Testing for the equality of two nonparametric regression curves

journal of statistical planning Journal of Statistical Planning and Inference 65 (1997) 293 314

ELSEVIER

and inference

Testing for the equality of two nonparametric regression curves Hira L. Koul a'~, Anton Schick b'* ~'Department of Statistics and Probability, Michigan State Universio,, East Lansing, M1 48b¢24-1027, US.t bDepartment qf Mathematical Sciences, Binghamton Uniters'it)', Binghamton, NY 13902-600, US'.-I

Received 22 January 1996: received in revised form 6 February 1997

Abstract This paper discusses the problem of testing the equality of two non-parametric regression curves against one-sided alternatives when the design points are common and when they are distinct. Two classes of tests are given for each case. One class of tests requires the estimation of the common regression function while the other class avoids this. This paper derives the asymptotic power of these tests for contiguous alternatives, obtains an upper bound on the asymptotic power of all tests under these alternatives and then discusses asymptotically optimal tests from these classes. As an example, in the distinct design case, a weighted covariate matched Wilcoxon Mann Whitney test turns out to be optimal against certain contingous alternatives if the error density is logistic. Optimal tests are also given at the normal error density. A simulation study investigates the behavior of these tests for moderate sample sizes. ~' 1997 Elsevier Science B.V. A M S class!lication: Primary 62G07; secondary 62G10 Keywords: Contiguous alternatives; Covariate-matched Wilcoxon Mann Whitney test: Covariate-matched two-sample t-test; Locally asymptotically most powerful: Locally asymptotically unbiased

1. Introduction This paper considers the p r o b l e m of testing the equality of two n o n p a r a m e t r i c regression curves against a one-sided alternative. More precisely, let lq a n d IL2 denote the two regression functions; then the p r o b l e m of interest is to test Ho: lq = Â~zagainsl the alternative H ~ : / q > tL2, where F~l > ltz m e a n s that l q ( x ) ~> ltz(X) for all covariate values x with strict i n e q u a l i t y for at least one x. Such a testing p r o b l e m arises when

* Corresponding author. 1Research partially supported by NSF Grant DMS-9402904 and by Humboldt Foundation. 0378-3758/97/S17.00 .~ 1997 Elsevier Science B.V. All rights reserved PIl S 0 3 7 8 - 3 7 5 8 ( 9 7 ) 0 0 0 6 3 - 3

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H.L. Koul, A. Schick/Journal of Statistical Planning and Inference 65 (1997) 293 314

one compares two treatments over the range of a covariate and expects the first treatment to be superior over the entire range. This is typically the case when the second treatment is just a placebo. In this setting the regression curves measure the treatment effects as a function of the covariate and the superiority of the first treatment is expressed t h r o u g h the alternative hypothesis Hi:/~1 >/~2. O u r analysis is carried out under the following set up. The observations consist of bivariate data (Xl.i, Yl.i), i = 1. . . . . hi, and ( X 2 , j , Y2.j),J = 1. . . . . n2, satisfying the relations

Yk,j=I~k(Xk,j)+ ek,j, j = 1, ...,rig,

k = 1,2.

(1.1)

Here the errors eL 1. . . . . el .... ~2,1 . . . . . e2,n2 are independent and identically distributed with density f a n d the covariates X1,1, ... , X1 .... X2,1 . . . . , X2,n2 are independent of the errors and satisfy either of the following assumptions.

Common design: The sample sizes n~ and n2 are equal, say n~ = n2 = n, and the design variables satisfy X L j = X2,j = X j , j = 1, ..., n, for independent r a n d o m variables X~ . . . . , Xn with c o m m o n density g. Distinct design: The design variables X 1 , 1. . . . . dent with c o m m o n density g.

X1 . . . . X2, 1. . . . . X2.,2 are indepen-

In each of these cases we shall propose two classes of tests. The first class uses a kernel estimate of the c o m m o n regression function under the null hypothesis. The second class avoids this by matching the covariates. T h r o u g h o u t w denotes a symmetric Lipschitz continuous density which is positive on ( - 1, 1) and vanishes o f f ( - 1, 1). Put

w,(x) = - w 1 a

(x) , x~,

a>0.

Let v, ~ and p denote measurable functions such that v >~ 0 and p is odd. These functions are used to generate various classes of tests. We shall discuss in Section 3 how to choose these functions. To describe the first class let fi be the kernel regression estimate

~j= 1 Yk,jW.(Xk,j - x) ~2k= 1 ~ j k 1 Wa(Xk,j -- X)

'

X E ~,

based on the pooled sample. Set

~k,j= Yk,j-- fi(Xk,j), j = 1, ...,nk,

k = 1,2,

(1.2)

so that gk,j mimics ek,j under the null hypothesis. F o r the common design case we propose the test which rejects the null hypothesis for large values of the test statistic

1 ~1 v(Xj)(~(~l,j)

'~1 -- N ~ j =

-- ~(~2,j))"

H.L. Koul. A. Schick/Journal q/" Statistical Planning and lnl~,rence 65 (1997) 293 314

295

F o r the distinct design case we propose the test which rejects the null hypothesis t\~r large values of the test statistic

-

x,/nl

+n2

~.i=1

--

- :

"

We shall now describe a class of tests that does not require the estimation of the c o m m o n regression function. For the common design case we consider the test which rejects the null hypothesis for large values of the statistic 1

T3

n

2 v(Xi)P(Y14v/2n =, 1

Y2..i).

This test mimics the one sample paired c o m p a r i s o n set up while incorporating the covariate via the function v. To motivate the following class of tests for the distinct design case, let Qk denote the distribution function of Yk, l , SO that Q k ( t ) = ]" F ( t - ltLk(X))g(x)dx, k = 1,2. with F denoting the distribution function of.tl The null hypothesis implies Q1 = Q_,- while under the alternative hypothesis, Q1 ~< Q2. Thus any distribution free two-sample test of Q1 = Qe against Q1 ~< Q2 based on the data YI.~ . . . . . Y1 // and Y2. t . . . . . Y_,.,. can be used to test the above hypotheses. A prominent such test is the Wilcoxon M a n n Whitney test which rejects for large values of the statistic 1

/ll

W -

itl 2

~ ~ (2I[Ye.j ~< Ya.,] - 1). l11 I 1 2

i -

1 .j -

1

Note that W nlt12i

nl

112

~

Y~ s i g n ( Y n i -

Y2,i)

almost surely.

1.j-1

This test ignores the covariates and one expects that there are tests based on all the observations that have greater power. One way of introducing the covariates in this test is by adjusting the statistic W using covariate-matching. To motivate this let M d ' , x) denote the conditional distribution function of Yk. 1 given Xk, 1 = x. so that M d y , x) = F(y - #k(x)), y e R, k = 1, 2. U n d e r the null hypothesis one has M1 = M:~, while under the alternative hypothesis M10', x)~< M 2 ( y , x ) for all y and x. This suggests using a locally weighted analog of W. A class of such tests can be based on the following statistics:

T4

kf'lln21~ 11 q- i<12 n l n 2 i= 1 j= 1

I (U(XI,i) ~- l'(X2 i))P(Y1 i '

Y2 ilwa(Xl.i

X2.i),

"

where a is a small positive n u m b e r depending on the sample sizes. The corresponding test rejects Ho for large values of T4. We refer to this test as the covariate-matched test.

