Journal of Statistical Planning and Inference 16 (1987) 63-74 North-Holland

63

A S Y M P T O T I C N O R M A L I T Y OF U - S T A T I S T I C S B A S E D O N T R I M M E D S A M P L E S *

Paul JANSSEN

Limburgs Universitair Centrum, Universitaire Campus, B-3610 Diepenbeek, Belgium

Robert SERFLING

Department o f Mathematical Sciences, The Johns Hopkins University, Baltimore, MD 21218, USA

No~l VERAVERBEKE

Limburgs Universitair Centrum, Universitaire Campus, B-3610 Diepenbeek, Belgium

Received 9 January 1986; revised manuscript received 1 May, 1986 Recommended by M.L. Puri

Abstract: Let Xnl <_ ... <_Xnn be an ordered sample of size n. We establish asymptotic normality of U-statistics based on the trimmed sample Xn, lunl+~<'" <<_Xn, n_l#nl where 0<a, fl<½. This theorem and its multi-sample generalization are illustrated by various statistics of importance for robust estimation of location, dispersion, etc. This unifies the flexibility of the class of U-statistics and the classical principle of rejection of outliers.

AMS Subject Classification: Primary 62E20; Secondary 62G35.

Key words and phrases: U-statistics; Trimmed samples; Robust inference; Nonparametric esti- mation.

1. Introduction

For robust estimation o f location, the ordinary sample mean is too sensitive to outliers. A classical and successful alternative is the trimmed mean, for which asymptotic normality was established by Bickel (1965). As discussed by Bickel and Lehmann (1975), for example, the trimmed mean remains relatively efficient with respect to the untrimmed mean even in the absence o f outliers.

Viewing the trimmed mean as simply the ordinary mean defined on a trimmed sample, we are motivated to consider other common statistics as well in this regard.

* Research supported by the U .S . Department of Navy under Office of Naval Research Contract No. N00014-79-C-0801.

0378-3758/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)

64 P. Janssen et al. / Asymptotic normality o f U-statistics

In this paper we study U-statistics in such fashion. The class of U-statistics, intro- duced by Hoeffding (1948), contains a wealth of statistics of interest in their own rights and also contains statistics which serve as approximations to statistics of more complicated type. A very significant broadening of the scope of robust statistical inference is achieved, therefore, by consideration of the class of U-statistics on t r i m m e d samples.

Specifically, the statistics we treat are defined as follows. Let X~, ..., X n be an i.i.d, sample from a d f F , let Xnl <- "" <--lYnn denote the ordered Xi ' s , let O< a, f l<½, and put nc,,B = n - [an] - [fin]. Let h(x l , ... ,Xm) be a 'kernel ' assumed (without loss of generality) to be a symmetric in its arguments. For each such kernel we consider the associated U-statistic defined on the (a, fl)-trimmed sample, i.e.,

UnaB= 2 h ( X n i . , ' " , X n i m ) , ( 1 . 1 ) c,,,,B

where C , aB denotes the set of m-tuples {(il , . . . , ira): [an] + 1 <_ i I < .... < i n < n - [fin]}.

(The cases a > 0, p = 0 and a = 0, B > 0 could also be considered but will be omitted for simplicity. The case 0: = fl = 0 corresponds to ordinary U-statistics based on full samples, for which there is already an extensive literature (see, e.g., Serfling (1980), Chapter 5).)

For m = 1 and the kernel h ( x ) = x , (1.1) gives the tr immed mean

n - [/~n]

f ( n a# = n gB 1 2 X , i , (1.2) i = [ an ] + 1

treated by Bickel (1965). For m = 2 and the kernel h ( x l , x 2 ) = I { x l +x2>0} we ob- tain a version of the Wilcoxon one-sample statistic treated by Saleh (1976). Evident- ly, these and only one or two other cases of (1.1) have been previously studied in the literature, despite the abundance of natural possibilities. For example, with m = 2 and h ( X l , X 2 ) = ½ ( X l - x 2 ) 2, we obtain the statistic

n - [ # h i

SnZa#=(n~B - 1) - l ~ ( X n i - f(nce#) 2, (1.3) i = [an] + 1

a quite natural robust analogue of the classical sample variance. This is included in a class of statistics which has been treated by BickeI and Lehmann (1976), Theorem 5, in the case of symmetric F. Also a Winsorized version of (1.3) has been treated by Jaeckel (1971) (see also Bickel and Doksum (1977), p. 375).

