonregular neighborhoods. The following proposition describes the empirical properties of these functionals in large samples.
Proposition 7 (Empirical Properties
Proposition
specified
l{y
<
y'}g{y\x){F{y\x)
-
as a stochastic process indexed by {y',x)
l{u
^/n
F{y\x))dy
^
on the
<
u'}g{u\x){F-'^{u\x)
G
yX,
I \{y < y']g{y\x)DG{y\x)dy,
Jy
in i°°{yX).
- F-\u\x))du
=»
l{u
<
u'}g{u\x)DG{u\x)du,
Ju
Ju
as stochastic process indexed by {u',x) G
The restrictions on
The
the conditions of
domains:
Jy
2.
Under
the following results are true with the limits being continuous
6,
V^ [
I.
of Integrated Curves).
UX,
the function g are the
linear functionals defined
in i°°{UX).
same
as in Proposition
above are useful building blocks
4-
for various statistics,
such as partial means, various moments, and Lorenz curves. For example, the conditional
Lorenz curve
is
L{u\x)
=
(
/
l{f
<
u}F-\t\x)dt'^/(^
I
F-\t
x)dt
17
which
is
statistics
a ratio of a partial
to the
Hadamard
mean.
differentiability of these
with respect to the underlying Q{u\x) immediately follows from the Hadamard
differentiability of the
rule.
mean
elementary functionals of Proposition 7 by means of the chain
Therefore, the asymptotic distribution of these statistics can be determined from
the asymptotic distribution of the linear functionals, by the functional delta method.
In particular, the validity of the bootstrap for these functionals
functional delta
method
is
preserved by the
for the bootstrap.
We next consider the empirical properties of the smoothed curves obtained by applying
the hnear smoothing operator
SF{y\x) :=
The
Jh{y
-
S
defined in (2.6) to F{y'\x) and F~^{u\x):
SF-\u\x) :=
y')F{y'\x)dy',
J
ks{u
following corollary immediately follows from Corollary 2
-
u')F-\u'\x)du'.
and the functional delta
method.
Corollary 4 (Large Sample Properties
Proposition
6,
of
Smoothed Curves). Under
the conditions of
in ^°°{yM),
MSF{y\x) -
SF{y\x))
as a stochastic process indexed by {y,x)
^/7^{SF-\u\x)
^ S[Dc{y\x)],
E yX, and
- SF-\u\x))
as a stochastic process indexed by {u,x)
E
in
^°°{UX),
^ S[bG{u\x)l
UX.
Thus, inference on the smoothed rearranged estimates can be performed without
excluding nonregular neighborhoods, which
validity of the bootstrap for the
is
convenient for practice.
Furthermore,
smoothed curves follows by the functional delta method
for the bootstrap.
3.
Theory of Rearranged Distribution Curves
The rearrangement method can
also be applied to rearrange cumulative distribution
curves monotonically by exchanging the roles of the quantile and probability spaces.
There are several situations where one might be faced with the problem of nonincreasing empirical distribution curves.
In an option pricing context, for example,
18
Ait-Sahalia and Duarte (2003) use market data to estimate a risk-neutral distribution.
Estimation error
may
cause the resulting distribution function to be non-monotonic. In
other cases the distribution curve
obtained by some inverse transformation and local
is
non-monotonicity comes as an artefact of the regularization technique. In other situa-
may
tions the particular estimation technique
and Yao, 1999).
Wolff,
We
not respect monotonicity
(see, e.g., Hall,
present an alternative solution to this problem that uses the
rearrangement method.
Here we do not present the conditional case
however, are exactly parallel to those presented in this
for conditional distributions,
Suppose we have y h^ P{y)
section.
for notational convenience. All derivations
as a candidate empirical probability distribution
curve, which does not necessarily satisfy monotonicity, with population counterpart
P{y). Define the following quantile curve
rQ
/•oo
Q[u)
l{P{y)< u}dy -
=
which
yC
is
The
In
what
tities
inverse of the quantile curve
is
Q
The
u}dy,
further
assume that the support of P{y)
it
is
can be treated analogously
first).
F{y)
which
we
follows,
+oo), so that the second term drops out (otherwise
[0,
to the
monotone.
l{P{y)>
J -oo
JQ
also
and
the rearranged probabihty curve
= M{y:Q{u)>y},
construction.