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ILL. Koul, A. Schick/Journal of Statistical Planning and inference 65 (1997) 293 314

In Section 2 we shall show that under Ho and under some mild additional assumptions on v, O,f, g and a, the statistic Tk is asymptotically normal with mean zero and variance rk, k = 1, . . . , 4. We also provide consistent estimators "~1. . . . . ~¢ of these asymptotic variances, so that the tests q~k = I(Tk > Z~'~k), k = 1, ..., 4, are of asymptotic level ~. Here and below z~ denotes the (1 - c0th percentile of the standard normal distribution. The asymptotic normality of T1 and T2 requires the design density g to be bounded and bounded away from zero on its compact support, ~9 to be Lipschitz continuous and p to be continuous, besides the square integrability of the underlying scores v and ~p. An advantage of the statistics T3 and To is that their asymptotic normality result is valid under more general conditions on the underlying entities. In particular it does not need g to have a compact support nor does it need p to be continuous. It should be perhaps emphasized that the tests based o n T 3 and T o avoid the estimation of the c o m m o n mean function. Cox et al. (1988), Eubank and Spiegelman (1990), Raz (1990), Hfirdle and Marron (1990), Hall and Hart (1990), King et al. (1991), and Kulasekera (1995), among others, address various forms of the above testing problem. All of these papers propose ad hoc tests of Ho against the two sided alternatives Pa ¢ ]A2 that depend on various nonparametric estimators of a regression function. None of these papers discusses any optimality theory. In Section 3 we study the local asymptotic powers of the tests based on T> ..., T¢ for sequences of contiguous alternatives. These alternatives are defined as follows. Fix a measurable function p and let A denote the set of all non-negative measurable functions 15 with ~ 152(x)g(x ) dx < vc. For 15E A, define the local alternatives

1 f nln2

#1 = 1J1,6 = 1"/ -Jr" -15 and H1 "~/ H1 ~- ~2

1 ~nine15.

,//2 = /-12.6 = // -- -H2 "~/H1 -~- t'~2

(1.3)

In the c o m m o n design case these alternatives simplify to /JX = ,//1.5 = ]1 -]- - -

1

15

and

/'/2 = /-/2,,-3 = P

1

15.

Perhaps it is worth mentioning that many existing tests in the literature cannot distinguish the above root-n types of neighborhoods from Ho. Using the Hfijek-Le C a m contiguity theory, we derive an upper bound on the asymptotic power of any asymptotic level • sequence of tests of H0 against the above alternatives. Then we discuss the optimality of our tests. For example, if the c o m m o n error density is normal, then the test based on the statistic 1 n__~l n~1 (Y2,j - Yl,g)wa(X2,i - X l , i ) / nln2 T~ = y nl + n: nln2 i j= is asymptotically optimal a m o n g all tests of the same level against the above alternatives with 15 = g. Note that T* is the test statistic To with v = 1 and p ( x ) = x.

ILL. Koul, A. Schick/Journal of &atislical Planning and ln.lerence 65 (1997) 293 314

29"7

Similarly, if the error density is logistic, then the test based on the covariate-matched Wilcoxon M a n n Whitney statistic

W*

f -nln2~

=

1

~1 ~~

~/ tll + ti 2 n~n2 i :

(2I[Y2.j

<~

j=

YI,i]

1)w,(X2 j

-- Xl,

i)

is asymptotically optimal a m o n g all tests of the same level against the above alternative with fi - ,q. Since Hln 2

1

~

sign(Yl.i

--

Y2, i ) w a ( X 2 , j -

Xl,i)

almost surely,

the test statistic W * is the test statistic T4 with t~ = 1 and p ( x ) - sign(x). Section 4 reports the numerical results of a simulation study for the tests based oll Tg and W * for various choices of error densities, design densities, local alternatives, bandwidths and sample sizes. These simulation results are consistent with the asymptotic theory at the moderate sample sizes considered and fairly stable over the range of bandwidths considered. Section 5 contains general results useful in deriving the above theory. They are generalizations of earlier work of Schick (1987) and are of independent interest. In the sequel, Pa and Ea refer to the probability measure and expectation associated with the local alternatives given at (1.3) above. For the sake of simplicity of notation we suppress the dependence of these entities on the pair (nl, n2). N o t e that P , and Eo correspond to the null hypothesis. T h r o u g h o u t X, s and Co denote independent r a n d o m variables with respective densities g , f and f Also, for any density h, we shall use !1• i1,, and { ",-}j, to denote n o r m and inner p r o d u c t in L 2 ( h ) .

2. Asymptotics under the null hypothesis In this section we derive the asymptotic distributions of our test statistics under the null hypothesis. T h r o u g h o u t we assume that ttl = / t 2 = It for some measurable function t~ and set ek. i

Yk, i - - t t ( X k , j ) ,

j=

1, ... , n k ,

k -

1,2.

(2.1)

We also assume that 0 ~. E U 2 ( X ) =

0
IIv II~ < < ,

=Ep2(S-ao)<

0 < Var(4,(c))

IIqJo II} < ~c.,

~,

with ~bo = t) - E~p(s) = t/J - ( ~ , 1)r. For our asymptotics we let n~ and n2 tend to inlinity and let a depend on nl and n2 in such a way that a --, 0.

ILL. Koul, A. Schick/Journal of Statistical Planning and In/erence 65 (1997) 293 314

298

Theorem 2.1. Suppose additionally that (1) f has zero mean and finite variance; (2) g is quasi-uniform on a closed interval o¢, i.e., g is bounded and bounded away.from 0 on o¢ and vanishes off J ; (3) p is continuous; (4) ~9 is Lipschitz continuous; and (5) (nl + n2)a 2 --+ or. Then the jbllowing hold. In the common design case the test statistic T1 satisfies 1

T1 - - ~ j = l L v(XJ)(~l(el,j) -- ~ ( e 2 , j ) ) -+- Opo(1)

(2.2)

so that T1 is asymptotically normal (0, rx ) with T1 =

Ev2(X)l//2(g)

=

IIv

rl2 II¢'o II}.

Moreover, a consistent estimator of ~l is given by

T1 = 1 L v2(Xj ) 1 L ( I / / ( ~ l , J ) nj=l j=l

ff/(~2,J)) 2"

In the distinct design case the test statistic

T 2 =

-

-

,Jnl +

--

7,

T 2

v(X,,j)t~(el.j)-

j=l

satisfies -- v(Xz,a)tP(ez,j)

nZj = l

+ Oeo(1)

(2.3)

so that Ta is asymptotically normal (0, r:) with

T2 = Var(v(X)g,(e)) = Ilv 112II~' II} - (v, 1}2(@, 1}}. A consistent estimator of t2 is given by ~2 -

1

v2(Xk,j)~12(&.j)

~

n l -[- n2 k = l j =

(1

1

E

V(Xk,j)~l(&,J)

-~- n2 k = l j = 1

ProoL Let 1

1

= a i n f g(t)

and

t~.J

0(x) - - -

2

~

n 1 ~- n 2 k = l

nk

y', w.(Xk,j -- X),

X e 0¢.

j= 1

Then EoO(X) = ~ g(x + at)w(t) dt ~> 2~/for all x c J eventually and supx~j 10(x) - E o 0(x)l = Opo(1) (cf. L e m m a 2.1.2 in Prakasa Rao, 1983). Consequently, Po ( inf O(x) < q) ~

(2.4)

ILL. Koul. A. Schick/Journal 0{' Statistical Planning and Inlbrence 65 (1997) 293 314

29')

F r o m this one concludes that Po(fi # fi) -+ O, where l

"~

....... Z ; = ~ Z~ ~- l Yk,?v~(Xk,j

1

2

~(X) = 11V(n77n:_,~g=1Zj=l

nt

wa(Xk, j

--X)

X ~ .¢.