Other methods of using trimming to produce robustness have appeared in the literature. For example, see the 'doubly tr immed standard deviation' of Bickel and Lehmann (1976), which is formulated for the case of symmetric F about k n o w n

center p and which involves ordering of the values [Xi- /~[ rather than the X i ' s .

Also see the ' t r immed standard deviation' introduced by Bickel and Lehmann (1979) and treated theoretically by Janssen, Serfling and Veraverbeke (1984). Further, con- sider the ' t r immed U-statistics', produced by trimming on the basis of ordered values of h(Xi , , ...,X/m), which are a special case of the generalized L-statistics

P. Janssen et al. / Asymptotic normality of U-statistics 65

treated by Serfling (1984). The present development, in which the trimming is ap- plied directly to the sample values Xi, is perhaps the most natural and reasonable way to implement a principle of rejection of outliers.

In Section 2 we establish asymptotic normality for statistics of form (1.1) (under suitable regularity conditions), thereby extending and unifying Bickel's result for the trimmed mean and Hoeffding's result for the case of untrimmed samples. We also cover the multi-sample case. The main results are given by Theorems 2.1 and 2.2.

Our method of proof utilizes recent work of Randles (1982) on U-statistics based on kernels having unknown parameters.

The remainder of the paper (Section 3) treats important examples: moment-type statistics, Wilcoxon-type statistics and Gini-like measures of location and spread.

2. General theorems

Our asymptotic normality result for the statistic UnaB defined by (1.1) will in- volve mean and variance parameters/zaB and a2B defined as follows. Correspond- ing to the given kernel h, we define for x, u, o e IR

I: m - I

h(X, X I , " ' , X m - 1 ) I-[ dF(xi), i = l

m

h(x , ," ' ,Xm) 1-I dF(xi), i = l

A (u, u) = Var g(X; u, o),

A ( u , o ) = - g ( u ; u,o), B(u,o)=g(o; u,o).

Then we define

PaB = P(F- I ( a ) ,F- l ( 1 - fl))

and

a a2# = m 2 { A aB + 2alz~B A a# - 2 flluaB B aB + a(1 - a )A 2B

+ fl(l - fl)B2# + 2a~Aa#BaB },

where AaB=A(F- l (a) ,F-1(1 -p ) ) , etc.

Assumptions. (A) F has a density f which is continuous and positive at F - 1 (¢~) and F - 1 ( 1 - B ) and is bounded in some J-neighborhoods of F - l ( a ) and F - l ( 1 - f l ) .

(B) For some a < F - l ( a ) and b > F - l ( 1 - ] 3 ) :

sup Ih (Xl,-.-,Xm)l-- g o < oo. a<--Xl, . . . , x m ~ b

(C) The function

66 P. Janssen et al. / Asymptotic normality of U-statistics

l I m - ! F- ' ( l - f l ) F-t0-,f) h(x, xl, ,Xm 1) l-I g(x) . . . . . . . _ JF-I(a) JF-l(a) i= 1

is continuous at F - 1 (a) and F - 1 (1 -/~).

d F ( x i )

Theorem 2.1. L e t Una,f be g iven b y (1.1). A s s u m e c o n d i t i o n s (A), (B) a n d (C) a n d

tha t ¢r2,f > O. Then

nl /2(Una, f_~a , f ) d N(0, 2 ' aa,f).

The proof will utilize a series of lemmas involving certain U-statistics closely related to U,a,f. For the given kernel h, and for each u < o ~ ~, we define an asso- ciated kernel

h(x l , . . . ,Xm; u, o) = (1 -- f l--Ot)-ml{U<--Xl, . . . ,Xm<-O}h(x 1, ... ,Xm)

and we denote by Un(u, o) the o r d i n a r y (i.e., defined on the full sample) U-statistic based on this kernel.

Then we have

(n) Una,f : (1 - tff - ~ ) m Un ( gn , [an] + I, Xn, n - [,fin] ) (2.1)

and we readily obtain:

Lemma 2.1. Un d e r c o n d i t i o n (B),

u.a , f = u . ( x . , t a.l + 1, x . , . _ t,f.l) + Op (n - 1 ).