It
should be clear at this point that the quan-
are exactly symmetric to
F
and
monotone by
F
is
Q
following improved approximation property
distribution function, then for
(J
|Fo(y)
where the inequality
and y h^ P{y)
is
strictly increasing.
is
-
all
p G
is
true for F: Let Fo{y) be the true
[l,oo],
F{y)\Pdy^
strict for
in the quantile case.
'
<
\Fo{y)
-
P{y)\''dy^
,
(j^
p E
(l,cx))
whenever the integral on the right
is finite
decreasing on a subset of positive Lebesgue measure,, while Fo(u)
This property
to hold in the population.
is
independent of the sample
size,
is
and thus continues
19
In the monotonically correct case, that
is
when P' {Q{u)) >
for all
u G
[0, 1], if
the
empirical distribution curve P{y) satisfies
V^[Piy)~P{y))^G{y)
where
in i°°{y),
G
is
a Gaussian process, then
in£°°([0,l]),and, in e^{y),
V^(F(y)-F(y))^G(y).
(3.1)
Results paralleling those of the previous section also follow for the monotonically
incorrect case. In particular,
we have
- Q{u))
^/^ {q{u)
^^"^G(y,(.))
=^
E^
fc=i
in £°°([0, 1]),
and
'
'*
K(u)
\
^(^<"'-'^')-(1f(^)
in ^°°{y),
is
where {yk{u),
bounded uniformly
for
A;
=
1, ...,
K{u)]
K(u)
twmk^
are the roots of P{y)
=
u,
(3.2)
u=P[y)
assuming K{u)
in u.
4.
4.1.
{ykH)\
Empirical Example. To
Illustrative Examples
illustrate the practical applicability of the
method, we consider the estimation of expenditure curves.
We
rearrangement
use the original Engel
(1857) data, from 235 budget surveys of 19th century working-class Belgium households,
to estimate the relationship between food expenditure and annual household income (see
Koenker, 2005). Ernst Engel originally presented these data to support the hypothesis
that food expenditure constitutes a declining share of household income (Engel's Law).
In Figure
3,
we show a
scatterplot of the Engel data on food expenditure versus house-
hold income, along with quantile regression curves with the qua.ntile indices {05,0.1,
0.95}.
We
see that the quantile regression lines
become
closer
and cross
...,
at low values of
20
income. This crossing problem of the Engel curves
we
also evident in Figure 4, in
is
which
plot the quantile regression process of food expenditure as a function of the quantile
index. For low values of income, the quantile regression process
The rearrangement procedure
is
clearly
non-monotone.
non-monotonicity producing increasing quantile
fixes the
functions. Moreover, the rearranged curves coincide with their quantile regression coun-
terparts for the middle values of income where there
In Figure
5,
we
plot simultaneous
90%
no quantile-crossing problem.
is
confidence intervals for the conditional quantile
function of food expenditure for different values of income (at the sample median, and the
5%
percentile of income).
We construct
the bands using both original quantile regression
curves and rearranged quantile curves based on 500 bootstrap repetitions and a grid of
quantile indices {0.10, 0.11,
....
0.90}.
We
obtain the bands for the rearranged curves
assuming that the population quantile regression curves are monotonically
that the
first
order behavior of the rearranged curves coincides with the behavior of the
original curves.
bands
lie
correct, so
The
shows that even
figure
for the
within the quantile regression bands.
low value of income the rearranged
This observation points towards the
maintained assumption of the monotonically correct case. The lack of monotonicity of
the estimated quantile regression process in this case
is
likely to
by caused by sampling
error.
We
find
more evidence consistent with the monotonically
which we plot the simultaneous confidence bands
and rearranged curves.
We
(with bandwidth equal to
ranged curves even
in the
for the
correct case in Figure
smoothed quantile regression
construct the band by bootstrapping the smoothed curves
The bootstrap bands
.05).
are valid for the
smoothed
rear-
monotonically incorrect case. The almost perfect overlapping
between the confidence bands points towards the monotonically correct
ingly,
6, in
case.
Interest-
smoothing reduces the width of the confidence bands, but does not completely
monotonize the quantile regression curves.
4.2.
Monte
Carlo.
We use the
following
Monte Carlo experiment, matching
closely the
previous empirical apphcation, to illustrate the estimation properties of the rearranged
curves in finite samples. In particular,
scale shift model:
Y=
Z{X)'a
+
we consider two designs based on the
{Z{X)'')')e,
where
e is
location-
independent of X, with the true
21
Figure
The
3.
scatterplot and quantile regression
food expenditure data.