X))'

Consequently, we only need to prove the desired statements with fi replaced by ft. To stress this we now write gi instead of ~i. First, consider the distinct design case. It follows from (5.15)in Example 5.4 below that sup.,-~, Ifi(x) - tl(x)[ = Oe(,(1). This, the Lipschitz continuity of ~ and the Weak Law of Large N u m b e r s give the consistency of "g2. We are left to prove that D 2 = Opt,( 1 ) w i t h

D2

=

/ nln {/ 1 "' r ( X l , i ) ( ~ ( g l . j ) ~ tll 4- FI2 \~11 ,.= ILl - '/J(elj))

l ~ u(x2..j)(O(ff,2.j ) _//j(,f?2,j))). i12 .j= 1 F o r this we apply Corollary 5.3 with n = n~ 4- n2; ¢i = (El.i, Y l . i ) and c i = ,.'s.'H~ | , ... , ,ql; {n,+j = ( X 2 , j , Y=,3) a n d c , , , + i = v/~;/n~"_ f o r .j 1, . . . . H~ w i t h nln=/(nl + n=); L((x, y), t ) = v(x)+(y - t); and /}(x, y ) = fi(x). One verifies (5.7) with K(x, y) = Cr(x) and C the Lipschitz constant of~b, (5.8) holds as 3~i]_ l ci = 0, and

f o r ,j =

s-

the conditions (5.9) (5.11) are verified in Example 5.4. Thus the assumptions of Corollary 5.3 are met and one can conclude that De = op,,(1). Next, consider the common design case. The consistency of {1 is verified as above. Thus it suffices to prove that 1

"

1

D1 - N,/~ j1'= E u(X j)(I/l(gl .j) --

ff-/(~2.j)) - - ~ x . . . 2 n .i = 1 l'(X

il(~(~q ..i) --)/J(e2. i))

Opt,(1).

F o r this we apply Corollary 5.3 with n = n 1 - tl 2, ~i = ( X l ,

L x,

3'1"

Y2)' t)

:

U(X)(I//(y

1 --

12

Y l . i , Y2,.i), cj = n t) - ~(y= - 0)/\/'5 and fi(x, Yl, 3'2) = fi(x). One verifies

(5.7) with K(x, y) = \ / 2 Cv(x) and C the Lipschitz constant of ~. Since

f ( ~ ( ) , : - fi(x)) - ~(Y2 -/'i(x))).f(3q - ll(x)).l()'2 -- ll(x))dyl dye = 0,

x c .~',

we obtain (5.8). The conditions (5.9) (5.11) follow from Example 5.4 applied with Y j = ½{Yl..i 4- Y2. i). Thus the assumptions of Corollary 5.3 are met and one can conclude DI = op,,(1).

[]

R e m a r k 2.2. The assumption that ~ is Lipschitz continuous can be weakened if we use instead of fi a leave-one-out estimator. M o r e precisely, if one replaces fi(X~, i) in

ILL. Koul, A. Schick/Journalof StatisticalPlanningand InJbrence65 (1997) 293-314

300

the definition of gk.j (see (1.2)) by ~k,j(Xk,j) =

E gm, lW.(Xm, i -- X~,j) Z w,,(X,.,i - Xk,j)

where the s u m m a t i o n is over all indices (re, i), i = 1. . . . . n,,, m = 1,2, with (m, i) ¢ (k,j) in the case of distinct design points and with i :#j in the case of c o m m o n design points, then the Lipschitz continuity of ~k can be weakened to f lO(Y + t) - O(y + s)lZ f (y) dy <<.M I t - sl 2,

s, t ~ ~,

for some constant M. A p r o o f of this can be given along the lines of the previous proof. The next result is immediate. Theorem 2.3. In the common design case n

T3 -- N/ 2n~jE~l v(X j)p(el'j - e2'j) @ °P°(1) so that T3 is asymptotically normal (0, "C3) with T3 = 1 g v 2 ( X ) p 2 ( ~ _ Co) = ½ IIv

~ p,.

A consistent estimate o f z3 is given by

"~3 =--l ~ v2(XJ) 2nl ~ D 2 ( y I , j _ g2,j). nj=l

j=t

To state the next theorem we need some m o r e notation. Let r(t) = E p ( t - e) =

f p(t -

R ( t ) = Ep2(t + ~ - e,o) =

y ) f ( y ) dy,

f p2(t + yl -

t

y2)f(yl)f(yz)

dyl dy2,

h~(x, y) = ½(v(x) + v(x + s))r(y + p ( x ) - tz(x + s))g(x + s),

t ~/t~,

x, y, s ~ [~.

Theorem 2.4. Suppose that nla ~ oc, n2a ~ 0% sup ~ ( v ( x ) + v(x + s))2R(#(x) -- l~(x + s))g(x + s)g(x) d x < o~

(2.5)

d

f o r some ~I > 0 and lira

iiIhs(x, y) -

s~0 3 J

ho(x, y ) 1 2 f ( y ) g ( x ) d x dy = 0.

(2.6)

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ILL. Koul, A. Schick/Journal of Slatistical Planning and MtkJrence 65 (1997) 293 314

Then in the distinct design case the test statistic: Ta, sati,@es Ta,

~/ tl 1 I-t"12 \I~ 1 j T 1

--~

j=/'1

j) ]

+ op,,(1) so that

is asymptotically normal (0. za,) with

Va,

"C4. =

Ep2(X)~j2(X)r2(g)

9

llv,q II,~ IrllT.

=

A consistent estimate of ra` is given by

{a"

2

j

1

u..)- + - -

(Os.-

-

112 j =

1

WhCV¢.? Ui, i

½(b'(Xl.i) + v(X2.j))p(Yl.i

1 "'U i,o = - - E u i . j , FI2 j= 1

-- Y 2 , i ) w . ( X l . i

1 "' = E [Ji,J n l i= l

g°j

and

-- X 2 , j ) ,

Uo.. -

1 ...... E E [Ji. j. 171172 i = 1 .j= 1

Proofi Let K,, d e n o t e the kernel a s s o c i a t e d with Ta,, i.e..

K,,(xl, Yl, xe, 3'2) = ½(v(xl) +

U(X2))j0(y 1 -- y2)Wa(X1 -- X2),

X1,-\'2" Y l , .l:2 • ~ .

