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y\co]-
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
IMPROVING ESTIMATES OF MONOTONE
FUNCTIONS BY REARRANGEMENT
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
Social Science Research Network Paper Collection at
http;//ssrn.com/abstract=9831 52
IMPROVING ESTIMATES OF MONOTONE FUNCTIONS BY
REARRANGEMENT
VICTOR CHERNOZHUKOVt
Abstract. Suppose
increasing,
and an
estimates /.
that a target function /o
original estimate
weakly increasing.
Many common
We show
ALFRED GALICHON^
IVAN FERNANDEZ- VAL§
:
— K
M'^
>
is
monotonic, namely, weakly
/ of the target function
available,
is
which
is
not
estimation methods used in statistics produce such
that these estimates can always be improved with no
harm using
rearrangement techniques: The rearrangement methods, univariate and multivariate,
transform the original estimate to a monotonic estimate /*, and the resulting estimate
closer to the true curve /o in
is
illustrate the results
common
We
metrics than the original estimate /.
with a computational example and an empirical example dealing
with age-height growth charts.
Key words. Monotone
function, improved approximation, multivariate rearrange-
ment, univariate rearrangement, growth chart, quantile regression, mean regression,
series, locally linear, kernel
AMS
Date:
methods
Subject Classification. Primary 62G08; Secondary 46F10, 62F35, 62P10
First version
is
of
December, 2006. This version
Andrew Chesher, Moshe Cohen, Emily
Gallagher,
is
of April 26, 2007.
Raymond
Guiteras,
We
Xuming
would
like to
thank
He, Roger Koenker,
Charles Manski, Costas Meghir, Ilya Molchanov, Steve Portnoy, Alp Simsek, and seminar participants
at
Columbia, Cornell, BU, Georgetown, MIT, MIT-Harvard, Northwestern,
UBC, UCL, and UIUC
for
very useful comments that helped improve the paper.
'*
Massachusetts Institute of Technology, Department of Economics
University College London,
CEMMAP,
search support from the Castle
CEMMAP
§
I
is
Krob
&
and The University of Chicago.
Operations Research Center,
El-mail:
vchern@mit.edu. Re-
Chair, National Science Foundation, the Sloan Foundation, and
gratefully acknowledged.
Boston University, Department
of
Economics. E-mail: ivanf@bu.edu.
Harvard University, Department of Economics.
E-mail:
galichon@fas.harvard.edu.
support from the Conseil General des Mines and the National Science Foundation
acknowledged.
1
is
Research
gratefully
1.
A common
problem
Introduction
in statistics is to
approximate an unknown monotonic function
on the basis of available samples. For example, biometric age-height charts should be
monotonic
in age;
econometric
demand
functions should be monotonic in price; and
quantile functions should be monotonic in the probability index.
possibly non-monotonic, estimate
is
available.
Suppose an
original,
Then, the rearrangement operation from
variational analysis (Hardy, Littlewood, and Polya 1952, Lorentz 1953, Villani 2003)
can be used to monotonize the original estimate. The rearrangement has been shown
to be useful in producing monotonized estimates of conditional
Neumeyer, and
Pilz 2006, Dette
and
Pilz 2006)
mean
functions (Dette,
and various conditional quantile and
probability functions (Chernozhukov, Fernandez- Val, and Galichon (2006a, 2006b)). In
this paper,
it
is
shown that the rearrangement
of the original estimate is useful not
The
only for producing monotonicity, but also has the following important property:
rearrangement always improves over the original estimate, whenever the
latter
is
not
monotonic. Namely, the rearranged curves are always closer (often considerably closer)
to the target curve being estimated. Furthermore, this
i.e.
it
improvement property
is
generic,
does not depend on the underlying specifics of the original estimate and applies
to both univariate
The paper
is
and multivariate
cases.
organized as follows.
issue in regression problems,
In Section 2.1,
we motivate the monotonicity
and discuss common estimates/ approximations of regression
functions that are not naturally monotonic. In Section 2.2,
we analyze the improvements
in estimation/ approximation properties that the rearranged estimates deliver. In Section
2.3,
we
discuss the computation of the rearrangement, using sorting and simulation. In
Section 2.4,
we extend the
analysis of Section 2.2 to multivariate functions. In Section
we provide proofs of the main
biometric age-height charts.
results. In Section 4,
we
present an empirical application to
We show how the rearrangement
the original estimates of the conditional
mean
monotonizes and improves
function in this example, and quantify
the improvement in a simulation example resembling the empirical one.
section,
we
In the
same
also analyze estimation of conditional quantile processes for height given age
that need to be monotonic in both age and the quantile index.
rearrangement to doubly monotonize the estimates both
We
3,
We
in age
show that the rearrangement monotonizes and improves the
apply a multivariate
and the quantile index.
original estimates,
and
quantify the improvement in a simulation example mimicking the empirical example. In
Section 5
we
2.
2.1.
summary and
eas of analysis
Estimates of Monotonic Functions.
A
unknown function
/o
is
to approximate an
available information.
unknown
a conclusion.
Improving Approximations op Monotonic Functions
Common
some
a
offer
In statistics, the
basic problem in
:
— R
M''
>
common problem
regression function, such as the conditional
ing an available sample. In numerical analysis, the
mean
is
many
ar-
on the basis of
to approximate an
or a conditional quantile, us-
common problem
is
to
approximate
an intractable target function by a more tractable function on the basis of the target
function's values at a collection of points.
Suppose we know that the target function
Suppose further that an
tonic.
Many common
original estimate
/*,
in
and
common
available,
which
is
not necessarily mono-
harm? The answer provided by
method transforms the
this estimate
is
monotonic, namely weakly increasing.
is
estimation methods do indeed produce such estimates.
estimates always be improved with no
the rearrangement
/
/o
is
original estimate to a
this
Can
paper
these
is
yes:
monotonic estimate
curve /o than the original estimate /
in fact closer to the true
metrics. Furthermore, the rearrangement
is
computationally tractable, and
thus preserves the computational appeal of the original estimates.
Estimation methods, specifically the ones used
into global
methods and
local
in regression analysis,
An example
methods.
of a global
can be grouped
method
is
the series
estimator of /o taking the form
where
Pfc„(x)
is
a fc„-vector of suitable transformations of the variable x, such as B-
splines, polynomials,
and trigonometric functions. Section 4
the context of an empirical example.
The estimate
b is
lists specific
examples
in
obtained by solving the regression
problem
n
6
= argmin5]p(y,-Pfc„(X0'6),
1
where {Yi,Xi),i
=
1, ...,n
=1
denotes the data. In particular, using the square loss
produces estimates of the conditional mean of
Yi
given Xi (Gallant 1981,
p('u)
=
u^
Andrews
1991, Stone 1994,
p{u)
=
{u
—
l{u
<
Newey
1997), while using the asymmetric absolute deviation loss
0))u produces estimates of the conditional u-quantile of Yi given Xi
(Koenker and Bassett 1978, Portnoy 1997, He and Shao 2000). Likewise,
in
numerical
analysis "data" often consist of values Yi of a target function evaluated at a collection
of
X
mesh
H->
=
points {Xi,i
f{x)
=
The
l,,n} and the mesh points themselves.