The
on food expenditure
household income
vs.
plot
fits
of the
Engel
shows a scatterplot of the Engel data
for
a sample of 235 19th cen-
tury working-class Belgium households. Superimposed on the plot are the
{0.05, 0.10, ...,0.95} quantile regression curves.
The range
displayed corre-
sponds to values of income lower than 1500 and values of food expenditure
lower than 800.
conditional quantile function
Qo{u\X)
Design
1
= Z{xya + {Z{Xy^)Q,{u).
includes a constant and a regressor, namely
has an additional nonlinear regressor, namely,
a
=
median(X).
We
select the
parameters
empirical example, employing the estimation
design
1
we
set
a =
(624,15,0.55) and
7
=
Z{X) =
Z{X) =
(1,.Y,
for designs 1
method
of
and
{1,X); and design 2
1{X >
2 to
a}
X), where
match the Engel
Koenker and Xiao (2002). For
(1,0.0013); and for design 2
we
set
a =
22
Income = 394 (1%
A.
quantile)
B.
Income = 452 (5% quantile)
0.0
Income = 884 (Median)
C.
Figure
D.
0.2
0.6
0.4
Income = 2533 (99% quantile)
Quantile regression processes and rearranged quantile pro-
4.
cesses for the Engel food expenditure data. Quantile regression estimates
are plotted with a thick gray line, whereas the rearranged estimates are
plotted in black.
(624.15,0.55,-0.003) and 7
Monte Carlo samples
of size
= (1,0.0017,-0.0003).
n = 235. To generate the
we draw observations from a normal
the residuals
X
e
= (V - Z(A')'a')/(Z(X)'7)
use designs
1
and
values of the dependent variable,
distribution with the
of the
in all the replications to the observations of
We
For each design, we draw 1,000
Engel data
income
same mean and variance
set;
in the
and we
fix
Engel data
as
the regressor
set.
2 to assess the estimation properties of the original
and
rear-
ranged quantile regressions under the correct and incorrect specification of the functional
23
A.
Figure
5.
Income = 452 (5%
quantile)
Simultaneous
Income = 884 (Median)
B.
90%
confidence bands for quantile regression
processes and rearranged quantile processes for the Engel food expenditure
data.
Two
bands
for quantile regression are plotted in light gray,
for
different values of the
income regressor are considered. The
rearranged quantile regression are plotted in dark gray.
form. Thus, in each replication,
Q{u\X)
we estimate the model
= Z{Xyp{u), Z{X) =
This gives the correct functional form
incorrect functional form for design
2,
for
that
design
is
1,
that
Q{u\X)
a nonlinear regressor). Accordingly, estimation error
sampling
and
whereas the bands
^
{l,X).
is,
Q{u\X)
=
Qo{u\X), and an
Qo{u\X) (due to the omission
for
design
error, while the estimation error for design 2 arises
1
arises entirely
of
due to
due to both sampling error
specification error. Regardless of the nature of the estimation error, Proposition 5
establishes that the rearranged quantile curves should be closer to the true conditional
quantiles than the original curves.
24
A.
Figure
tile
Income = 452 (5%
6.
Simultaneous
regression processes
for the
quantile)
90%
B.
Income = 884 (Median)
confidence bands for the smoothed quan-
and the smoothed rearranged quantile processes
Engel food expenditure data. The bands
for the
smoothed
original
curves are plotted in light gray, whereas the bands for the smoothed rear-
ranged curves are plotted
in
dark gray. The smoothed curves are obtained
using a bandwidth equal to 0.05.
In each replication,
monotonize
we
fit
this curve to get
a linear quantile regression curve Q{u\X)
=
X (3{u)
F~^{u\X) using the rearrangement method. Table
1
and
reports
measures of the estimation error of the original and rearranged estimated conditional
quantile curves using different
a value,
We
X =
norms
{p
=
xq, that corresponds to the
select this value
1,2, 3,4,
5%
and
oo),
with the regressor fixed at
quantile of the regressor
X
{X =
452).
motivated by the empirical example. Each entry of the table gives
a Monte Carlo average of
i/p
U':={
/
\QQ{u\xo)-Q{u\xoWdu
25
for
Q{u\xo)
indices
Both
u
=
XqP{u) and Q{u\xo)
=
F~^{u\xq).
We
evaluate the integral using a net of
of size .01.
and
in the correctly specified case
in the misspecified case,
we
find that the
rearranged curves estimate the true quantile curves Qo{u\X) more accurately than the
original curves, providing a
4%
to
15%
reduction in the estimation/approximation error,
depending on the norm.