D e n o t e the integral a p p e a r i n g in (2.5) by H(s). It follows from (2.5) that

EoUi. I = E o K ~ ( X I . I ,

YI,I,X2.1,Y2.1)=~a

H(at)w2(t)dt=O(a

that K, is a n t i s y m m e t r i c in the 3'2, Xl, Y~ ). Thus, for all positive a,

Note

sense

that

IL

(2.71

K , ( x l , 3 : l , x 2 . y2} :'=

-- K,,(x2,

EoTa, = EoK,,(X1.I, Y1.1, X2.1° Y2,1) = 0 and

k,,(x, y) = EoK,,(x, y, X2.1, X2.1, Y2.1) = - EoK.(X2.1, Y2.1, x, y), Direct c a l c u l a t i o n s show t h a t

k,(x, y) = f h,t(x, y - tl(x))w(t) dt,

x, y • N.

~[hus

t:(X.,.J)'q(Xm..i)r(a,..J) = ko(X.,.i' ¥ " 4 ) '

j = 1. . . . . rim.

m = 1, 2.

N o w write T4 =

~/ , , ~

.2

< j=l

ka(Xl.j

YI,j) ---

~, k,,(X2, j, Y 2 . j )

,7. > 1

+ R.

x, y~_ JR.

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302

where Eo R2 -

1

-Eo[Ka(XI,1, Y1,1,X2.1, Y 2 , 1 ) - k a ( X l , 1 , n 1 -{- F/2

Y1.1)

-~- ka(X2,1, Y2,1)3 2 1

2

nl + ne E ° K " ( X I ' I '

y l , 1 , X2,1, Y2,1) = 0 (

1

(hi + n2)a

)=0(1),

as (nl + n2)a -~ o~. It follows from (2.6) that Eolk,(Xm, 1, Ym, 1 ) - ko(Xm, 1, Ym, 1)] 2 --* 0,

m = 1,2,

(2.8)

and this implies that

In,he ~n,+n2 (1 n~--1 ka(Xl,j, \Tj~'~ =~~~(+/1n2n l~ 1

1 ~,

YI,j) ----g12 J:1- ka(X2,j, Y2,j)

)

j~-'l k°(Xl'j' gl,j) --~221 j~-i n~ k°(X2'j' g2,j ) -}- OPo(1).

Combining the above completes the proof of the first statement. It follows from (2.8) and the Weak Law of Large Numbers that k2,(Xl,j, YI,j) ---- r~ + OPo(1) and -nlj-1

- 1k2(X2,j, Yz,j) = z4 + opo(1). n2j=l

Thus the consistency of 44 follows if we show that -

E ([7/o -- ka(Xl,j, Yl,j)) 2 = OPo(Â) and hi j-1 -

1

n2j=l

(g.,j -Jr-ka(X2,j, Y2,/)) 2

Opo(1);

but these statements follow from the bounds Eo(UI,o -- ka(Xl,i, Y1,1)) 2 ~ 1 EoU12,1 and n2 Eo(~7. 1 -Jr-ka(X2, i, Y2,1)) 2 ~ L E o U 2 , I. n2

These bounds are easily verified after one realizes that

ka(Xl,1, YI,I) = E o ( U I , j I X I , I , Yl,1)

and

ka(X2,1, Y2,1) = -- Eo(Ui, 1 IX2,1, Y2,1)-

This proves the second statement and completes the proof.

[]

ILL. Koul, A. S c h i c k / J o u r n a l Of Statistical Planning and h¢/erence 65 (1997) 293

314

3(}5;

Remark 2.5. If v and g are bounded, t~ is continuous, and r and R are Lipschitz.. continuous, then (2.5) and (2.6) are implied by lim,~o (]It(X + s) It(xj[2,q(x) dx = 0.

3. Asymptotic power considerations In this section we first obtain an upper bound on the asymptotic power of any sequence of asymptotic level-z~ tests of Ho against the sequence of contiguous alternatives specified at (1.3). We also derive the asymptotic power of the above tests at these alternatives and discuss asymptotic optimality of our tests. From now on we shall additionally assume that the error density f h a s finite Fisher information for location, i.e., f is absolutely continuous with a.e. derivative f ' and El20:) = [[ll[~. < cJ,: with

.f'0:) .1"(~:) "

I(:~)-

For 5 ~ A, let A(5) denote the log-likelihood ratio 2

A(6) = Z k

na

2

tlk

~, l o g f ( Y k , j - - ttk.,*(Xk.j)) -- ~, ~ Iogf(Yk.i--t,(Xk.i)). l j-1

k :1./=1

Note that this log-likelihood ratio does not depend on the design density .q. We have the following local asymptotic normality result. L e m m a 3.1. Under (1.3) and the above conditions on .L

wheye

1 II .~ j _. 1

(~(X2. i)l(e2.,i /

is asymptotically normal under Po with mean 0 and variance [l/[l~ I1~ I 2. Proof. This follows from standard arguments (see e.g. HS.jek and Sid'flk, 1967).

Eli

In the c o m m o n design case F(b) simplifies to

1 3(Xi)(l(el j) --l(e2 i)). F(6) - V/2~n .i= 1 "

From L e m m a 3,1 and the N e y m a n - P e a r s o n lemma we immediately get the following result.

ILL. Koul. A. Schick/Journalof StatisticalPlanning and Inference 65 (1997) 293 314

304

Corollary 3.2 (Power bound). Let t ..... be a test of no against H1 such that lim sup Eot ..... <~ nt , n 2

for some 0 < et < 1. Then, for every 6 e A, lim sup Eat ..... <~ 1

-

~(z=

116 I1~ Ill II,),

-

(3.1)

n l , n2

where • denotes the standard normal distribution function and z~ its (1 - ~)-quantile. The next two results are consequences of the L A N Condition and Le Cain's Third L e m m a (see Hfijek and Sidfik, 1967). Corollary 3.3. Let T be a statistic such that

T=

~ ( 1

~ h(X,j,e,,j)

Hi ~- n2 \ H i j~-I

'

l

~ h(X2,j, e2,j))+Opo(l),

n2 j~-I

where h is a function such that h(X, c) has zero mean and finite positive variance r. Then, under Pa, T is asymptotically normal (va, r) where % = Eh(X, e)6(X)l(e). Consequently, a test ch which satisfies 6, = l [ T > z=x/7] + OPo(1) has local asymptotic power //(6) = lira Each = 1 - ¢b(za - ~,(6)H6I]gHII[~),

6 ~ A,

?ll,• 2

where 7(6) is the correlation co(fficient of h(X, c) and 6(X)l(c). Corollary 3.4. Consider the common design case. Let T be a statistic such that 1

"

W -- x ~ H j~-i

h(Xj, el,j, e2.j) + Opo(1),

where h is a function such that h(X, e, Eo) has zero mean and finite positive variance r. Then, under Pa, T is asymptotically normal (va, z) with va = 1 Eh(X, e, eo)a(X)(l(c) -/(Co)). Consequently, a test 4) which satisfies ch = I [ T > z~.x/7 ] + %o(1) has local asymptotic power H(6)= l i m E a c h = 1 - ~ ( z ~ nl ,n2

7(6)H6[l~llllP,~)), 6 ~ A ,

where 7(6) is the correlation coefficient of h(X, c, Co) and

6(X)(l(,:)

- l(co)).

R e m a r k 3.5. Let ch be a test of asymptotic level :~ which has local asymptotic power /1(6) = 1 - ,~(z= - w(6)lla [lolllll,),

6~ A.