Pk^[x)'h are widely used in data analysis due to their
series estimates
good approximation
properties and computational tractability However, these estimates need not be naturally
monotone, unless
explicit constraints are
added into the optimization program
example, Matzkin (1994), Silvapulle and Sen (2005), and Koenker and
Examples
of local
methods include kernel and
Ng
(for
(2005)).
locally polynomial estimators.
A
kernel
estimator takes the form
n
=
f{x)
where the
loss function
arg min
J^
p plays the same
high-order, kernel function, and
Wand
X
I—>
Wip{Yi -b),
/i
>
is
Wi
=
/
K
role as above,
Y —
\
'
i
j
K{u)
is
a standard, possibly
a vector of bandwidths
and Jones (1995) and Ramsay and Silverman (2005)).
,
(see,
The
for
resulting estimate
Dette, Neumeyer, and Pilz (2006)
f[x) needs not be naturally monotone.
that the rearrangement transforms the kernel estimate into a monotonic one.
show here that the rearranged estimate
whenever the
local
latter
is
not monotonic.
method (Chaudhuri
1991,
necessarily improves
The
example.
upon the
We further
original estimate,
locally polynomial regression
Fan and Gijbels 1996).
show
is
a related
In particular, the locally linear
estimator takes the form
(/(x), d{x))
=
argmin
6eR,deR
The
are
V] Wip{Yi - 6 ~[
resulting estimate x ^—> f{x)
may
also
d{Xi
- x))^
Wi
=
K —^—
n
[
\
be non-monotonic, unless explicit constrains
added to the optimization problem. Section 4
illustrates the
non-monotonicity of
the locally linear estimate in an empirical example.
In
summary, there are many
tistics
attractive estimation
and approximation methods
in sta-
that do not necessarily produce monotonic estimates. These estimates do have
other attractive features though, such as good approximation properties and computational tractability.
Below we show that the rearrangement operation applied to these
estimates produces (monotonic) estimates that improve the approximation properties of
the original estimates by bringing
arrangement
is
them
closer to the target curve. Furthermore, the re-
computationally tractable, and thus preserves the computational appeal
of the original estimates.
The Rearrangement and
2.2.
Case.
In
what
follows, let A'
Approximation Property: The Univariate
its
convenient to take this interval to be
X
mapping
K, a bounded subset
to
distribution function of
f{X) when
r (x)
Without
be a compact interval.
=
A:"
of E.
X
loss of generality,
is
Let f{x) be a measurable function
[0,1].
=
Let F/(y)
J.^
< y}du
^{fiu)
on
follows the uniform distribution
:= Qf{x) := inf {y E
it
R
:
>
Fj{y)
denote the
[0, 1].
Let
x]
be the quantile function of Ff{y). Thus,
Hf{u) < y}du
f*{x) :=inf <^y e
>
X
X
This function /*
called the increasing rearrangement of the function /.
is
Thus, the rearrangement operator simply transforms a function / to
tion /*.
That
X ~ [7(0,
is,
X
It is
1).
i—>
f*{x)
is
the quantile function of the
also convenient to think of the
random
its
quantile func-
variable
f{X) when
rearrangement as a sorting operation:
given values of the function /(x) evaluated at x in a fine enough net of equidistant
points,
way
is
we simply
point of this paper
Proposition
is
1.
Let
a bounded subset of M.
For any p G
is
the following:
fo'-X—^Kbea
measurable function, an
1.
function created in this
the rearrangement of /.
The main
K
The
sort the values in an increasing order.
initial
[l,oo], the
This
weakly increasing m.easurable function in x, where
is
the target function.
Let f
:
X
-^
K
another
be
estimate of the target function /q.
rearrangement of f denoted f*
,
,
weakly reduces the estimation
error:
i/p
\Ip
UX
2.
nx)-fo{x)
Suppose that there
that for all x
dx
<
fix)
-
foix)
dx
(2.1)
X
exist regions
E Xq and
x'
Xq and Xq, each of measure greater than 5
6 Xq we have
that
(i) x'
>
x, (ii) /(x)
>
f{x')
+
t,
>
0,
such
and
(Hi)
>
fo{x')
+
fo{x)
p E
strict for
(
1
,
for some
e,
oo)
>
e
Nam.ely, for any
.
where
rjp
=
first
inf{|t'
—
+
t'|^
part of
\v'
—
t\^
all v, v',
t, t'
oo]
,
\v
—
\[
Jx
—
t\^
in the set
-
fix)
\v'
K
—
dx —
foix)
t'\P}
and
such that
>
rjp
>
v'
5r]p
(2.2)
for p E
v
+
e
and
with
(1, oo),
t'
>t + e.
proposition states the weak inequahty (2.1), and the second part
tfie
the original estimate f{x)
the target function
—
is
i/p
<
dx
states the strict inequahty (2.2).
X
increasing on
/o(x)
if
For example, the inequahty
is
strict for
p G (l,oo)
if
decreasing on a subset of PC having positive measure, while
is
/o(a;) is
throughout). Of course,
ity,
[1
.
the infimum taken over
The
p G
the gain in the quality of approximation
i/p
^
p
nx)-fo{x)
UX
Then
0.
is
is
strictly increasing
constant, then the inequality (2.1) becomes an equal-
as the distribution of the rearranged function /*
the original function /, that
we mean
(by increasing,
Ft,
=
is
the same as the distribution of
Ft.
This proposition establishes that the rearranged estimate /* has a smaller estimation
norm than
error in the L^
This
size
is
a very useful and generally applicable property that
and of the way the
An
the original 'estimate whenever the latter
original estimate
indirect proof of the
/
weak inequality
is
X
K, a bounded subset
to
not monotone.
independent of the sample
obtained.
is
(2.1) is a
simple but important consequence
of the following classical inequality due to Lorentz (1953): Let q
mapping
is
and g be two functions
Let q* and g* denote their corresponding
of M.
increasing rearrangements. Then,
L[q*{x),g*{x),x)dx<
for
any submodular discrepancy function
f*{x), g{x)
=
fo{x),
and g*{x)
almost everywhere, that
L{v,w,x)
=
\w
—
v\^
is
proposition above. For p
In Section 3
of the
is,
=
f^ix).
(2.1).
:
M^
for
p G
is
Set q{x)
in
=
f{x), q*{x)
our case foix)
own rearrangement.
its
[1,00).