Next we show that the leading term in (2.2) may be approximated by

Tna,f = U. ( F - l ( a ) , F - I(1 -- B)) -- l ta , f + u(X. , tanl + 1, Xn, n - [,f"l)"

Lemma 2.2. Under conditions (A) and (B),

T.aB= Un(Xn,[an]+ l,Xn, n-[Bn]) + Op(n-1/2).

(2.2)

Proof. We make direct application of Theorem 2.8 of Randles (1982), by which (2.3) holds if Randles' Conditions 2.2 and 2.3 are fulfilled. First we note that, by classical central limit theory for order statistics (or by (2.7) and (2.8) below).

(Xn, tanl + 1,Xn, n_Lanl) - ( F - l ( c t ) , F - l (1 - fl)) = O p ( n - l/z). (2.4)

This is Randles' Condition 2.2 specialized to our setting. Next we note that by condi- tion (B) there exists M1 < oo such that for all xl,-..,Xm and all (u, o) in some neigh- borhood of ( F - l ( a ) , F - 1(1 -B)) ,

Ih(x1, . . . , x m , u , o ) - h ( x 1, . . . , X m ; F - l ( o t ) , F - l ( 1 - f l ) ) i < - M l . (2.5)

(2.3)

P. Janssen et al. / Asymptotic normality o f U-statistics 67

Now let K be a neighborhood of (F- l (a) , F - l ( 1 - f l ) ) which is contained in the rec- tangular neighborhood of (F-l(ct),F-l(1-fl)) in which (A) and (B) hold. For (u, v) e K and for a sphere D centered at (u, o) with radius d, such that D C K, we have

sup ]h(xa,...,Xm; u',v')-h(Xl,. . . ,Xm; u,o)[ (u',o')eD

_<(1 - f l - a)-mlh(xl, ... ,Xm)[ I m

• ~ I{u-d<-xi <-u+d} I-I l{u-d<_xj<_v+d} i=1 jq:i

m

+ ~ I { o - d < x i <-o+d} II I{u-d<xj<__o+d} . i=1 j:~ i

Then, using (A) and (B), we obtain

E I sup [h(Xl , . . . ,Sm;u;o ' ) -h(Xl , . . . ,Sm; u,o)l 1 (u" v')~D

<--m(1--fl--a)-mL\Ou-dOu-d Ju-d Jo-d Ju-d/lhli=|

< 2mM0(1 - f l - a) -m [F(o + d) -F(u - d)] m-1

• { [F(u+d) -F(u-d)]+ [F(o+d)-F(o-d)]}

<-MEd, (2.6)

for suitable choice of constant ME not depending on choice of D. By Randles' Lemma 2.6, his Condition 2.3 follows from our (2.5) and (2.6). Thus our lemma follows• []

L e m m a 2 .3 . Under condit ion (B) ,

Un(F-l(a),F- 1 (1 - f l ) ) - P,~B

=m ~ [g(Xi;F_l(a),F_l(l_fl))_lua#l+op(n_l/2)" n i = l

Proof. This is immediate from the projection theory of ordinary U-statistics (see, e.g., Serfling (1980), Chapter 5). []

Lemma 2.4. Under conditions (A) and (C),

/~ (An, t,-,] + 1, X,,, ,, _ La,,]) - ,u,~p

= mAaB [a - F n ( F - 1 (a))] + mBaB [ 1 - f l - F n ( F - 1 (1 - fl))] + Op ( n - 1/2),

where F n denotes the usual s a m p l e d f o f X 1 , . . . , X n .

68 P. Janssen et al. / Asymptotic normality o f U-statistics

Proof. By condition (A) and a result of Ghosh (1971) on Bahadur representation of order statistics, we have

Xn, lcm]+ 1 - F - l (a) = a - F n ( F - l ( a ) ) + Op(n-1/2), (2.7) f ( F - l ( a ) )

and

Xn.n_[flnl - F - l(1 - / 3 )= 1 - / 3 - F n ( F - l ( 1 -/3))

f ( F - l (1-/3)) + Op(n-!/2). (2.8)

By conditions (A) and (B) again along with condition (C),

and

~-~BU I (u, o) = (F- I(a),F- I(l -fl)) = m f ( F - l (ct))Aa#,

cglu = m f (F - 1(1 -/3))Ba#, O0 (u,o)=(F-t(ct),F-l(l-fl))