ILL. Koul, A. Schick/Journal o.1"Statistical Planning and ln[i'rence 65 (l!)97) 293 314

?05

It follows from Corollary 3.2 that 7(~) ~< 1 for all ,5 • A. If 7(60) = 1 for some ~o • A, then the test q5 achieves the power b o u n d at ~ = do and is thus locally asymptoticall) most powerful a m o n g s t all tests of asymptotic level ~ against the local alternatives indexed by ~ = ,50, (short LAMP~(6~)). If 7(6) ~> 0 for all ~ e A, t h e n / / ( ~ ) ~> ~ for all ~5• A and the test 4 is locally asymptotically unbiased. We shall now use the above results to obtain the local asymptotic power of tests. based on the test statistics T1, . , . , T4 and to determine asymptotically optimal tests.

The common desi¢ln case: First consider the test (]51 = I[T1 > z ~ / ? l ], where ~ is. as in T h e o r e m 2.1. Under the assumptions of Theorem 2.1 the conclusions of Corollary 3.3 are satisfied with T = TL and h(X,~:)= t~(X)~Po(~:). The correlatior coefficient of c(X)O0(c) and /i(X)I(~:) is )'1(~$) -

@, 6),,

~0, l)r

(5~A.

I1~,I1~ I1~ tl, II4'0 llf ILlllf' Hence 41 has asymptotic level ~ and its local asymptotic p o w e r / / 1 is given by //~ ((~) = I - 4,(z~ - ~'1(,~)116II. II/llt),

e3e A.

This test is locally asymptotically unbiased if (~/Jo, l)/. >~O. Of course, if we know ./'and if I is Lipschitz continuous, then we can take ~ = 1 and get ( 0 , / ) i > 0. As 0 is Lipschitz-continuous by assumption and EI(~:)= 0, we calculate (~o, l ) t = E0(~:)I(~:) = - j' ~P(y)f'(y) dy = .(O'(Y)f(Y) dy = E~p'(l:). Thus 4~ is locally asymptotically unbiased if and only if E0'((:) ~> 0. The latter holds if 0 is also nondecreasing as in this case ~'(t:) ~> 0 almost surely. Thus we r e c o m m e n d the use of a nondecreasing i f / i s unknown. It follows from the C a u c h y Schwarz Inequality that 71(~3)= 1 if and only if (~(X) = by(X) and tP0(e,) = cl(f:) almost surely for positive constants b and c. Thus, if f i s k n o w n and I is Lipschitz-continuous, we can take ~/J = cl for some positive c and obtain that the test q~l is LAMP~(bt,) for every h > 0. If f is unknown, we can choose a density j~ which has a nondecreasing and Lipschitz-continuous score function /o and set ~ = Io. The resulting test q~l will then be LAMP~(bv) for all positive h if f =.[;~. For example, if ~p(~:) = c, then 01 is L A M P~(bv) for all b > 0 and every normal density f with mean 0. If t~(c,) = (1 e x p ( - c)/(1 + e x p ( - c)), then the test ~/)1 is LAMP~(br) for all b > 0 and for the logistic density f Next consider the test 4)3 = I[T3 > z~x/"~3] with f3 as in T h e o r e m 2.3. Then the conclusions of Corollary 3.4 are satisfied with T = T 3 and h(X, ~, c0) - t:(X)f~(c -- ~:o1, and the correlation coefficient of c(X)p(r. - co) and (5(X)(/i~:) - I(~:o)) is

.,3(,5) = x/2~<,-,~ 1> I , 0 •/i. Iir II, I1~ I1~,~/p, I[lll j. Consequently, 43 has asymptotic level ~ and its local asymptotic power I/3 is given by //3((~)

=

1 - - qb(Z~ - -

),3(6)1[(5110i1111~.), ~3• A.

306

ILL. Koul, A. Schick/Journal of Statistical Planning and Inference 65 (1997) 293 314

Thus the test 05a is locally asymptotically unbiased if (r, l } f >~ O. Again, this is the case if r = l or if r is nondecreasing and Lipschitz-continuous. The former condition m a y not be possible even if f is k n o w n as p is required to be odd. The latter condition can be guaranteed by choosing p to be nondecreasing and Lipschitzcontinuous. As p , = Ep2(~: - Co) = E(r(e) - r(eo)) 2 + Po = 2 IIr II} +

po

with Po = E(p(e - C,o) - r(e) + r(eo)) 2, one obtains that

o i / 211rll} 7~(a) = Ilvllol[611o IIr [Is Illl[s ~/ 211,'11}+ po

9

,SEA.

I f p o > 0, then 73(~) < 1 and q~3 c a n n o t be LAMP=(6) for any a. If f has finite variance we can take p(e) = e and obtain P0 = 0 and r(e) = g - Ee. In this case we see that 73(6) = 1 if and only if 6(X) = b y ( X ) and l(e,) = c(e - E(c,)) almost surely for positive constants b and c. The latter can only happen if f is a normal density. This shows that if p(e) = e, and f i s a n o r m a l density, then 053 is LAMP~(bv) for all positive b.

T h e distinct design case. First consider the test (~2 = I I T 2 > z~ x/'~2 ], where ~2 is as in T h e o r e m 2.1. U n d e r the assumptions of T h e o r e m 2.1 the conclusions of Corollary 3.3 are satisfied with T = T2 and h(X, e,) = v(X)4'(e) - Ev(x)4'(e). Here the relevant correlation coefficient is (v, 6}~(4', 1}f

72(6) = ~

r16 I1~ Illlls

with

~2 = V a r ( v ( X ) ~ ( e ) ) .

Hence (~2 has asymptotic level ~ and its local asymptotic p o w e r / / 2 is given by /72(~ ) = 1 -

4~(z~ -

?'2(~)11611o11/11s), Oe

A.

This test is locally asymptotically unbiased if ( ~ o , 1}f >~ O. As shown above this can be always achieved by taking 4' to be nondecreasing. Thus we r e c o m m e n d the use of a nondecreasing 4' if f is unknown. As El(e) = 0 we see that ( ~ , l } I = (4'0, l } f and get

7~(6) = ?'1 (~)

IIv I1~II 4,o IIs 2 ~/Ilvlo/14'o IP} +

(E4'(~,))2Var(v(X))

Thus, if E4'(e) = 0, then 72(~) = 71(~) for all 6 ~ A. F o r example, if f i s k n o w n to be symmetric a b o u t zero and 4' is taken to be odd, then we obtain E4'(e) -- 0. Of course, in view of condition (1) of T h e o r e m 2.1, we can always guarantee E 4 ' ( e ) = 0 by choosing 4'(g) = ~. If f is k n o w n and I is Lipschitz-continuous, we can take 4' = I and obtain that the test (/)2 is LAMP=(bv) for every b > 0. If f is unknown, we can choose a density f0 which has a nondecreasing and Lipschitz-continuous score function

lt.L. Koul, A. Schick/Journal 0"/"Statistical Planning and lnlbrence 65 (1997) 293 314

~07

/o and set ~ = lo. The resulting test 02 will then be LAMP~(bv) for all positive h if .l'=./i~. F o r example, if O(e) = ~:, then 4)2 is LAMP~(bv) for all b > 0 and every normal density f with mean 0. If ~(c) = (1 - c x p ( - c})/(1 + e x p ( - c)L then the test ~)_, is LAMP=(hv) for all b > 0 and for the logistic density .[i Finally consider the test q54 = I[T4 > z~-v/f4], where "~4 is as in T h e o r e m 2.4. Under the assumptions of T h e o r e m 2.4 the conclusions of Corollary 3.3 are satislied with T = T4 and h(X, c) = c(X)g(X)r(cl; the correlation coetficient of c(X)0tX)r(c) and 3(X)/(c)is ;'4(0) -

(t:g,a)~ (r,/)j. I!.qv!l~ 1i6 ll~ lit li,~ ll/ll.r'

~5e A;

hence 4~4 has asymptotic level ~ and its local asymptotic power

H4(b) =

I -

40(z= -

5,4(,5)[I,511,,

,Ii!.r ),

b~

[14 is given by

A.