Table
Estimation Error of Original and Rearranged Curves.
1.
Correct Specification
Design 2
Incorrect Specification
Original
Rearranged
Ratio
Original
Rearranged
Ratio
L'
6.79
6.61
0.96
7.33
7.02
0.95
L2
7.99
7.69
0.95
8.72
8.20
0.93
L3
8.93
8,51
0.95
9.85
9.12
0.92
L'
9.70
9.17
0.94
10.78
9.86
0.91
L°°
17.14
15.32
0.90
19.44
16.44
0.85
Design
1
5.
Conclusion
This paper analyzes a simple regularization procedure
for
estimation of conditional
quantile and distribution functions based on rearrangement operators.
Starting from
a possibly non-monotone empirical curve, the procedure produces a rearranged curve
that not only satisfies the natural monotonicity requirement, but also has smaller
mation error than the original curve. Asymptotic distribution theory
rearranged curves, and the usefulness of the approach
is
is
esti-
derived for the
illustrated with
an empirical
example and a simulation experiment.
^
Massachusetts Institute of Technology, Department of Economics and Operations
Research Center, University College London,
CEMMAP,
and The University of Chicago.
E-mail: vchern@mit.edu. Research support from the Castle Krob Chair, National Science
Foundation, the Sloan Foundation, and
§
CEMMAP
is
gratefully acknowledged.
Boston University, Department of Economics. E-mail: ivanf@hu.edu.
26
I
Harvard University, Department of Economics. E-mail: galichon@fas.harvard.edu.
Research support from the Conseil General des Mines and the National Science Foundation
is
gratefully acknowledged.
27
Appendix
A.l.
Proof of Proposition
Proofs
note that the distribution of Y^ has no atoms,
First,
1.
A.
i.e.,
Pr[y^
since the
=
=
y]
number
Fv[Q{U\x)
=
=
y]
Pr[[/ e
=
of roots of Q{u\x)
y
{ueU
is finite
Next, by assumptions (a)-(b) the number of
& root of Q{u\x)
-.u is
under
(a)
-
(b),
critical values of
and
=
U~
Q{u\x)
is
y}]
=
0,
Uniform(W).
hence
finite,
claim (1) follows.
Next, for any regular
l{Q{u\x)
/
<y]du=
where Uo{y\x) :=
we can write F{y\x)
y,
and {uk{y\x),
in increasing order.
first
term
0}{uk+i{y\x)
'^K{y\x){y\->-))-
for
/c
=
1,
<y]du+
...,K{y\x)
<
l{Q{u\x)
oo} are the roots oiQ{u\x)
<
y}
=
1
=
l{Q'{ui:+i{y\x)\x)
Uk{y\x)); while the last term simplifies to 1{Q' {uK{y\x){y\x)\x)
An
y]du,
y
Hence
on the interval [uk^i{y\x),Uk{y\x)].
in the previous expression simplifies to Y^i^Iq
—
<
I
Note that the sign of Q'{u\x) alternates over consecutive Uk{y\x),
determining whether l{Q(y|a:)
the
\{Q{u\x)
/
J^
as
<
0}(1
>
-
additional simplification yields the expression given in claim (2) of the
proposition.
The proof of claim
(3) follows
that at any regular value y the
by taking the derivative
number
of solutions
of expression in claim (2), noting
K{y\x) and sign{Q' {uk{y\x)\x)) are
locally constant; moreover,
>(
x_
sign{Q'{uk{y\x)\x))
'''^^^^^~
I
\Q'iu,{y\x)\x)\
Combining these
To show
derivative,
Theorem
facts
we
get the expression for the derivative given in claim (3),
the absolute continuity of F{y\x) with /(y|x) being the
Radon-Nykodym
=
J^^dF{y\x), cf
it
suffices to
show that
for
31.8 in Billingsley (1995).
centered on the critical points
yl}f{y\x)dy
= /^^
y^^
each
y'
G
3^x,
Jl^f{y\^)dy
Let V^^ be the union of closed balls of radius
\y*, and
define
y^
l{y e yl}dF{y\x). Since the set of
=
Then, J^
points 3^^, \ y*
l{y G
yx\V{^.
critical
t
is
finite
28
and has mass zero under F{y\x), J^^l{y G yl}dF{y\x)
Therefore,
Claim
l{y € yi}f{y\x)dy
/_^;^
we have that F{y\x) =
= F [F~'^ [u\x)\x) =
the equation u
Claim
We
(5).
function of
a
-t-
have Y^
(3Q{U\x)
=
=
!Q^s\x)<y^^
Q~'^
s
—
Q{s\x)
>
^s<Q-Hy\x)^^
=
is
{F~^ [u\x)\x) yields F~^{u\x)
(3Y^, for
>
/3
therefore inf{y
0, is
t
^
0.
increasing and
Q~\y\^)- Inverting
=
Q{u\x).