Thus, by the multivariate version of Young's form of Taylor's theorem (e.g., an im- mediate extension of Theorem C on page 45 of Serfling (1980)),

II (u, o) -Pa~ = m f ( F - l (~)) Aa~ (u - F - 1 ( a ) )

+ m f ( F - 1 (1 -/3))Ba,a(o - F - ! (1 -/3))

+ o(ll(u, o ) - (F- l (a), F - 1 (1 -/3))D- (2.9)

Applying (2.4), (2.7) and (2.8) in (2.9), we obtain the desired result. []

Proof of Theorem 2.1. Define

q/(x; u, o) = m { [g(x; u, o) - p(u, o)] + A (u, o)[F(u) - I {x < u} ]

+ B(u, o)[F(o) - I { x < o}]}.

Then, combining Lemmas 2.1-2.4, we may write

n Una#-lga#=n -1 ~ qY(Xi; F - l (o t ) ,F - l (1 - f l ) )+Op(n- l /2) .

i=1

Finally, note that ~u(X1,F-1(a),F-I(I" - f l ) ) has mean 0 and vai'iance traB .2 []

Remark A. By Lemmas 2.1 and 2.2, we may write

, . - 1/2 Una,a -,ua# = Un + On + op (n ), (2.10)

where U* is an ordinary U-statistic with mean 0 and asymptotic variance parameter m2A,~,a, and O* is a function of (two) order statistics. It can happen that aa~>0 but one of these two components is negligible, namely U* if Aa,a=O and O* if

P. Janssen et al. / Asymptotic normality of U-statistics 69

A ~ = B a ~ = O. For example, in the case of the kernel h(xi,x2)~-COS(X 1 --X2) with X l

uniform on [0, 27r] and a + il = ½, we have the Rayleigh statistic defined on tr immed samples with A~a>0 but AaB=Ba#=O. The case that U* is non-negligible (i.e., Aap>0) can occur even when the ordinary U-statistic based on the original kernel h is degenerate. That is, a U-statistic which has nonnormal limit distribution when defined on the full sample can have a normal limit distribution when defined on a tr immed sample. An illustration can easily be obtained by calculating the expressions lUa~, d a,a, Aa,s, Ba B for the kernel h(xl, X 2) = XI'X 2 and noting that the corresponding (full sample) U-statistic is degenerate i f f E(X1)= 0.

It is straightforward to extend Theorem 2.1 to the case of multi-sample U- statistics. For simplicity, we consider the 2-sample situation• Let X~, ... ,Xn, be an

i.i.d, sample from dfF1 and II1, ..., Yn2 i.i.d, from F 2. Let h(Xl, ...,Xm,;Yl, "",Ym2) be symmetric within blocks and consider

2 ( n i - [ a i n i ] - [ f l i n i ] ) - ! u.ap= H

i=1 mi

where

~, h(X,,,i,,...,Xn,im,; Y,,2Jt,'", Yn2jm2) (2.11) Cna,8

Chap = { (il, -.., ira, ; J l , " " ,Jm:): [ a l nil + 1 _< il < " " < ira, <- nl - [ill nil

and [a2n2] + 1 _< Jl < " " <Jm: <- n2 - [12n2] }.

For u < o e R and u ' < o ' e ~, define associated kernels

h(Xl, "",Xm~; Yl, "",Ym2; u, o; u', o')

= H ( 1 - f l i - O t i ) - m i h ( X l , ' " , X m , ; Y l , ' " , Y m 2 ) i=1,2

• I{U<---X 1, ... ,Xml <-- O, U'<--yl, . . . , ym2 <_ 0 '} .

Denote by g~(x; u, o, u', o') the conditional expectation of this kernel given XI =x , and by g2(Y; u, o, u', o') the conditional expectation given Y1 =Y. Put

lt(u, o, u', 0') =Egl(Xl; u, o, u', o') =Eg2(Yl; u, o, u; 0'),

A l (u ,o ,u ' ,o ' )=Var gl(Xl; u,o,u' ,o') ,

A2(u, o, u', o') = Var g2( Yl ; u, o, u', o').