00, the first part follows
The
R.
i-^
Now, note that
the target function
submodular
=
L
we provide a proof of the strong
weak inequality
L{q{x),g{x),x)dx,
/
Jx
X
This proves the
by taking the
first
limit as
p
=
—
fo{x)
Moreover,
part of the
^ 00.
inequality (2.2) as well as the direct proof
direct proof illustrates
how
reductions of the estimation
Moreover, the
error arise from even a partial sorting of the values of the estimate /.
direct proof characterizes the conditions for the strict reduction of the estimation error.
The
following immediate implication of the above finite-sample result
The rearranged estimate
emphasizing:
original estimates /. For
p G
[l,oo],
sequence of constants a„, then
A„
|/o(s)
[J^,
be used
=
-
[/^ \fo{x)
-
equidistant points in
[0, 1].
[0, 1]
or (2) a sample of
<
computed
=
and the second
deterministic,
is
stochastic.
Thus,
The number
of evaluations
random
sity function of the
B
of the following
u e
=
P^
zero over a neighborhood of f*{x).
of 0p(l/\/5), as
As shown
B
number
a given
for
{x G A"
f'{x)
:
Thus, the density
=
if
0}
f{x)
is
is
F'^{t) is
=
on /
is
K
is
a.
bounded;
x'
equivalent to the
B.
Suppose that the den-
1), is
bounded away from
in particular,
E
(2,3)
ijhfr
bounded away from zero
>
x.
we
Its
if
G
f'{x)
-^
:
is
bounded away from
there
is
infinity.
an x such that f'{x)
=
Approximation Property: The Multivariate
consider multivariate functions /
bounded subset
of M.
The notion
:
A''^
-^
of monotonicity
K, where
we seek
to
A""^
=
impose
We say that the function / is weakly increasing in x if f{x') > f{x)
The notation x' = {x\, ...,x'^) > x = {xi, ...,Xd) means that one vector
the following:
whenever
of draws B, the
t}.
Cfise. In this section,
and
is
continuously differentiable and the number of elements
The Rearrangement and
[0, 1]*^
U{0,
size
method
^f oo, where the rate follows from the results of Knight (2002).
Interestingly, the density has infinite poles at {i
2.4.
X~
when
is
first
Then f*{x) can be computed with the accuracy
m)=
and f{x)
sample of
in the
The
Chernozhukov, Fernandez- Val, and Galichon (2006a), the density of f{X),
in
denoted Fj{t), exists
in
methods can
can be approximately-
1,...,B}.
can depend on the problem.
variable f{X),
some
Opia^^.
complexity of computing the rearranged estimate f*{u) in this way
complexity of computing the sample u-quantile
for
1,...,B} be either (1) a net of
at point
sample {f{Xj),j
as the u-quantile of the
=
A„
Op{an)
draws from the uniform distribution
i.i.d.
Then the rearranged estimate f*{u)
=
!{x)\^(hi^l'P
j*{x)fdv)^l^
computing the rearrangement. Let {X^^j
for
worth
also
/* inherits the Lp rates of convergence from the
Computation of the Rearranged Estimate. One
2.3.
on
if
is
weakly larger than the other
is
j
=
1,
/ on
its
what
In
d.
...,
follows,
we use the notation f{xj,X-j)
j-th argument, Xj, and
of monotonicity above
mapping
Xj
i-^
is
f{xj,X-j)
components, that
in each of the
all
is,
>
x'j
to denote the
Xj for each
dependence of
other arguments, X-j, that exclude Xj.
The notion
equivalent to the requirement that for each j in l,...,d the
is
weakly increasing
in Xj, for
each x^j in X'^^^.
Define the rearranged operator Rj and the rearranged function fj{x) with respect to
the j-th argument as follows;
[ l{f{xr,x.j)<y}dxr >Xj
fnx):=R,of{x):=mf\y:
This
is
the one-dimensional increasing rearrangement applied to one-dimensional func-
The rearrangement
tion Xj h^ f{xj,X-j), holding the other arguments x^j fixed.
is
applied for every value of the other arguments x_j.
Let
TT
=
(jTi, ...jTTd)
be an ordering,
i.e.
a permutation, of the integers
1, ...,d.
Let us
define the yr-rearrangement operator i?^ and the 7r-rearranged function f*{x) as follows:
f:ix) :=
For any ordering
to
all
tt,
R,
o f{x) := R^,
c.oR^^o
f{x).
the 7r-rearrangement operator rearranges the function with respect
of its arguments.
As shown below, the
resulting function fn{x)
is
weakly increasing
in x.
In general, two different orderings
tt
and
it'
of l,...,d can yield different rearranged
functions f*{x) and f*i[x).
Therefore, to resolve the conflict
done with
we may
different orderings,
collection of distinct orderings
tt,
consider averaging
we can
among rearrangements
among them:
letting 11
define the average rearrangement as
(2-4)
/*(^)^=m5I/^*(^)'
where
|n|
be any
denotes the number of elements in the set of orderings
approximation error of the average rearrangement
is
11.
As shown below, the
weakly smaller than the average of
approximation errors of individual 7r-rearrangements.
The
following proposition describes the properties of multivariate yr-rearrangements:
Proposition
able in X.
2. Let the target function fo
Let f
:
X'^ -^
K
he a
:
X'^ -^
K
be
weakly increasing and measur-
measurable function that
is
an
initial
estimate of the
target function /q. Let f
K
X'^ -^
:
he
including, for example, a rearranged
1.
For each ordering
I, ...,d,
of
tt
another estimate of
f with respect
the
it
which
fo,
For any j
(a)
2.
to the j-th
in l,...,d
arguments. Then,
to som.e of the
-rearranged estimate f*{x)
Moreover, f*{x), an average of n -rearranged estimates,
in X.
and any p in
is
rearrangement of f with respect
[l,oo], the
argum.ent produces a weak reduction in the approximation error:
1/p
|/» - f^{x)\Ux
<
\rAx)-f,{x)Ydx
X'i
mation error of
(2.5)
x<^
n -rearranged estimate f*{x) of f{x) weakly reduces
(h) Consequently, a
\f:{x)
V.J
u
>
-
1/p
\KA - fo{x)\Mx
<
fo{x)\'dx
X
(2.6)
x"-
Suppose that f{x) and fo{x) have the following properties:
X
C
Q,
C X,
and Xj
each of measure 5
such that for
all
x.j e X^j, we have that
for some
e
>
=
x
(i) x'j
>
0,
{xj,x^j) and x'
>
Xj, (ii) f[x)
>
C
and a subset X_^
=
with
{x'-,x^j),
f{x')
+
e,
there exist subsets
and
X'^~^
any p €
[1,
6 Xj,
x'j
<
X''
the
G Xj,
foix)
+
e,
tVp
1/p
=
>
(Hi) fo{x')
Xj
oo],
\f;{x)-fo{x)\''dx
rip
of measure
,
0.