Q{U\x) has quantile function F~^{u\x).
= a+
as
J^;^ dF{y\x).
by noting that at the regions where
(4) follows
one-to-one,
=
fiy\x)dy
] r_'^
J^^dF{y\x)
t
:
Pr(a
The
+
quantile
PV^
<
y)
>
= a + (3F~^{u\x).
u}
Claim
(6) is
Claim
(7).
immediate from claim
The proof
(3).
of continuity of F{y\x)
subsumed
is
Proposition 3 (see below). Therefore, for any sequence
F{y\x) uniformly in
F{y\x)
=
y,
and F{y\x)
u has a unique root y
is
=
continuous.
Xt
A. 2.
of the proof of
—^ x we have that F{y\xt)
F~^{u\x), the root of F{y\xt)
e.g.,
=
Ut,
x.
i.e.,
—
Since
yt
=
van der Vaart and Wellner
D
Proof of Propositions
Lemma
1
Let Ut —^ u and Xt -^
F'^{ut\xt), converges to y by a standard argument, see,
(1997).
in the step
1,
2-7. In the proofs that follow
we
repeatedly use
will
which establishes the equivalence of continuous convergence and uniform
convergence:
Lemma 1. Let D and D' be complete separable metric spaces, with D compact. Suppose
f D ^> D' is continuous. Then a sequence of functions fn'-D^D' converges to f
uniformly on D if and only if for any convergent sequence x^ ^> x in D we have that
:
fni^n) -^ f{x).
Proof of
Lemma
See, for example, Resnick (1987), page
1:
Proof of Proposition
for
u e B^{uk{y\x)) and
l{Qiu\x)
for all k
e
I, ...,
+
2.
We
for small
tht{u\x)
have that
enough
<y}<
+
any
tht{u\x)
^
>
0,
D
there exists
e
>
>
l{Q{u\x)
K{y\x); whereas for ah u
l{Q{u\x)
t
for
2.
+ t{h{uk{y\x)\x) -
[JkB^{ui,{y\x)), as
<y} =
l{Q{u\x)
<
i
^
y}.
0,
5)
<
y},
such that
29
Therefore,
f^ l{Q{u\x)
^ ""y
+
tht{u\x)
l{Q{u\x)
f
~-
< y]du -
+ t{h{u,{y\x)\x) -
<
5)
-
y}
<
l{Q{u\x)
y}
^^
i
JB,{uk(y\x))
k=l
< y}du
J^ l{Q{u\x)
which by the change of variables
y'
—
Q{u\x)
equal to
is
K{y\x)
^
dy\
\Y.
fc=l
where Jk
the image of B^{uk{y\x)) under u
is
possible because for
Fixing
>
e
for
0,
-
t{h{uk{y\x)\x)
small enough, Q{-\x)
e
i
—
>
we have that
0,
Y^
ES'^^
lQ'"u[l£')t^^
is
Jk
fl
[y,y
+
5
—
5)]
=
^
[y,y
—
Uk{y\x).
^
^
Lemma
result holds uniformly in {y,x)
Take a sequence of
1.
uniformly continuous on K, and
tinuous on K.
To
{yt,Xt) in
pact sets [Ki,
K{y\x)
(1) follows
=
...,
^
Since 5
>
can be
K
that converges to (y,x) G
...,
some
Km)
of Uk{y\x) (implied
(1)
the function {y,x)
the function (y,x)
K
i-^-
K{y\x)
integer kj
>
K, then
the
i-^
\o'(u''(
\x)\x)\
uniformly con-
is
excludes a neighborhood of critical points
such that the function K{y\x)
by noting that the
the sets {Ki,
E K, a compact subset of yP^*, we
0, for all (y,
x)
is
number
of
com-
constant over each of these
sets,
therefore can be expressed as the union of a finite
Km)
kj for
(2)
see (2), note that
{y\yx,x € ^), and
i.e.,
t{h{uk{y\x)\x)
^(1) bounds (A.l) from below.
preceding argument applies to this sequence, since
is
—
than
^lO greater
\Q'{uk{y\x)\x)\
+
is
arbitrarily small, the result follows.