This test is locally asymptotically unbiased if (~0, 1)l >~ 0. As shown above this can be always achieved by taking p to be also nondecreasing and Lipschitz-continuous. Thus we r e c o m m e n d the use of such a p. ,Arguing as above, we see that the test 4~4 is LAMPAbvg) for every t, > 0 if r(c) = el0:) almost surely for some positive c. F o r the normal density we achieve this by taking p(c) = ~; for the logistic density we achieve this by taking p(c) = sign(c). Thus the test q54 with p(c) = sign(c) is LAMP~(cv,q) for all positive c if./is the logistic density. In particular, the test based on covariate-matched Wilcoxon M a n n Whitncy statistic T4 = W* corresponds to ~/).~ with p ( c ) = sign(c) and c = 1, and is thus L A M P d c q ) for all positive c if f is the logistic density. Similarly, the test 4)a with p(c) = ~: is L A M P ~ ( c w ) for all positive c if f is a normal density,. Of course, if q is k n o w n and b o u n d e d away from 0 on its support, then we can take v = 3o/.q to optimize the local asymptotic power for the local alternatives with ~5 ~,.

4. Simulations In this section we shall report the results of simulations performed for the covariate-matched Wilcoxon M a n n Whitney test, i.e., the test q)4 with v(x) = 1 and p(x) = sign(x), and the covariate-matched two-sample t-test, i.e., the test (/)4 with v(x) = 1 and p(x)= x. We studied the performance of each test for two different design densities 9, namely uniform on (0, 1) and exponential with mean 0.5:

~ll(x)--I[O
g2(x)=2exp(

2x)l[x>O];

three error densities, namely standard normal, double exponential and logistic: .li(x) =

,--1 exp(-- ½x2),

N/"2K

.]i(x) = ½ exp(-- Ixl),

exp( - x)

.

.13(x)= (1 + e x p ( - x)) 2'

ILL. Koul, A. Schick/Journal of Statistical Planning and Inference 65 (1997) 293 314

308

and several choices of bandwidths and sample sizes, n a m e l y bandwidths a = 0.3, 0.4, 0.5, 0.6 for nl = n2 = 30 and bandwidths a = 0.2, 0.3, 0.4, 0.5 for nl = n2 = 50. In all cases we t o o k / ~ ( x ) = 3 arctan(5x - 2) and w(x) -- ~16 (1 X2)2I[Ix] < l]. -

-

In the tables below we report the M o n t e Carlo power based on 5000 repetitions for the a b o v e choices and the following four local alternatives:

61(x)=0,

62(x)= 1, ~53(x) = 2 x

and

6~(x)=x+l.

We used ~ = 0.05. The simulations were d o n e in S-plus. Table 1 reports M o n t e Carlo powers for the covariate-matched W i l c o x o n - M a n n W h i t n e y test for the aforementioned choices of

9 , f 6, a, nl and n 2. Table 2 does the

Table 1 Covariate-matched Wilcoxon Mann-Whitney Test 91

fl

f2

n~ = n2 = 30 0.4

0.5

0.6

0.2

0.3

0.4

0.5

61 62 63 64

0.050 0.203 0.206 0.357

0.046 0.220 0.222 0.358

0.046 0.192 0.198 0.329

0.046 0.180 0.188 0.311

0.048 0.214 0.215 0.383

0.050 0.224 0.232 0.392

0.041 0.209 0.219 0.366

0.046 0.206 0.218 0.350

0.050 0.252 0.252 0.429

/51

0.042 0.174 0.179 0.289

0.045 0.165 0.169 0.274

0.050 0.175 0.178 0.276

0.047 0.159 0.165 0.252

0.051 0.182 0.183 0.308

0.048 0.182 0.186 0.303

0.047 0.177 0.183 0.295

0.041 0.161 0.169 0.279

0.050 0.218 0.218 0.365

0.040 0.117 0.119 0.189

0.045 0.128 0.129 0.198

0.050 0.125 0.127 0.190

0.052 0.131 0.136 0.192

0.044 0.126 0.125 0.188

0.049 0.130 0.132 0.197

0.048 0.124 0.127 0.194

0.051 0.130 0.130 0.200

0.050 0.143 0.143 0.218

62

,51 /52 63 /54

92

fl

n1 =

nl = nz = 50

30

H4(6)

0.4

0.5

0.6

0.2

0.3

0.4

0.5

64

0.041 0.177 0.108 0.251

0.040 0.181 0.113 0.252

0.046 0.168 0.113 0.243

0.043 0.170 0.114 0.239

0.041 0.187 0.105 0.259

0.047 0.195 0.117 0.268

0.041 0.184 0.112 0.264

0.049 0.194 0.124 0.266

0.050 0.212 0.111 0.279

i51 62 /53 64

0.043 0.160 0.097 0.215

0.044 0.149 0.096 0.206

0.051 0.167 0.110 0.219

0.046 0.153 0.108 0.207

0.041 0.163 0.099 0.217

0.051 0.169 0.108 0.230

0.048 0.154 0.101 0.207

0.049 0.163 0.111 0.213

0.050 0.185 0.102 0.240

61

0.039 0.118 0.081 0.148

0.044 0.115 0.085 0.153

0.045 0.116 0.088 0.150

0.045 0.115 0.089 0.151

0.041 0.114 0.077 0.149

0.045 0.113 0.080 0.145

0.045 0.121 0.089 0.154

0.050 0.121 0.087 0.160

0.050 0.126 0.082 0.154

61 ,52

/52 f3

n 2 =

0.3

/53

J2

//4(6)

0.3

/53 /54

f3

nl = n2 = 50

/53 64

H.L. Koul, A. Schick/Journal q/" Statistical Planning and h#erem'e 65 (1997) 293 314