(a) Then, for
where
the approxi-
the original estimate:
1/p
PCj
weakly increasing
weakly increasing in x.
is
\Ip
3.
measurable in x,
is
\f{x)
-
fo{x)\^dx-r]p6u
(2.7)
X''
+
mi{\v -t'Y
infimum taken over
(b) Further, for
-t\P -\v -t^" -\v'
all v, v'
,
t, t'
an ordering n
rearranged function, f{x)
the function f{x)
\v'
and
=
=
R^^^^ o
in the set
K
-
t'\T],
such that
(ttj, ...,7rfc, ...,7rd)
...
o
and
with
R^^ o f{x) (for k
>
v'
tt^
=
>
rjp
v
=
for p e (l,oo), with
+
j,
e
and
let
f
t'
>
t
+
e.
be a partially
d we set f{x)
=
f{x)).
If
the target function /o(x) satisfy the condition stated above, then,
for any p 6 [l,oo],
1/p
\f:{x)
Ux<^
- fo{x)rdx
1/p
<
\f{x)
X''
-
fQ{x)\^dx-r]p5i
(2.8)
10
4-
The approximation error of an average rearrangement
is
weakly smaller than the
average approximation error of the individual n- rearrangements: For any p E [1,00],
\f*{x)-fo{xWdx
/
^
- n
'
'
\f:{x)-f,{x)Ydx
This proposition generalizes the results of Proposition
to the multivariate case,
1
also demonstrating several features unique of the multivariate case.
TT-rearranged functions are monotonic in
of the arguments.
all
any argument improves the approximation properties
improvement
strict
is
when the rearrangement with
formed on an estimate that
is
increasing in the
(2.9)
VJx'i
7760
We
see that the
The rearrangement along
of the estimate.
Moreover, the
respect to a j-th argument
is
per-
decreasing in the j-th argument, while the target function
is
same j-th argument,
in the sense precisely defined in the proposition.
Moreover, averaging different 7r-rearrangements
is
better (on average) than using a single
TT-rearrangement chosen at random. All other basic implications of the proposition are
similar to those discussed for the univariate case.
Proofs of Propositions
3.
3.1.
Proof of Proposition
The
1.
part establishes the
first
in part the strategy in Lorentz's (1953) proof.
The second
the result stated in the proposition.
Proof of Part
1.
We
assume
—
part establishes the strong inequality.
/() and
l)/r, s/r], s
—
l,...,r.
on the 5-th
integer in
1,
...,
/o(-)
/s,
equal to the value of
Let us define the sorting operator S{f) as follows: Let £ be an
interval.
r such that /^
> /^
for
some
m
>
I.
Hi
does not
exist, set
i exists, set S{f) to be a r-vector with the f-th element equal to fm, the
equal to
fg,
and
all
are simple
For any simple /(•) with r
/ denote the r-vector with the s-th element, denoted
steps, let
inequality, following
focuses directly on obtaining
at first that the functions
functions, constant on intervals ((s
/(•)
The proof
weak
S{f)
=
/.
If
m-th element
other elements equal to the corresponding elements of /. For any
M^
—
L{fm, fod
+
submodular function L
:
>
1R+,
by
/^
>
/„, /om
>
foe
and the definition of the
submodularity,
L{f,, fom)
<
Life, foe)
+
L{fm, fom).
11
Therefore,
we conclude that
/
using that
we
Mx))dx < [
L{Sif){x),
completely sorted, that
= S
composition f*
o
...
o S{f)
By
.
L{r{x), fo{x))dx < f
JX
Jx
Furthermore, this inequality
A' to
number
sufficient finite
rearranged, vector /*.
is,
weakly reduces the approximation
mapping
(3.1)
integrate simple functions.
Applying the sorting operation a
/
L{f{x), fo{x))dx,
Thus, we can express /* as a
error. Therefore,
L{f{x),
Mx))dx.
(3.2)
-JX
7~r^'
finite times
extended to general measurable functions
/(•)
and
/o(-)
by taking a sequence of bounded simple functions f^^\-) and /q
ii'
(•)
The almost everywhere
converging to /() and /o(-) almost everywhere as r -^ oo.
convergence of f^^\-) to /(•) implies the almost everywhere convergence of
function
finite
repeating the argument above, each composition
L{§_^^{f), fo{x))dx < [
is
we obtain a
of times to /,
its
quantile
the quantile function of the limit, /*()• Since inequahty (3.2) holds
f*^'^\-) to
along the sequence, the dominated convergence theorem implies that (3.2) also holds for
D
the general case.
Proof of Part
1.
We
A'o,
that
Let us
2.
first
consider the case of simple functions, as defined in Part
take the functions to satisfy the following hypotheses: there exist regions Xq and
each of measure greater than 5
(i)
x'
>
X,
(ii)
f{x)
>
f{x')
+
>
e,
0,
such that for
and
(iii)
/o(x')
>
the proposition. For any strictly submodular function
r?
where the infimum
t'
>t +
We
=
inf {L(i;',
is
taken over
t)
+
L{v,
all
t')
-
L{v,
t)
all
-
x G
fo{x)
L R^
:
L{v'
v,v',t,t' in the set
,
K
and
P^o
+
e,
for e
^ R+
t')}
x'
>
E.
Xq,
>
we have
specified. in
we have that
0,
such that
v'
>
v
+
e
and
e.
can begin sorting by exchanging an element f{x), x e Xq, oi
element f{x'),
The
total
sort
all
x'
G Xq, of r-vector
/.
This induces a sorting gain of at least
mass of points that can be sorted
of these points in this way,
r- vector
in this
way
is
at least
6.
We
r]
/ with an
times 1/r.
then proceed to
and then continue with the sorting of other
points.
12
After the sorting
is
completed, the total gain from sorting
5r].
That
is,
L{r{x)Jo{x))dx < f L{f{x)Jo{x))dx-5T].
JX
IX
We
at least
is
then extend this inequality to the general measurable functions exactly as in the
D
proof of part one.
Proof of Proposition
3.2.
Proof of Part
/*(a:)
We
1.
The proof
2.
prove the claim by induction.
We
being a quantile function.
—
true in d
1
consists of the following four parts.
dimensions.
If so,
The claim
then consider any d
>
2.
Xj,
(i
=
1
Suppose the claim
by
is
then the estimate f{xj,X-j), obtained from the original
estimate f{x) after applying the rearrangement to ah arguments
argument
true for
is
must be weakly increasing
in
x_j for each
of x, except for the
rc_j
Thus,
Xj.
for
any
x'_j
>
x_j,
we have that
f{X„x'_^)
Therefore, the
random
> f{X„x_j)
variable on the
for
X,
~
U{0,
(3.3)
1).
random
of (3.3) dominates the
left
variable on
the right of (3.3) in the stochastic sense. Therefore, the quantile function of the random
variable on the
dominates the quantile function of the random variable on the
left
right,
namely
f*{xj,x'_j)
>
Moreover, for each x^j, the function Xj
We
being a quantile function.
in all of its
arguments
Proof of Part 2
is
weakly increasing by virtue of
conclude therefore that x
at all points x
G
The claim
X'^.
i-^
fj{x)
of Part
1
is
weakly increasing
of the Proposition
now
D
/
By
(a).