To show that the
use
of variables
one-to-one between B^{uf.{y\x)) and J^.
-h{uk{y\x)\x)
^
made
The change
Q{-\x).
i-^
5)1 and \Q'{Q-\y'\x)\x)\ -^ \Q' {uk{y\x)\x)\ as Q-\y'\x)
Therefore, the right hand term in (A.l)
Similarly
is
^{y'\x)\x
\Q' [Q
J Jkn\y,v-t{Huk(y\x)\x)^6)]
i
E Kj and
j
limit expression for the derivative
by the assumed continuity of h{u\x)
in
G
is
{1,
...,
M}.
Likewise,
continuous on each of
both arguments, continuity
by the Implicit Function Theorem), and the assumed continuity of
Q'{u\x) in both arguments.
D
30
Proof of Proposition
quantile operator
For a fixed x the result follows by Proposition
and by an application of the Hadamard
of the proof below,
1
3.
shown by Doss and
Gill (1992).
2,
by step
differentiability of the
Step 2 establishes uniformity over
xeX.
Step
1.
Let /C be a compact subset of 3^^*. Let
to a point, say (y, x). Then, for every such sequence,
\yt
-
-^
y|
be a sequence
(y^, Xt)
Ct
:=
^||/!-t||oo
in
K, convergent
+ ||<?(-kt) — <5('k)||oo +
and
0,
\F{yt\xu ht)
-
<
F{y\x)\
f
[l{Q(n|x,)
+ tht{u\x) < yj -
l{Q{u\x)
<
y}]du
Jo
<
f l{\Q{u\x)-y\<et}du -^0,
(A.2)
Jo
where the
last step follows
By
function of Q{U\x).
continuous in {y,x).
is
which
in turn implies
from the absolute continuity of y
setting ht
Lemma
1
=
2.
We
F{y\x), the distribution
the above argument also verifies that F{y\x)
implies uniform convergence of F{y\x,ht) to F{y\x),
by a standard argument^ the uniform convergence
F~^{u\x,ht) -^ F~^{u\x), uniformly over
Step
i—>
have that uniformly over
K* where K*
is
,
of quantiles
any compact subset oiUX*.
A'*,
F[F-\u\xM\:cM)-F[F-M-M)\x) ^
^^(^-:(,|,^
,^)|,)
^
^^,^^
(A.3)
t
= Dh{F-\u\x)\x) +
using Step
over
1,
K* by
,
Proposition
2,
o{\),
and the continuity properties of Dh{y\x). Further, uniformly
Taylor expansion and Proposition
F{F-\u\xM)\x) - F{F-\u\x)\x) ^
1,
as
i
^(j.-i(-^|^)|^)
—
>•
0,
-^'^H^>
M - i^-HH^) ^
^^^^^
(A.4)
and
(as
-will
be shown below)
F{F-\u\x,ht)\x,ht)
as
i
-*
(A.3).
0.
The
^See, e.g.,
Observe that the
left
hand
- F{F-\u\x)\x)
1
in
^^_^^
side of (A. 5) equals that of (A.4) plus that of
result then follows.
Lemma
_^^^^
Chernozhukov and Fernandez- Val (2005).
31
only remains to show that equation (A. 5) holds uniformly in K*. Note that for any
It
right-continuous cdf F,
where F{
—
)
we have
denotes the
strictly increasing cdf
that u
of F,
left limit
F{xo—) =
i.e.,
F, we have that F{F~^{u))
F{F-\u\x,ht)\x,ht)
0<
< F{F~^{u)) <
=
u
+
— F(F~^(n) — ),
F{F~^{u))
lim3;f3;o
F{x). For any continuous,
Therefore, write
u.
- F{F-\u\x)\x)
t
u
<
+
F{F~^{u\x,
— F{F~^{u\x, ht) -
ht)\x, ht)
—
|x, ht)
u
t
F{F-\u\x,
<
- F{F-\u\x, ht) -
ht)\x, ht)
\x, ht)
t
(^)
[F{F-\u\x,ht)\x,ht)
- F{F-\u\x,ht)\x)]
t
[F{F-\u\x,
ht)
-
- F{F-\u\x, ht) -
\x, ht)
\x)]
t
'=^
—
where
DH{F-\u\x,ht)\x) - DhiF-\u\x,
we use that F{F~^{u\x,
as
i
is
continuous and strictly increasing in
V
The
0,
following
Lemma
then
if
in (1)
J Gn
Proof of
Lemma
sition 2
—^
and
y,
lemma, due to Pratt (1960),
< Gn and
2. Let \fn\
JG
finite,
Lemma
2.
then
ht)\x)
in (2)
will
suppose that fn
J
fn —*
ht)
=
-
\x)
+
o(l)
F{F~^{u\x,
=
ht)
o(l),
—
we use Proposition
\x) since
2.
