Charles University and National Institute of Public Health, Czech Republic .... model (proved by Koenker and Bassett (1978) and by Ruppert and Carroll. (1980) ) ...
© 1994 Gordon and Breach Science Publishers
Nonparametric Statistics, Vol. 3, pp. 201-222 Reprints ava il able directl y from the publisher Photocopying permitted by license only
Printed in the Uni ted States of A merica
REGRESSION QUANTILES AND TRIMMED LEAST SQUARES ESTIMATOR IN NONLINEAR REGRESSION MODEL JANA JUREČKOVÁ and BOHUMÍR PROCHÁZKA Charles University and National Institu te of Public Health , Czech Republic
Consider the non li near regression model
Y,
=
g(x, , 0) + ci
i= 1, ... , n
(1)
R"c, 0 = (Bo, e,' ... ' ep)' EG (compact in Rp+I) and with g of the form g(x, 0) = Ba+ BP ) which is continuous. twice differentiable in 0 and monoton e in compo nents o f 0. U nder some further regu larity conditions, we prove the consistency, asymptotic representation an d asymptot ic norma lity of regression quantil es in model (1). This enables to construct various consiste nt L-estimators in the model; we consider (i) linear combi na tions of selected regressio n quantiles and (ii) th e a-trimmed least sq u ares estimator of 0 . For both we obtain the asymptotic representations and asymptotic distributions. The latter estimator is ill ustrated on three sets of medical data; the models being used are either mixtures of two expo ne ntial functi ons or a logistic function . respectively.
with x,
E
g(x, 8 1 ,
•• • ,
KEYWORDS: Nonlinear pseudodimension .
regression.
regression
quantile.
trimmed
least
squares
es tim ator,
1. INTRODUCTION
Consider the nonlinear regression model
Y,
=
g(x;, 0) +
E;,
i= 1, ... , n
(1.1)
where Y = (Y1 , . .. , Y,,) ' is a vector of observations, X; E žť c IR%- are given vectors, i = 1, ... , n, E 1 , ••• , E are i.i.d. errors with a positive (but generally unknown) density f and 0 = ( e0, e1 , . . . , eP) ' E [RP+ 1. We assume that the function g(x, 0) could be written in the form g(x, 0) = e0 + g(x, (8 1 , . . . , ep)' ) with appropriate function g of x and (8 1 , . . . , eP)'; hence the component 8 0 of the parameter 0 is an intercept. The most popular estimator of parameter in a nonlinear regression model, the least-squares estimator (LSE) , was studied by a host of authors. The asymptotic behavior of LSE was studied, e.g. , by Jenrich (1 969), Ivanov (1976) , Wu (1981), Schmidt and Zwanzig (1983) , Prakasa Rao (1985, 1986a, b, 1987) Lauter (1989), among others. A nother natural estimator, the L 1 -estimator, was studied , e.g. , by Oberhofer (1982) , Richardson (1987) and Weiss (1991) , who proved its consistency and 11
201
J. JUREČKOVÁ and B. PROCHÁZKA
202
asymptotic normality under various regularity conditions. Koenker and Bassett (1978) introduced the concept of a-regression quantile and a -trimmed leastsquares estimator in the linear regression model. These estimators soon became very popular even among applied statisticians, because their idea is very natural, they are computationally appealing and represent a straightforward extension of the empirical a -quantile and of the a -trimmed mean from the location to the linear regression model. Moreover, Gutenbrunner and Jurečková (1992) showed that the variables, dual to regression quantiles in the parametric linear programming sense, could be interpreted as regression rank scores and they also extend the duality of order statistics and ranks from the location to the linear regression model. The basic definition of the a -regression quantile can be naturally extended to the nonlinear regression model as well as to other models. The extension to the nonlinear regression model was already considered in Chen (1987) and Procházka (1988); Koenker and Park (1992) elaborated the pertaining computational algorithms. However, though the extension is natural and straightforward, the problem is that of consistency of such estimators. Chen (1987) first linearized the regression function and then used the well-known asymptotic properties of regression quantiles and trimmed LSE in linear model. In the present paper, we shall prove the consistency and the asymptotic normality of a -regression quantiles and of a -trimmed LSE in the model (1.1) with a function g monotone in every component of 0 and satisfying some other regularity conditions; yet, this model will cover, e.g., a mixture of two exponentials and the logistic regression , as illustrated on real data. The asymptotic properties are mostly derived with the aid of chaining technique applied to the empirical processes connected with the regression quantiles. Moreover, we shall use the standard uniform asymptotic linearity technique in a similar way as it was used in Ruppert and Caroll (1 980) for the trimmed LSE in the linear model. In Section 2 we formulate the regularity conditions and the main results of the paper: the consistency, the asymptotic representation and the asymptotic normality of regression quantiles in model (1.1) . The results are proved in Section 3. T he consistency and the asymptotic normality of a-trimmed LSE as well as of a linear combination of several selected regression quantiles are studied in Section 4. In Section 5 we illustrate the procedures on real data sets.
2. REGRESSION QUANTILES We shall work with the regression model (1.1) where Y E IR X; E ž!t c IR't, i= 1, .. . ) n and 0 = (80 , 81, ... ) 8p)' E ec w +1. The regression quantile and the trimmed LSE will be studied under the following regularity conditions: (A.1) The parameter space 8 and the space 2e of x are compact. (A.2) The function g(x, 0): 2e X 8 ~ IR 1 could be written in the form 11
,
g(x, 0) where
g is
=
80 + g(x, (81, .. . ) 8p) ') ,
a function of x and of (8 1 ,
... ,
8P)' .