Let Iu, AI ,A 2 denote the evaluations of these quantities F21(a2),F21(1-B2)). Further let A I , B I , A 2 , B 2 denote of p(u, o, u', 0') with respect to u, o, u ' and (FTl(al ) ,Fl l (1- f l l ) ,F21(ot2) ,F21(1- f l2) ) . Set Bi /mi~(F7 I( 1 -f l i)) for i= 1,2, assume that

at (FI l(al),F~ 1(1 - i l l ) , the partial derivatives

v' respectively, evaluated at

Ai=,4i /mi f i (Fi- l (ai ) and Bi=

70 P. Janssen et al. / Asymptotic normality o f U-statistics

Hi

nl +n2

and define

0-2__

Then we have:

' h i (i= 1,2) as min(nl, n2)--* oo,

m 2 [A i + 2o t igZ i _ 2f l i laBi + oti(1 - Oti)A 2 i=1,2 '~i

+ fli(1 -Bi)B 2 + 2ai[3iAiBi].

Theorem 2.2. Assume that Fi (i = 1, 2) and h satisfy (analogues of) conditions (A), (B), (C) and assume ix2>0. Then, as min(nl, n2)-~oo such that ni /(nl+n2)-"~ i (i = 1, 2),

d (nl + n2)l/2(Unap-!.0 ---* N(0, 0"2).

The proof is similar to that of Theorem 2.1. Also, from the above formula for the asymptotic variance, it is easy to recognize what the asymptotic variance para- meter should be in the c-sample case for c > 2.

Remark B. Bickel (1965), in treating the trimmed mean, used a different method of proof, which was also adapted and followed by Saleh (1976) with some gaps in the development. The results in Theorems 2.1 and 2.2 can also be obtained from this approach, based on conditioning, under a similar set of conditions. However, the precise formulation of these conditions is not immediate using the previous litera- ture.

3. Examples

3.1. Central moments

Here we consider robust (i.e., trimmed sample) versions of the classical measures of location, dispersion, skewness, kurtosis, etc. Since all central moments may be represented as U-statistics (Hoeffding (1948), p. 295), Theorem 2.1 yields the appro- priate results. In particular, let us symmetrically trim the sample (u = fl) and con- sider the trimmed mean (1.2) and the trimmed variance (1.3). Let us also confine attention to df's which are symmetric about 0. Then Theorem 1.2 yields

nl/2~naa d, N(0, o.2(a, l _ a ) ) , (3.1)

with

o ,ol o ,=, l " JF- ! (a)

(3.2)

P. Janssen et al. / A s y m p t o t i c normal i t y o f U-statistics 71

which corresponds to Bickel (1965), and

1 / 2 ~ 2 d n -,ua ~ N(a(a, 1 - a), a2(a, 1 - a)),

with

and

p(a, 1 - a) = (1 - 2 a ) -1 ? ~F-t(I-a)x2f(x) dx J F - I(a)

(3.3)

(3.4)

a22(a, 1 - a) = (1 - 2a)- x4f(x) dx- x2f(x) d 3 F - I(a) \ OF- I(ct)

_ (F-'(1-a) 4ct(F- l (a)) 2 x2f(x) dx

J F - l (a)

+ 2a(1 - a)(F- 1 (0~))4], (3.5)

which corresponds to Bickel and Lehmann (1976), formula 3.3, with a - 0 and fl = 2a. This explicit result for the asymptotic variance makes it possible to compare sEaa with competitors such as the usual sample variance and the mean absolute deviation (see for example, Bickel and Lehmann (1976), Tables 3.1 and 3.2). Theorem 2.1 also provides similar applications to the cases of asymmetric F and/or asymmetric trimming.

3.2. Wilcoxon-like statistics

In the one-sample case, assume F has density f symmetric about A and con- sider testing A = 0 versus A > 0. This can be formulated as a problem of testing P{XI + X2 > 0} = ½ versus P{X1 + X2 > 0} > ½, with corresponding test statistic

(:)' V.= E I{Xi+Xj>O}, 1 <_i<j<_n

which is asymptotically equivalent to the one-sample Wilcoxon statistic. Trimming leads to consideration of

(n-2[anl) -1 u..a= E I{x.,+x.j> 0}.