F{y\x)
D
be very useful to prove Proposition
^
f
o,nd
Gn
—>
G
4.
almost everywhere,
Jf
D
See Pratt (1960).
3 (Boundedness and Integrabihty Properties). Under the hypotheses of Propo-
and
3,
we have
that for all {y,x)
E
yX
\DhMx,t)\<\\ht\U,
(A.6)
and
Jo
f
where for any Xt^>-xEPi,ast—^Q,
A{y\xt,t) -^ 2\\h\\^f{y\x) for a.ey
ey
and / A{y\xt,t)dy -^ /
Jy
Jy
2\\h\\o^f{y\x)dy.
32
.
Lemma
Proof of
To show
3.
(A. 6) note that
sup
<
|5ft,(y|x,t)|
immediately follows from the equivariance property noted
The
inequality (A. 7)
for a.e y
||/).t||oo
E
and
y
=
Xf
^> y
any
Proposition 2
the case
yt
^y,
when
>
0;
Claim
in
x G X,
>
(5) of
lS.{y\xt,t)
Proposition
-^
>
f{y\x)
0;
and
1.
2||/i||oo/(y|a;)
2 respectively with functions h[{u\x)
=
trivially otherwise).
-^
2||/i||oo/(y|a;) for a.e
and
trivially otherwise)
/S.{yt\x, t)
f{y\x)
—
when
—\\ht\\oo (for the case
Similarly, that for
(for
any
for
by applying Proposition
follows
h'f{u,x)
That
is trivial.
(A.8)
||/ii||oo
x ^
follows
A!
by
.
Further, by Fubini's Theorem,
=
Note that
ft{u)
.<_.2||/it||oo.
enough
i,
2||^||oo.
By Lemma
A. 3.
and
2||/i4||(x)
converges to
any
y[
—
>
y'
<
and
for
y'}g{y\x)Dh{y\x).
x^
is
t
—
>
continuous
-^ m{y\x,y').
We
3.
^
y'}g{y\x)Dht{y\x,t) and
to demostrate that
/ m{y\x,y')dy,
We
we need
(A.IO)
have that \mt{y\xt,yt)\
a.e.
and the
Moreover, by Proposition
to
show that
m{y\xt,yt)dy -^
dominated by
—
Jy
in {x,y').
for
integral of
2, for
y[
bounded,
for
which converges
almost every y we have
Lemma
any
is
—
»
y'
2.
and
Xt
—
>
x
(A.ll)
m{y\x,y')dy.
Jy
Jy
have that m{y\xt,y[)
||/ii||ooo!w
J^
2||/i||tx)-
we need
(1),
conclude that (A.IO) holds by
In order to check continuity,
We
trivially, 2
<
for small
2||/!.t||oo
x
number by Lemma
Tnt{y\xt.,y[)
=
ft{u)
Then,
0.
To show claim
some constant C, by CA(y|xt, t) which converges
to a finite
-^
u,
Define mt{y\x,y') := l{y
4.
/ mt{y\xt,y't)dy
limit
as
2||/i||oo
Jy
and that the
almost every
the right hand side of (A. 9) converges to
Proof of Proposition
m{y\x,y') := l{y
for
2
Moreover,
/t(")
:
^
m(y|x,y') for almost every
\\g\\oo\\h\\oof{y\xt),
which converges to
y.
Moreover, m{y\xt,yt)
||5||oo||^||oo/(y|3;) for
is
almost every
33
y,
and, moreover, /^ \\g\\o^\\h\\^f{y\x)dy converges to
Lemma
holds by
2.
To show claim
l{u
<
(2), define m^t{'<^\x,u')
=
<
l{u
mt{u\xt,u[)du —^
/
and that the
limit
^(ulxj)!!/?,^!!^),
which converges to
We
We
g'('u|a:)||/i||oo
theorem. Moreover, by Proposition
u.
—
2,
we have that
m{u\xt,u[)du ^^
m{u\xt,u[)
integral of
The
of R,
2.
we need
to
show that
as
t
-^
for
any
u[
—
>
u'
and
lemma
0.
an approximation
is
to
[1,
is
a function
mapping
a non- decreasing function
Qq{u). Let FQ[y)
~
(7(0,1).