\f*{xj,x_j)
-
Proposition
1,
fo{xj,x_j)\'' dxj
Now, the claim
sides.
follows
For p
Proof of Part 2
to f{x)
=
we have that
<
/
for
each X-j,
\f{xj,x_j)- fo{xj,x_j)\''dXj.
(3.5)
Jx
Jx
we
f*{xj,X-j)
i-^
(3.4)
(0, 1).
by induction.
follows
both
X^
f*{xj,x^j) for each xj E
=
by integrating with respect to x^j and taking the p-th root
of
by taking the limit as p -^
D
oo, the claim follows
(b).
R^ra o f{x),
We
first
apply the inequality of Part 2(a) to f{x)
then to /(x)
recursively generate a sequence of
stated in the Proposition.
oo.
=
R„^_, o R^^ o f{x), and so on.
weak
=
f{x), then
In doing so,
inequalities that imply the inequality (2.6)
O
13
Proof of Part 3
inequality (3.5),
in
Proposition
1
For each x_j G X'^~^ \ X^j, by Part 2(a),
(a).
and
for
Part
2,
each X-^ G A'_j, by the inequality for the univariate case stated
we have the strong
\f*{xj,x^j)- fo{xj,X-j)\ dxj
<
defined in the
r}p is
(3.5) over
of
inequality
\f{xj,x^j)- fo{xj,x_j)\ dxj-r]p6,
/
(3.6)
Jx
IX
where
we have the weak
same way as
x_j € X'^"^ \ X^j, of measure
measure
u,
Proposition
in
1
—
1.
Integrating the
and the strong inequality
z/,
(3.6) over X^j,
we obtain
\f;ix)-fo{x)\'dx<
X<^
The claim now
\f{x)-fo{x)\^dx-VpSi^.
/
J X<i
(3.7)
D
follows.
Proof of Part 3
(b).
As
Part 2(a), we can recursively obtain a sequence of weak
in
improvements
inequalities describing the
in
approximation error from rearranging
quentially with respect to the individual arguments.
The
se-
Moreover, at least one of the
inequalities can be strengthened to be of the form stated in (3.7),
of the claim.
weak inequality
from the assumption
resulting system of inequalities yields the inequality (2.8), stated in
D
the proposition.
Proof of Part
4.
We
can write
nx)-fo{x)
i/p
i/p
p
1
dx
dx
Ux'^
Jx'i
Tren
(3.
V-
1
'
where the
last inequality follows
i/p
Tre
n
f:{x)-Mx)
^
dx
X''
by pulling out l/|n| and then applying the triangle
D
inequality for the Lp norm.
4.
In this section
We
Illustrations
we provide an empirical application
to biometric age-height charts.
show how the rearrangement monotonizes and improves various nonparametric
esti-
mates, and then we quantify the improvement in a simulation example that mimics the
empirical application.
14
An
4.1.
Empirical Illustration with Age-Height Reference Charts. Since
introduction by Quetelet in
mon
tlie
19th century, reference growth charts have become com-
These charts describe the evolution
tools to asses an individual's health status.
of individual anthropometric measures, such as height, weight,
across different ages. See Cole (1988) for a classical
Koenker, and
He
their
and body mass index,
work on the subject and Wei, Pere,
(2006) for a recent analysis from a quantile regression perspective and
additional references.
To
illustrate the properties of the
of growth charts for height.
It is
clear that height should naturally follow
Our data
relationship with age.
rearrangement method we consider the estimation
consist of repeated cross sectional
an increasing
measurements
of
height and age from the 2003-2004 National Health and Nutrition Survey collected by
the National Center for Health Statistics.
centimeters, and age
factors that
to
might
is
Height
is
measured as standing height
in
recorded in months and expressed in years. To avoid confounding
affect the relationship
between age and height, we
US-born white males age two through twenty. Our
final
restrict the
sample
sample consists of 533 subjects
almost evenly distributed across these ages.
Let
Y
X
and
denote height and age, respectively. Let
tional expectation of
given
(1)
X=
x,
Y
where u
given
is
X=
x,
and Qy{u\X
the quantile index.
for several quantile indices
tional quantile process
X
f->
x
^-^
(CQP)
is
x
I—> £'['K|X
(5y[u|A"
=
x], for
/o(x)
tion (zi,x) I—» fo{u,x)
u
is
=
=
H-^
Qy{u\X =
x]; in
the conditional quantile functions
I—*
Qy[u\X =
x].
The
h->
fo{x)
is
natural monotonicity requirements
The CEF x
k-+
and the
£'[y|X
CQP
=
x]
and the
{u,x) h^ (5y[u|X
CQF
=
x]
u.
target functions using non-parametric ordinary least squares or quan-
regression techniques and then rearrange the estimates to satisfy the monotonicity
requirements.
(e)
the entire condi-
in the third case, the target func-
should be increasing in both age x and the quantile index
tile
(3)
the second case, the target function x
x) should be increasing in age x,
We estimate the
Y
for height given age. In the first case, the target function
5%, 50%, and 95%; and,
(u,x)
denote the condi-
x]
functions of interests are
(5%, median, and 95%), and
for the target functions are the following:
X
(2)
=
x) denote the u-th quantile of
The population
the conditional expectation function (CEF),
(CQF)
=
i?[y|J'(^
Fourier,
We
and
consider (a) kernel, (b) local linear,
(f)
flexible Fourier
(c) spline, (d)
global polynomial,
methods. For the kernel method, we provide a
fit
15
on a
cell-by-cell basis,
with each
cell
corresponding to one month. For the local linear
method, we choose a bandwidth of one year and a box
we use cubic
sines
For the sphne method,
B-splines with a knot sequence {3,5,8, 10, 11.5, 13, 14.5, 16, 18}, following
Wei, Pere, Koenker, and
polynomial.
kernel.
He
(2006). For the global polynomial
method, we
fit
a quartic
For the Fourier method, we employ eight trigonometric terms, with four
and four
For the flexible Fourier method, we use a quadratic polynomial
cosines.
and four trigonometric terms, with two sines and two
tion of the conditional quantile process,
we use a
cosines.
Finally, for the estima-
net of two hundred quantile indices
{0.005,0.010, ...,0.995}.
In the choice of the parameters for the different methods,
select values that either
have been used
in the previous empirical
work or give
we
rise to
specifications with similar complexities for the different methods.