[an] + 1 <_i<j<_n - [an]

From Theorem 2.1 it follows by routine calculations that, under the null hypothesis A =0 ,

( l+4ct "~ nl/Z(unaa-½) d ' N 0, 3(l_----~a)zj, (3.6)

a result previously found by Saleh (1976). In the two-sample case, let P = {(F, G) :F(x)<_ G(x), all x} and consider testing

7 2 P. Janssen et al. / Asymptot ic normality o f U-statistics

F = G versus Fg:G, or in turn consider testing P { X I < Y I } = ½ versus P{XI < Yl} <½. The usual Wilcoxon-Mann-Whitney statistic has the following formulation in the trimmed-sample case:

1 n, -- [ctnl ] n2 - [an2l

U~IH2 ~ = E 2 (n I - 2[an I l ) ( n 2 - - 2[otn 2]) i = t~/'/i ] + l ] = [~t/2] + 1

which can be studied by our Theorem 2.2, which yields

1 1 +4ct "~ nl/2(Un,n2a_½) d N 0, 2 ( 1 - 2 ) 12 (1 -2a ) : J (3.7)

under the null hypothesis, where n = n l + n2 and 2 = lim n l/n. This result can also be found in Hettmansperger (1968).

3.3. Gini-like measures o f location and dispersion

For estimating location, Yanagawa (1968) (see also Heilmann (1980)), proposed to modify the sample mean, which can be expressed as

.~'~= ~] median(Xi, Xj) 1 <_i<j<_n

to

Ln= 1 <_i<j<k<-n

median(Xi, Xj, Xk).

In similar vein, we consider as competitors o f the trimmed sample mean the statistics Una# of the form (l. l) corresponding to the kernels given by

h ( X l , . . . , X m ) = median(xl, ..., x m) (3.8)

for m = 3, 4, . . . . Asymptotic normality is provided by Theorem 2.1. Let us note that naive computation of the statistic U,,aB corresponding to (3.8) would require O(n m) steps. It is of interest, therefore, that for the kernel (3.8) Una B may also be repre- sented as an L-statistic,

n- [#nl Una• = E c n i X n i , ( 3 . 9 )

i = [anl + 1

where, with (g)=0 if a<b,

1 - i - [ a n ] - 1 ~ f n - [ / / n ] - i~ ]

+ [~ml ) \ [½(m-1)] ] J

(3.10)

Thus, given the cni's (which itself is a computational problem, of course), one can compute the statistic via the formula (3.9) in O(n log n) steps.

P. Janssen et al. / Asymptotic normality o f U-statistics 73

For estimating dispersion, Heilmann (1980) proposed to modify Gini's mean dif- ference, which is the U-statistic based on kernel h ( X l , X 2 ) = I X l - x 2 1 , but which also can be expressed as

Gn= ~ range(Xi, Xj) (3.11) 1 <_i<j<_n

tO

G*= E range(Xi, Xj, Xk) • (3.12) l<i<j<k<_n

In similar vein, we consider as competitors to G n and G* the class of statistics Un~ of form (1.1) corresponding to the kernels given by

h (Xl, ..., Xm) = range(x1, ..., Xm) (3.13)

for m = 3, 4, . . . . Asymptotic normality is provided by Theorem 2.1. It is well-known that the statistic Gn may be written as an L-statistic (see, e.g.

Serfling (1980)) and this is shown by Heilmann (1980) to be true also for G*. It is easily seen that in fact, this is true in general for the statistics UnaB based on the kernel (3.13), including the case a = f l = 0 (untrimmed sample). The relevant cni's for a representation of form (3.9) are found to be

I (i_[an]_l (n C n i = ( ~ ) - [k, m - 1 ) _ . -[Pml-m_l i / j . (3.14) )]

An alternative way of generalizing the classical Gini's mean difference is by con- sidering the class of U-statistics based on the kernels of form

h ( x I , x 2 ) = Ix1 - x2l p, (3.15)

for p > 0. This gives the 'p-th power measures' considered by Bickel and Lehmann (1979). However, results of Boos (1979) indicate that the case p = 1 is highly com- petitive to the cases p ~ 1. It is of interest to examine the class of 'p-th power measures' on trimmed samples, which is now possible via Theorem 2.1.

Acknowledgement

The authors greatly appreciate the referees' resulted in several important clarifications.

constructive comments, which

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