=
Moreover,
U
We
conclude that (A. 13)
U
:=
5:
(0, 1) to
mupping
= Fq\u) =
\Qoiu)~Q*{u)fdu
r
U
K,
to
K
inf{y G
r
<
1
\Q,{u)
uu
a bounded subset
Think of Q{u)
.
the distribution
M
:
Fqiy)
>
u}.
i/p
- Q{u)f du
provided (1) p G (l,oo), (2) Q{u)
that has positive Lebesgue measure,
increasing on U.
converges to the
|5(ti|xt)|||/!.||oo
oo],
this inequality is strict
a subset of
almost every value of
for
J^l{Q{u) < y}du denote
Let Q*{u)
i/p
uu
which converges
be used to prove Proposition
Q{u)
Q{U) when U
Then, for any p G
|g(u|3;j)|||/i||oo,
Moreover, for smaU enough
u.
Furthermore, the integral of
will
that
that Qo{u)
function of
(A. 13)
^,
D
2.
Assume
4.
-
by dominated convergence theorem.
|g'(u|a;)|||/i||oo
Lemma
m{u\x,u')du.
/
m{u\x,u') for almost every
>
dominated by
following
Lemma
as
is
|5(u|a;)|||/i||oo
holds by
m{u\x,u') for almost
>
Ju
—
have that m{u\xt,u[)
u to
—
X
>
/
t,
bounded by
by the dominated convergence
mt{u\xt.,u[)
Lemma
conclude that (A. 12) holds by
is
Furthermore, the integral of
u.
5^(u|a;)||/i||oo
Ju
We
x
(A. 12)
have that mt{u\xt,u[)
for a.e.
In order to check the continuity of the limit,
Xf
Xt —^
and
m{u\x,u')du,
/
continuous in {u',x).
is
u'
=
Ju
converges to the integral of
5(u|a:t)||/it||oo
and
u'}g{u\x)Dht{u\x) and m{u\x,u')
u'}g{u\x)Dh{u\x). Here we need to show that for any u[ -^
Ju
every
Conclude that (A.ll)
||g||oo||/i||cx:>-
and
is
decreasing on
(3) the true function
Qo{u)
is
34
Proof of
Lemma
A
4.
lemma
direct proof of this
Chernozhukov, Fernandez- Val, Galichon (2006a).
Q
to Lorentz (1953): Let
G
and
1
of
helpful to give a quick indirect
It is
lemma
proof of the weak inequality contained in the
given in Proposition
is
using the following inequality due
U
be two functions mapping
K, a bounded subset
to
Let Q* and G* denote their corresponding increasing rearrangements. Then, we
of R.
have
L{Q*{u),G*{u))du<
/
any submodular discrepancy function L
for
G*{u)
=
=
\w
—
inequality, please refer to
For p
4.
=
submodular
v\^ is
i—
>•
E+
In our case,
p e
for
is its
G{u)
=
Qq{u)
=
own rearrangement.
For the proof of the strict
[l,oo).
Chernozhukov, Fernandez- Val, Galichon (2006a), Proposition
oo, the inequalities follows
Proof of Proposition
A. 4.
R^
:
Qoiu) almost everywhere. Thus, the true function
Moreover, L{v,w)
1.
L{Q{u),G{u))du,
/
by taking
limit as
This proposition
5.
is
—
p
>
oo.
D
an immediate consequence of Lemma
D
A. 5.
Proof of Proposition
method
less
(e.g.
This Proposition simply follows by the functional delta
6.
van der Vaart, 1998). Instead of restating what this method
it
takes
space to simply recall the proof in the current context.
To show the
The sequence
hn' -^
first
of
part, consider the
maps
map
satisfies gn'iy,x\hr,')
h in (.°°{UX*), where h
is
continuous.
gn{y,x\h)
—
>
It
^{Q{u\x) -
process indexed by (y,x), since
Conclude similarly
for the
Proof of Proposition
to the proof of Proposition
6.
second part.
—
y/n{F{y\x,n'^^'^h)
Dh{y\x) in £'^{K)
follows by the
ping Theorem that, in £°°{K), gn{y, x\y/n{Q{u\x)
A. 6.
is,
- Q{u\x)))
for
-
F{y\x)).
every subsequence
Extended Continuous Map=»
Ddylx)
as a stochastic
Qiu\x)) => G{u\x) in £°°{K).
D
This follows by the functional delta method, similarly
7.
D
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