The panels A-F
of Figure
1
show the
original
and rearranged estimates of the con-
ditional expectation function for the different methods.
All the estimated curves have
trouble capturing the slowdown in the growth of height after age sixteen and yield non-
monotonic-curves for the highest values of age.
The
Fourier series have a special difficulty
approximating the aperiodic age-height relationship. The rearranged estimates correct
the non-monotonicity of the original estimates, providing weakly increasing curves that
coincide with the original estimates in the parts where the latter are monotonic. Moreover, the rearranged estimates necessarily
by the theoretical results derived
original estimates.
We
earlier,
improve upon the original estimates,
since,
they are closer to the true functions than the
quantify this improvement in the next subsection.
Figure 2 displays similar but more pronounced non-monotonicity patterns for the
estimates of the conditional quantile functions.
in delivering curves that
The rearrangement again performs
well
improve upon the original estimates and that satisfy the natural
monotonicity requirement.
Figures 3-7 illustrate the multivariate rearrangement of the conditional quantile process
(CQP) along both the age and the
dimensions the original estimate,
its
its
quantile index arguments.
age rearrangement,
its
We
plot in three
quantile rearrangement, and
average multivariate rearrangement (the average of the age-quantile and quantile-age
rearrangements).
We
also plot the corresponding contour surfaces.
show the multivariate age-quantile and quantile-age rearrangements
(Here,
we do not
separately, because
16
they are very similar to the average multivariate rearrangement.)
tour plots that, for
non-monotone
index.
all
The contour
see from the con-
CQP
methods considered, the estimated
of the estimation
and non-monotone
in age
We
in the quantile
is
index at extremal values of this
plots for the estimates based on the Fourier series best illustrate the
non-monotonicity problem.
We
see that the average multivarite rearrangement fixes the
non-monotonicity problem, and delivers an estimate of the
CQP
that
monotone
is
in
both the age and the quantile index arguments. Furthermore, by the theoretical results
of the paper, the multivariate rearranged estimates necessarily improve
upon the
original
estimates.
4.2.
Monte-Carlo
improvement
Illustration.
in the
The
following
Monte Carlo experiment
quantifies the
estimation/approximation properties of the rearranged estimates
The experiment
relative to the original estimates.
closely
matches the empirical appli-
cation presented above.
we consider the design where
Specifically,
function plus a disturbance
regressor
5),
1{A"
e,Y = Z{X)'P + e, and
the disturbance
X. The vector Z{X) includes a constant and a piecewise
of the regressor
>
10}
•
X
with three changes of slope, namely
{X -
10),
1{X >
15}
•
{X -
Y
the outcome variable
15)).
Z{X) =
is
equals a location
independent of the
linear transformation
(1,
X, 1{X
>
5}
•
(X —
This design implies the conditional
expectation function
E[Y\X]
= Z{Xyp,
(4.1)
and the conditional quantile function
QY{u\X)
We
select the
= Z{Xyf3 +
Q,{u).
(4.2)
parameters of the design to match the empirical example of growth charts
in the previous subsection.
Thus, we set the parameter P equal to the ordinary least
squares estimate obtained in the growth chart data, namely (71.25, 8.13, —2.72, 1.78,
—6.43). This parameter value and the location specification (4.2) imply a model for
and
CQP
that
is
monotone
in age over the
CEF
range of 2-20. To generate the values of the
dependent variable, we draw disturbances from a normal distribution with the mean and
variance equal to the
in the
mean and
growth chart data.
We
variance of the estimated residuals,
fix
the regressor
X
e
= Y—
Z{X)' [3,
in all of the replications to
be the
17
observed values of age in the growth chart data
CEF
and
CQP
In Table
1
we estimate the
using the nonparametric methods described in the previous section.
we
=
1,2,3,4 and oo) for the original
We
also report the relative efficiency
report the average LP errors (for p
estimates and the rearranged estimates of the
of the
In each rephcation,
set.
ratio of the average error of the rearranged estimate
two estimates, measured as the
to the average error of the original estimate.
Monte Carlo average
CEF.
We
W error as the
calculate the average
of
i/p
U': =
\f{x)~Ux)Tdx
.,
X
where the target function /o(x)
is
the
CEF
either the original nonparametric estimate of the
For
all
CEF
of the
methods considered, we
more accurately than the
= x], and
CEF or its
Efl^lX
original curves, providing a
we report the average L^
increasing rearrangement.
find that the rearranged curves estimate the true
average error, depending on the method and the norm
In Table 2
the estimate /(x) denotes
2%
to
84%
reduction in the
values of p).
(i.e.
errors for the original estimates of the conditional
quantile process and their multivariate rearrangement with respect to the age and quantile
We
index arguments.
also report the ratio of the average error of the rearranged
estimate to the average error of the original estimate.
The average Lp
error
is
the
Monte
Carlo average of
-
LP
= II
U
i/p
\f{u,x)- fo{u,x)\Pdxdu
JX
where the target function fo{u,x)
is
the conditional quantile process
Qy{u\X =
x),
and
the estimate f{u, x) denotes either the original nonparametric estimate of the conditional
quantile process or
its
multivariate rearrangement.
erage multivariate rearrangement only.
The
We
present the results for the av-
age-quantile and quantile-age multivariate
rearrangements give errors that are very similar to their average multivariate rearrange-
ment, and we therefore do not report them separately. For
we
all
the methods considered,
find that the multivariate rearranged curves estimate the true
than the original curves, providing a
4%
depending on the method and the norm.
to
74%
CQP
more accurately
reduction in the approximation error,
18
we report the average Lp
In Table 3
error for the univariate rearrangements of the
conditional quantile function along either the age argument or the quantile index ar-
gument.
We
also report the ratio of the average error for these rearrangements to the
average error of the original estimates. For
these rearranged curves estimate the true
of the
all
CQP
methods considered, we
more accurately than the
find that
original curves,
providing noticeable reductions in the estimation error. Moreover, in this case the re-
arrangement along the age argument
is
more
effective in reducing the estimation error
than the rearrangement along the quantile index. Furthermore, by comparing Tables 2
and
3,
we
also see that the multivariate rearrangement provides an
the individual univariate rearrangements, yielding estimates of the
much
CQP
that are often
closer to the true process.
5.
Suppose that a target function
original estimate of this function,
tion
improvement over
Conclusion
known
is
which
is
to
be weakly increasing, and we have an
not weakly increasing.
methods provide estimates with such a property.
We show
Common
estima-
that these estimates
can always be improved using rearrangement techniques. The univariate and multivariate rearrangement
The
resulting
methods transform the
monotonic estimate
is
metrics than the original estimate.
original estimate to a
monotonic estimate.
common
in fact closer to the target function in
We
illustrate these theoretical results
with a com-
putational example and an empirical example, dealing with estimation of conditional
mean and
quantile functions of height given age.
and improves the
whether
this
original
The rearrangement both monotonizes
non-monotone estimates.