X E
f!t ,
0E8
REGRESSIO N Q UANTILES IN NO NLINEAR REGRESSION
(A.3) Every set (y 1 , x 1 ) , uniquely 0 E 8 such that
(Yp+ 1 ,
. . . ,
Y1 = g(x 1, 0),
Xp+i )
203
of p + 1 different points determines
i= 1, ... ' p + l.
(A.4) The function g(x , 0) : fJť X 8 ~ ~ 1 is strictly monotone in every argument of 0 = (8 0 , 8 1 , . . . , ep)'. Moreover, g is twice differentiable in the components of 0 and the first and the second derivatives are bounded by K, O< K< oo, uniformly in fJť and 8 . (A.5) The errors E 1 , . . . , E are independent, identically distributed with the density f such that f (x) = f (-x) and O n 0 11
1
11
k1 11 02 - 01li
2
: : :; -1 L [g(x1, 02) - g(x1, 01)] 2 :::::; k2 11 02 - 01li 2 n
1~1
where li · li stands for the Euclidean norm. (A.8) Denote D 11 = D,i(x) = (d 1/Y,:,? ;; with
d
=
[ag(x1, 0 + T)J
.~o'
arj
lj
i=l ,
„ n:
j =O, ... ' p.
We assume that lim 11 ~ x (l / n)D;,0 11 = Q(x) = Q, where Q is a positively definite (p + 1) X (p + 1) matrix. Moreover, we assume that 1
11
- L ll d1ll 4 = ó'(l) as n~ x n 1~1
where d; is the ith row of D 11 , i= 1, ... , n. The conditions (A.1)-(A.8) are consistent with those usually imposed in the nonlinear model. (A.2 ) means that the first component of 0 is an intercept and this , in turn , guarantees that the a 1 , a 2 quantiles are distinguishable for a 1 oF a 2 . (A.7) is one of the conditions guaranteeing the uniqueness of estimator. Following Koenker and Bassett (1978). we define the a-regression quantile in the modeJ (1.1) aS the VeCtOr E w + I which minimizes
REMARK.
ean
11
L Pa(Y1-
(2.1)
g(x1, t)) :=min
i=l
with respect to t
E
8 , where
Pa(Z) = lz l {al[ z > OJ + (1- a)I[z r
for any r >O. Proof Let t ~O ; denote 8;(t) O. Then
(1
=
g(x;, 6 + t) - g(x;, 6) and assume first that 8;(t)
8,( t )
Os.Epn;(w,t)= - E
)
i/!a(E;a -u)du
8,( t )
=
0
1
F(F - 1 (cx)+u)du
0
(15,( t ) ( "
= Jo
2:
( D,( t ) ( ''
1
Jo / (F - (cx)) du du + Jo
(u
1
Jo Jo f' (F - (cx) + w) dw du du
= if(F - 1 (cx))(8;(t)) 2 + ť7( 1 d; t l 3 )
(3.7)
and an analogous proposition we obtain in case 8;(t) r, regarding (A.7) ,
E
0, 0 +t
(3.11 ) E
0. Thus taking O
oo,
(4.2)
where
(eo+ L wjF - 1(aJ , B1, ... ' ek k
E=
)'
1 ~1
and k
k
w;wj(a ; Aaj -a;a j) Q - 1.
V =~ j~ f(F- 1 (a;))f(F
1
(aj))
(4.3)
(ii) The idea of a -trimmed LSE in nonlinear regression model was already mentioned in Procházka (1987) , but without any asymptotic considerations. Following Koenker and Bassett (1978) , we may characterize 0T in the follo wing way: Fix a, O< a< t and calculate the regression quantiles 0" and 01 _ " . Put á; =
f [g( x„ iÍ a )
(4.8)
00
the asymptotic representation n1 12(6T -
n
where a(e ) = ("'(E)] + l'p (l)
(4.21)
11
n-
1/2"' L.J
D nianiEi A
-
-
n-
l /2W*(A A ) Ui, L.>.2 -
.
(1)
(4.22)
+ 011 (1)
(4.23)
ftp
i=l
n - 1 D~Á11 D 11 = U*(A 1 , A2 ) = (1 - 2a )Q
where
Dn; is the (p + 1)
X
=
([a2g(x;, 0 + T)J__ ) . ·-
ar, ark
,- o
(4.24)
1,k - o, ... .p
(p + 1) matrix of the second derivatives , i= 1, ... , n.
Proof of Theorem 4.2. Consider the behavior of the left-hand side of (4.6) for lit - 011 = n - 112 c, C fixed , O< C < cxi . By the Taylor expansion at t = 0 we rewrite the left-hand side of ( 4.6) in the form n
D~ÁnE - D~ÁnDn(t - 0) + ~ E;Dn;Dn;(t - 0) + Op( llt* - 0 f ) i=l
(4.25)
J. JUREČKOY Á and B. PROCHÁZKA
214
with t* between 0 and t. Hence, the scalar product of (4.25) with (t - 0) can be estimated from above as
(t - 0)'D~ÁnE - (t - 0)'D;,ÁnD(t - 0) n
+ (t- 0)'
2: c;Dn;an;(t -
0) + O'P(llt - 0 11 lit* - 0 11)
i=l
::::: CO'p(l) - (1 - 2a)C 2 (Amin/2) + C 2 úp(n - 112 )
(4.26)
what is negative with probability 2'.l - s (s> O arbitrary given) , for n 2'. n 0 and for sufficiently large C, where Amin> O is the minimal eigenvalue of Q. Hence, by Theorem 6.3.4 in Ortega and Rheinboldt (1970) , there exists the root 0~ of ( 4.6) satisfying 11 0~ - 0 11 ::; n - l/2c (4.27) with pro ba bili ty 2'.l - s for n 2'. n 0 and for some O< C