309

Table 2 Covariatc-matched

.1~

t[~

.12

t[~

II4(D)

ni = ~12 = 5 0

(1.3

0.4

0.5

0.6

0.2

0.3

(1.4

{1.5

31 c$, 5 D4

0.048 0.225 (/.22(/ 0.382

0.043 0.210 0.210 0.365

0.046 0.205 0.207 0.345

0.049 0.194 (/.193 0.315

0.(/51 0.248 0.246 0.411

0.051 0.237 0.236 0.402

0.044 0.220 0.220 0.385

0.056 (t.221 0.224 {1.365

0.050 0.259 0.259 (1.442

~$~
0.044 0.162 0.161 0.267

0.047 0.161 0.161 0.253

0.050 0.158 0.156 0.251

0.054 0.156 0.157 (/.243

0.047 0.16(I 0.160 0.259

0.052 0.170 0.173 0.271

0.051 I).168 0.169 0.264

0.048 0.160 (/.160 0.249

/).(t5/) 0.174 0.174 (/.279

~$~ ~5~ D ~t4

0.048 0.128 0.126 (I.192

0.047 0.128 0.126 0.194

0.050 0.126 0.127 0.186

0.049 0.127 0.127 0.181

0.046 0.123 0.123 0.178

0.046 0.129 0.130 0.196

0.053 0.134 (I.135 (/.2(t3

0.053 (1.134 (/.133 0.197

0.050 0.137 0.137 0.207

q~

ft

t-test

nl = n2 = 30

gi

f~

two-sample

~11 = n2 = 30

llLlfil

~7, = ~1~ = 50

(1.3

0.4

0.5

(1.6

(1.2

0.3

0.4

)5

<$~ D, <53 ,54

0.047 0.202 0.122 0.280

0.(147 0.196 0.125 0.279

0.054 0.184 0.128 0.260

0.047 0.164 0.114 (/.232

0.(/44 0.204 0.114 (I.281

0.(/45 (/.205 0.122 0.293

0.046 0.203 0.128 (!.287

(t.(/52 0.184 (/.124 0258

0.05o 0.21~ (/.113 (!.2s7

~$1 <52 D3 D~

0.043 0.154 0.1(/4 0.205

0.048 0.152 0.103 0.199

0.051 0.143 0.102 0.190

0.052 0.155 0.115 0.204

0.040 0.146 0.090 (/.188

0.051 0.159 0.108 0.205

0.051 0.158 0.110 (/.2:12

0.047 0.139 0.096 0.188

11.05( 0.151 0.09(, I).19(,

~5~ D, ~53 D4

0.044 0.114 0.083 (/.147

0.043 0.115 0.083 0.146

0.048 0.116 0.083 0.153

0.050 0.123 0.095 0.161

(/.(/44 0.116 0.078 0.145

(t.047 (/.118 0.084 (t.158

0.050 0.128 0.092 (/.162

//.05(1 0.131 (/.097 (/.165

0.05(, 0121 0,1/8(~ {}14-

same for the covariate-matched t-test. As comparison we list the appropriate local asymptotic power in the last column of each table. The numerical values of the local asymptotic power /7~(~$) for the local alternative d appearing in the last column of each table were calculated using the formula

H4(~$) = 1 - q~ z~ -- A({$)x/~l ~-

re(x)

dx

I

for the covariate-matched Wilcoxon M a n n - W h i t n e y test and H,,(($) = I -- 4,(z~ -- A(d)/crs,)

H.L. Koul, A. Schick/Journal of Statistical Planning and Inference 65 (1997) 293 314

310

for the covariate-matched two-sample t-test, where crz is the standard deviation of the error density f and A(6) = ~oo f i ( x ) g 2 ( x ) d x ( ~ 9 3 ( x ) d x ) 1/2 •

It can be seen from the tables that the fraction of rejections in 5000 repetitions are already fairly close to their asymptotic values at the moderate sample sizes chosen. The power performance is fairly stable across the range of the chosen bandwidths. As expected from the asymptotic theory, the covariate-matched W i l c o x o n - M a n n -Whitney test clearly outperforms the covariate-matched two-sample t-test for the double exponential error density f2. lf9 = g~ and ~ = 62, then the covariate-matched two-sample t-test is asymptotically optimal for the normal density f~, while the covariate-matched Wilcoxon M a n n - W h i t n e y test is asymptotically optimal for the logistic error density f3. However, the differences in the asymptotic powers are small. For 9 = g~ and ~ = 62, the covariate-matched two-sample t-test has better Monte Carlo power than the covariate-matched Wilcoxon Mann Whitney test at f = f l , while their Monte Carlo powers seems to be the same a t f = f 3 . The Monte Carlo powers for both tests are much closer to their asymptotic powers for the design density 92.

5. Technical details

Let (~2, ~1, P) and (S, ffi, Q) be two probability spaces and let ~1. . . . , ~, be measurable functions from O to S which are i.i.d.Q. Let cl, ..., cn be real numbers satisfying n I Cj 2 1. Set Zj= T =

cjh(~j), j=l

where h(s) = h(s, ~1, ... , ~0), s e S, for some measurable function h from S "+ 1 to N. We shall now give conditions that imply T = Op(1). The following lemma is a generalization of Schick's (1987) L e m m a 3.1. His lemma uses cj = 1 / ~ . Inspecting its proof reveals that it can be modified as follows to cover non-constant weights. L e m m a 5.1. T h e f o l l o w i n 9 conditions imply T = op(1):

j=l

cjfa(,)dQ(s)= Op(1),

E f h 2 ( s ) dQ(s) ~ 0,

ci[~(~j) - ~j(~j)] = op(1), j=l

(5.1) (5.2)

(5.3)

H..I. Koul, A. Schick/'Journal

i Eflh.j(s)-/~(s)l dO(s) 2

i =

of Statistical Planning and ln/erence 65 t1997) 293 314 --, 0,

311

(5.4)

1

w h e r e hlj(S) = E ( h ( s ) I ~l'~ "'" ' ~j

1' ~ j + l . . . . .

~gl)" S ~ S, j = 1. . . . .

n.

Remark 5.2. As in Schick (1987) one can show that the following condition is equivalent to (5.3) and (5.4). There are measurable functions hi from S" to (R such thal ii

AI=

c . i [ h ( ~ i ) - hj(¢j)] = Op(1),

{5.5)

i= 1 gl 2

--

i E ;Ih.j(s)j=

h(s)l 2 d Q ( s ) - * -

0,

(5.(1,

1

with hi(s} = hi(s, ?,1. . . . . g j - l , ~ j + l

. . . . .

~n), S e S, j = 1. . . . . n.

Corollary 5.3. Let L be a measurable function ji'om S × R into g~, fi be a measurable .function ji'om S to ~ and fi be an estimate of ft. Suppose that

[g(s,x)--g(s,y)l<<.M(s)lx-yl,

seS, x.y~R,

15.7)

for some Q-square integrable function M and i .i=

1

c i ~ 'tL(s, fi(s)) - L(s, fi(s))} dQ(s) --- Op(1), J

F, = sup EI/}(s) -- [3(s)l 2 ~ 0, s~S

V2 = i

(5.:g) (5.9)

ID(~j) -/~j(~j)l 2 = %(1),

(5.101

sup E[]~(S)-

(5.11)

j- 1 /'3 = i

~j(S)l 2 --~ 0

.j= 1 seS

Jot estimates fii = fi.i(', ~1 . . . . . ~j 1, ~.j ~ l . . . . . ?~,) qf fi. Then 11