It
would be interesting to determine
improved estimation/approximation property
of monotonization.
We
carries over to other
methods
leave this extension for future research.
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Constrained
Wiley
of polynomial splines
estimation," Ann. Statist., 22(1), 118-184,
With
&
statistical inference,
Sons],
and
Wiley Series
in
Proba-
Hoboken, NJ.
their tensor products in multivariate function
discussion by Andreas Buja and Trevor Hastie and a
rejoinder by the author.
ViLLANI, C. (2003):
Topics in optimal transportation,
American Mathematical
Wand, M.
P.,
and M.
Applied Probability.
vol.
58 of Graduate Studies in Mathematics.
Society, Providence, RI.
C.
Jones
Chapman and
(1995): Kernel smoothing, vol. 60 of
Hall Ltd., London.
Monographs on
Statistics
and
20
Wei, Y., a. Pere, R. Koenker, and X. He
growth charts,"
Stat. Med., 25(8),
1369-1382.
(2006):
"Quantile regression methods for reference
21
Table
L^ Estimation/Approximation Error
1.
of Original
and Rear-
ranged Estimates of the Conditional Expectation Function,
1,2,3,4, and oo. Univariate Rearrangement.
LI
p
LI
^'r/li
.
Lr
L'o
Lr/^o
B. Local Polynomial
A. Kernel
1
3.69
1.33
0.36
0.79
0.76
0.96
2
4.80
1.84
0.38
1.00
0.96
0.96
3
5.81
2.46
0.42
1.17
1.13
0.96
4
6.72
3.12
0.46
1.33
1.28
0.96
00
16.8
9.84
0.58
2.96
2.81
0.95
D. Quartic
C. Splines
1
0.87
0.81
0.93
1.33
1.19
0.89
2
1.10
1.02
0.93
1.64
1.46
0.89
3
1.31
1.22
0.93
1.89
1.68
0.89
4
1.52
1.39
0.92
2.10
1.87
0.89
00
3.72
3,19
0.86
4.38
3.79
0.87
E. Fourier
F. Flexible Fc urier
1
6.57
3.21
0.49
0.73
0.72
0.97
2
10.7
3.79
0.35
0.91
0.89
0.97
3
15.2
4.24
0.28
1.06
1.04
0.98
4
19.0
4.59
0.24
1.18
1.16
0.98
oo
48.9
7.79
0.16
2.44
2.40
0.98
Notes:
The
table
is
based on 10,000 replications.
Lq
is
the LP error of the original estimate.
L^
is
the LP error of the rearranged estimate.
for
p
=
22
Table
U
2.
Estimation/ Approximation Error of Original and Rear-
ranged Estimates of the Conditional Quantile Process,
and
for
p
=
1,2,3,4,
Average Multivariate Rearrangement.
oo.
p
JP
^RR
L'o
TV
ITP
^RRl^O
LI
Lrr
Lrr/^o
B. Local Polynomial
A. Kernel
1
5.35
3.13
0.58
1.21
1.09
0.91
2
6.97
4.37
0.63
1.61
1.46
0.91
3
8.40
5.49
0.65
2.03
1.84
0.91
4
9.72
6.49
0.67
2.48
2.24
0.91
00
34.3
26.4
0.77
12.3
10.4
0.84
D. Quartic
C. Splines
1
1.33
1.20
0.90
1.49
1.35
0.90
2
1.78
1.60
0.90
1.87
1.69
0.90
3
2.30
2.03
0.88
2.23
1.99
0.89
4
2.92
2.50
0.86
2.62
2.29
0.87
oo
16.9
12.1
0.72
12.6
8.61
0.68
F. Flexible Fourier
E. Fourier
1
6.72
4.18
0.62
1.05
1.00
0.96
2
13.7
5.35
0.39
1.38
1.31
0.95
3
20.8
6.36
0.31
1.72
1.63
0.95
4
26.7
7.25
0.27
2.12
1.98
0.94
oo
84.9
21.9
0.26
10.9
9.13
0.84
The
Notes:
Lq
is
based on 1,000 replications.
the LP error of the original estimate.
is
^'rr
table
.
'^
^^^ L^ error of the average multivariate rearranged estimate.
23
Table
of
tlie
3.
L^ Estimation/ Approximation Error of Rearranged Estimates
Conditional Quantile Process, for p
=
l,2,3,4,oo. Univariate Re-
arrangements.
p
^Ru
^k
A
^rJ^o ^rJ^o
L%,
^l.
^Rul^O
^'kr/^O
B. Local Polynomial
Kernel
.1
5.35
3.13
1.00
0.58
1.20
1.10
1.00
0.91
2
6.97
4.37
1.00
0.63
1.60
1.47
1.00
0.91
3
8.40
5.49
1.00
0.65
2.01
1.85
0.99
0.91
4
9.72
6.49
1.00
0.67
2.45
2.26
0.99
0.91
oo
34.3
26.4
1.00
0.77
11.8
10.8
0.96
0.88
D. Quartic
C. Splines
1
1.31
1.21
0.99
0.91
1.49
1.35
1.00
0.91
2
1.75
1.63
0.98
0.91
1.87
1.69
1.00
0.90
3
2.24
2.08
0.97
0.90
2.22
2.00
0.99
0.90
4
2.80
2.59
0.96
0.89
2.60
2.30
0.99
0.88
oo
14.4
13.9
0.85
0.82
11.9'
9.11
0.95
0.72
E. Fourier
F. Flexible Fourier
1
6.71
4.19
1.00
0.62
1.04
1.01
0.99
0.96
2
13.7
5.36
1.00
0.39
1.36
1.32
0.99
0.96
3
20.8
6.37
1.00
0.31
1.70
1.65
0.99
0.96
4
26.7
7.26
1.00
0.27
2.08
2.02
0.98
0.95
oo
84.9
22.2
1.00
0.26
10.0
9.86
0.92
0.91
The
Notes.
Lq
is
table
is
based on 1,000 replications.
the LP error of the original estimate.
L^
is
the LP error of the estimate rearranged in age x.
!/;;_
is
the Lf error of the estimate rearranged in the quantile index u.
.
24
A.
CEF
B.
(Kernel)
CEF
Age
Age
C.
CEF
(Splines)
D.
Age
E.
CEF
(Local Pol.)
CEF
(Quartic)
Age
(Fourier)
F.
.^'
CEF
(Flexible Fourier)
-a-
X
Age
Figure
1.
Nonparametric estimates of the Conditional Expectation
Function (CEF) of height given age and their increasing rearrangements.
Nonparametric estimates are obtained using kernel regression (A)
linear regression (B), cubic
(D), Fourier series (E),
and
B-sphnes
series (C), a four degree
flexible Fourier series (F).
,
locally
polynomial
25
A.
CQF: 5%, 50%, 95%
(Kernel)
B.
CQF: 5%, 50%, 95% (Local
Age
Age
CQF: 5%, 50%, 95% (Splines)
C.
D.
CQF: 5%, 50%, 95%
Age
CQF: 5%, 50%, 95%
E.
2.
(Quartic)
Age
(Fourier)
F.
CQF: 5%, 50%, 95%
Age
Figure
Pol.)
(Flexible Fourier)
Age
Nonparametric estimates of the 5%, 50%, and 95% Condi-
tional Quantile Functions
rearrangements.
(CQF)
of lieight given age
and
their increasing
Nonparametric estimates are obtained using kernel
re-
gression (A), locally linear regression (B), cubic B-splines series (C), a
four degree polynomial (D), Fourier series (E), and flexible Fourier series
(F).
26
A.
CQP
(Kernel)
B.
-|
0-0
CQP: Contour
1
1
1
1
0.2
0.4
0-6
0.8
r1.0
quantile
C.
CQP: Age Rearrangemenl
D.
0.0
E.
G.
I
I
0.2
0.4
CQP: Quantile Rearrangement
F.
CQP: Average Quantlle/Age Rearrangement
H.
,
CQP: Contour (R-Age)
r
o.e
CQP: Contour (R-Quantile)
CQP: Contour (RR-Quantile/Age)
^
1
1
1
_,
'
R
o
-,
":
.,.
--
'
1
1
0-0
0.2
0.6
0.4
quantile
Figure
3.
Kernel estimates of the Conditional Quantile Process (CQP)
of height given age
E
and
their increasing rearrangements.
Panels
C and
plot the pne dimensional increasing rearrangement along the age
quantile dimension respectively; panel
rearrangement.
G
and
shows the average multivariate
27
A.
COP
B.
{Local Pol.)
CQP: Contour
quantile
C.
CQP: Age Rearrangement
D.
CQP: Contour (R-Age)
quantile
E.
CQP: Quantile Rearrangement
F.
CQP: Contour {R-Quantlle)
a
jr_^
^S:
Wi
'^—
o
w.
in
;;./>
_
-
02
0.0
0,4
0.8
0,6
quantile
G.
CQP: Average Quantile/Age Rearrangement
H.
a
-
lO
_
CQP: Contour (RR-Quantile/Age)
V
-V
I
"L
,
?
o
_
-X
—
:::^:__
^
^•^-—
^^=^;-^:^
^^^^i^n::!:
^6=V.
.„
-N
^~"
i:;-
—
=^
~-
'
MO
;7-
:
^
2
0.4
6
-
8
1.0
quantile
Figure
(CQP)
and
4.
Locally linear estimates of the Conditional Quantile Process
of height given age
E plot
and
their increasing rearrangements. Panels
C
the one dimensional increasing rearrangement along the age and
quantile dimension respectively; panel
rearrangement.
G
shows the average multivariate
28
A.
C.
E.
G.
(Splines)
B.
COP: Age Rearrangement
D.
CQP: Quantile Rearrangement
F.
CQP: Average Quantile/Age Rearrangement
Figure
Process
Panels
COP
5.
Cubic B-splines
(CQP)
C and E
H.
and
CQP: Contour (R-Age)
CQP: Contour (R-Quantlle)
CQP: Contour (RR-Quantile/Age)
series estimates of the
of height given age
COP: Contour
Conditional Quantile
their increasing rearrangements.
plot the one dimensional increasing rearrangement along
the age and quantile dimension respectively; panel
multivariate rearrangement.
G
shows the average
29
A.
COP
B.
(Quartic)
COP: Contour
quantile
C.
CQP: Age Rearrangement
D.
CQP: Contour (R-Age)
'^
20
1
15
1
10
1
'"^-—
aga
^
zuzz^
no —
'
;;(]
.
0,0
',>
0,4
0.2
—
0.6
quanlile
E.
CQP: Quantile Rearrangement
F.
J—
'^
c
7~
CQP: Contour (R-Quantile)
C
^
_
"
,
"
'"
..-
'
:o:>
—
1
r
0.2
0.0
...,
1
1
,
0.6
0.4
quantile
G.
CQP: Average Quantile/Age Rearrangement
H.
s
i2
ff
-
-
s
.
u.
-
CQP: Contour (RR-Quantile/Age)
l-^-^^
V
^^^rrm^
v^
/~~^
.
'so
_\
^^^^^^^^nizzi:^::;"'"~~--$
znzzz"--^^
^^==:::=:
V
0.0
Figure
tile
6.
Process
ments.
0.2
4
6
8
Quartic polynomial series estimates of the Conditional Quan-
(CQP)
Panels
of height given age
C and E
and
their increasing rearrange-
plot the one dimensional increasing rearrange-
ment along the age and quantile dimension
the average multivariate rearrangement.
respectively; panel
G
shows
30
A.
CQP
-|
0.0
C.
CQP: Contour
B.
(Fourier)
r-
1
1
1
1
0.2
0.4
0.6
0.8
CQP; Age Rearrangement
D.
1.0
CQP: Contour (R-Age)
quantile
E.
CQP; Quantile Rearrangement
F.
CQP: Contour (R-Quantile)
quanlile
G.
CQP: Average Quantile/Age Rearrangement
H.
a
CQP: Contour (RR-Quantile/Age)
-
--.„-,
-^«>--
--^:^:r—-^
lO
_
o
_
„.-
^hi^^h?^r~^^^
^i^=^9^=
-1
(CQP)
and
7.
0-2
0-4
1
r
0.6
Fourier series estimates of the Conditional Quantile Process
of height given age
E plot
u
^P^i;:^^^:::rz:^
0.0
Figure
.
and
their increasing rearrangements. Panels
C
the one dimensional increasing rearrangement along the age and
quantile dimension respectively; panel
rearrangement.
G
shows the average multivariate
31
A.
COP (FlexIbJe
Fourier)
B.
COP: Contour
___^..^
T^n:^
-l_
:)
—
0.0
C.
E.
G.
COP: Age Rearrangement
COP: Quantile Rearrangement
F.
8.
H.
.
flo
—
0.6
0,4
D.
CQP: Average Quantile/Age Rearrangement
Figure
0.2
-^
_
0.8
1.0
COP: Contour (R-Age)
CQP: Contour (R-Ouantile)
CQP: Contour (RR-QuantJIe/Age)
Flexible Fourier form series estimates of the Conditional
Quantile Process
ments. Panels
(CQP)
C and E
of height given age
and
their increasing rearrange-
plot the one dimensional increasing rearrangement
along the age and quantile dimension respectively; panel
erage multivariate rearrangement.
G
shows the av-
Date Due