~.j(L(~j, fi(~01 - L(~j,/~(~jl)) -- %(1t. j= 1

Proof. In view of L e m m a 5.1 and R e m a r k 5.2 it sufficies to verify (5.1), (5.2), (5.5) and (5.6) with h(s) = L(s, fi(s)) - L(s, fi(s)) and hj(s) = L(s, fij(s)) - L(s, fi(s)). Of course, (5.1) is assumed. The other conditions follow from the bounds E ~ ~2 dQ <~ 5 M2 dQF~, A2~ <~ E ~ = I c~M2(~j)F2 and A 2 ~ f M 2 dQ F 3. [] E x a m p l e 5.4. Consider the regression model

Yj=tt(Xi )+~,

j = 1. . . . ,n,

H.L. Koul, A. Schick/Journal of Statistical Planning and Inference 65 (1997) 293-314

312

where the e r r o r variables el . . . . . e, are i n d e p e n d e n t with zero m e a n s a n d c o m m o n v a r i a n c e a 2, the c o v a r i a t e s are i n d e p e n d e n t with c o m m o n density 9 a n d are independ e n t of the errors. W e a s s u m e t h a t # a n d g are as in T h e o r e m 2.1. Let n o w ~/= k i n f ~ ~ g(x) a n d p u t __

fi(x)

1

n

n2i=l •

Yjwa(Xj-

x)

(~E;=,w,(Xj_x))v

~

and

ill(x)

=

1

n~,j#i

Yjwa(Xj__ x)

(1Ej#i%(Xj_x))v~

l

,

XCz~g- '

for s o m e positive n u m b e r a d e p e n d i n g on n such t h a t a --+ 0 a n d na 2 --+ ~ as n --+ oo. Then sup E[fi(x) - #(x)[ 2 --* 0,

(5.12)

xel

Elfij(Xj) -- fi(Xj)[ 2 --+ O,

(5.13)

sup Elfi(x) - fij(x)l 2 -, O,

(5.14)

j=l

j = 1 xe.¢

and sup ]fi(x) - #(x)l = Op(1).

(5.15)

xe.e

Let Ir#JIo~ = s u p x ~ I/~(x)l a n d Ilwjlo~= sup,~aw(t). T h e n E ( Y ] ) ~< a 2 + Irpdr~, for j = l . . . . . n. It is easily c h e c k e d that [f~(x)-f~j(x)l<(If~(x)l+lgjl)llwll~/ (naq). T h u s (5.14) follows from (5.12) a n d n a 2 ~ oo. Similarly, I f i ( x ) - fij(x)J <~ (IYjI + Ifij(x)l)llwll~/(naq), a n d (5.13) follows from (5.12), (5.14) a n d na 2 ~ . Let n o w

f ( x ) - ~ 2 ~ = , # ( X j ) w . ( X j - x) 1

n;W~jj:~-))-V7

and

O(x) =

(~2j=

1_

Z %(Xj-

x).

Hj=I

Then Ifi(x) -- #(x)l <

sup

s~t~J

[#(s)-p(t)]+

O(x) jl/tljo~ 0 ( x ) v q

1

'

x 6 J.

I.~ rl < a

Since # is u n i f o r m l y c o n t i n u o u s on the c o m p a c t set J , we thus o b t a i n from an a r g u m e n t similar to (2.4) that

l i m =s u p E s uxp~[ f i (0x ) - p ( x ) 1 2 <" " 2 l t p l l Z l i m . s u p P ( i n f O\(:x, ~) <, ~ l )

(5.16)

Finally, c o n d i t i o n i n g on X1, . . . , X , shows that

El~(x) - ~(x)l 2 <

g g l2W .2( X 1 - - X ) < nt] 2

C o m b i n i n g (5.16) a n d (5.17) yields (5.12).

o-2HwH2Rllglr~ naq 2

(5.17)

H.L. Koul, A. Schick,/Journal of Statistical Planning and lnfi:rence 65 (1997) 293 314 In v i e w of (5.16), we o b t a i n (5.15) if we s h o w t h a t IlAli,

31 :;

= sup,.~ : A ( x ) = op(l~.

where

A(x)=OTvO(x))(fi(x)--

fL(x)) - -

1

~_ e.?v.(Xj - x),

xe,¢.

H j--1

Let L d e n o t e the l e n g t h of the i n t e r v a l / . / i n t o m i n t e r v a l s [1 . . . . .

a n d m be the i n t e g e r p a r t of n ~2. D i v i d e

I,, of e q u a l l e n g t h s L / m a n d d e n o t e their m i d p o i n t s by

x l . . . . , x,,,. T h e n

IIA li,

LK sup IA(xi)] + sup sup IA(x) - A(xi)l ~ sup IA(x~)l + - sup B{\e), i xel, i Ram i

(5.18) w h e r e K d e n o t e s the L i p s c h i t z c o n s t a n l o f w a n d

B ( x ) = - - ~ Icjl/l-lXj - xl < a + L / m ] , na .j = 1

xEI.

O n e verifies t h a t E A Z ( x ) <~

20"2 I1w [12~ II~ [I ~ Ha

and

E(B(x)-

E B ( x ) ) 2 <~

20"2(. + L/'m)II~J II t1(3 2

(5.19) Since na 2 ~ vc, we h a v e na/m --+ Gc a n d ma ~ ~_,. This, the C h e b y s h e v I n e q u a l i t y a n d the b o u n d s in (5.19) yield t h a t supi IA(xg)l = Op(1) a n d supi IB(xi)l = supi IgBIx~)l +

Op(1)

O p ( l ) . In view of(5.18) a n d ma--~ o~, we o b t a i n the desired result IIA !l,

op(1).

Acknowledgements T h e first a u t h o r w o u l d like to t h a n k P a u l S p e c k m a n for d r a w i n g a t t e n t i o n to the p r o b l e m d i s c u s s e d in this p a p e r .

References Cox. D.. Koh, g., Wahba, G., Yandell, B.S., 1988. Testing the (parametric) null model hypothesis m (semiparametric) partial and generalized spline models. Ann. Statist. 16, 113-119. Eubank, R.L., Spiegelman, C.H., 1990. Testing the goodness-of-fit of a linear model via nonparamet"ic regression techniques. J. Amer. Statist. Assoc. 85, 387 392. Hall, P., Hart, J.D., 1990. Bootstrap test for difference between means in nonparametric regression. J. Amer. Statist. Assoc. 85, 1039 1049. H/tjek, J., Sid/~k, Z., 1967. Theory of Rank Tests. Academic Press, New York. H~irdle, W., Matron, J.S., 1990. Semiparametric comparisons of regression curves. Ann. Stalist. 18.63 ~';9.

314

H.L. Koul, A. Schick/Journal of Statistical Planning and Inference 65 (1997) 293 314

King, E., Hart, J.D., Wehrly, T.E., t991. Testing the equality of two regression curves using linear smoothers. Statist. Probab. Lett. 12, 239 247. Kulasekera, K.B., 1995. Comparison of regression curves using quasi-residuals. J. Amer. Statist. Assoc. 90, 1085-1093. Prakasa Rao, B.L.S., 1983. Nonparametric Functional Estimation. Academic Press, Orlando. Raz, J., 1990. Testing for no effect when estimating a smooth function by nonparametric regression: a randomization approach. J. Amer. Statist. Assoc. 85, 132 138. Schick, A., 1987. A note on the construction of asymptotically linear estimators. J. Statist. Plann. Inference 16, 89 105. Correction: (1989), 22, 269 270.