Goodness-of-fit statistics based on weighted Lp-functionals

STATISTICS& PROBABILITY LITTERS ELSEVIER

Statistics & Probability Letters 35 (1997) 261-268

Goodness-of-fit statistics based on weighted Lp-fUnctionals Ibrahim A. Ahmad Division of Statistics, Northern Illinois University, DeKalb, IL 60115, USA Received 1 February 1994; received in revised form 1 December 1996

Abstract

A class of test statistics is introduced which is based on the Lp-distance between distributions. It is shown to be asymptotically normal both under the null and under the alternatives. The class is easily adaptable to multivariate distributions and to situations when parameters are estimated. It can also be modified to handle distributions defined on circles or when the data are sampled minima. (~) 1997 Elsevier Science B.V.

1. Introduction

The (weighted) Lp-functional of a distribution function F and its hypothetical counterpart F0 is defined as follows: Up(if) =

F

[F(x) - Fo(x)]P~(Fo(x)) dF0(x),

(1.1)

oo

where ~b(u) is a known function on [0, 1] such that Up(qQ < ~ and cxD> p>~l. To test the hypothesis H 0 : F = F 0 , known versus H1 : F ¢ F 0 one needs a sample X1 . . . . . X,, from F and we form the empirical version of U(~b) by plugging Fro(x), the empirical distribution, in place of F in (1.1) to get

Um,p(O ) =

F

[Fro(x) - F(x)]PO(Fo(x)) dF0(x).

(1.2)

The classical goodness of fit problem is to reject H0, if [Um,p(~b)[ is large enough. This procedure began when p = 2 and i f ( t ) = 1 by Cramrr-vonMises and when p = 2 and ~k(t)=[t(1 - t ) ] - I by Anderson and Darling; cf. Shorack and Wellner (1986) for details. The general Lp-case was discussed by Csorgo and Horvath (1988) where it was shown that mp/2Um,p(~) converges in distribution to a limit for a large class of weight functions ~b. This limiting distributions are usually not normal with unique exception in the case ~k(t)=[t(1 - t ) ] -O+p/2) (cf. Csorgo and Horvath, 1993). Also, methods of proofs employed are not easily adaptable to the multivariate case or to the case when parameters are estimated, leaving much to be added. Thus, in the present investigation we present an estimate of Up(if), say, Up,~(~k), such that ml/20p,~,(~l) is asymptotically normal both under H0 and under H1. Further, under H0, a test based on 0p, r(~) is distributionfree and consistent. In addition, the results presented here can be used equally whether the data are univariate 0167-7152/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved PH S0167-7152(97)00021-7

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I.A. Ahmad / Statistics & Probability Letters 35 (1997) 261-268

or multivariate and may be adapted to the case when parameters are estimated using, e.g., the work of Randles (1982). Besides being an extension to the well-known L2-functionals, using the Lp-functionals Up(0) allows us to provide for testing H0, a class (in p ) of test statistics (whether using Um,p(~k) or Up,~(~b)), thus one can choose the value of p that maximizes the asymptotic power of the test for a specific alternative. This is particulary plausible with our procedure since ml/2Um, p,~(~k) is asymptotically normal under the alternative. All we need to do is calculate the alternative asymptotic mean and variance of 0p,~(~). In Section 2, we describe the estimate 0p.e(~k) and derive its null and nonnull asymptotically normal limiting distribution. We also demonstrate how Up,~(ff) is calculated and how it can be modified for circular distributions, cf. Watson (1961). Finally, we show that our procedure can be used unaffected for any choice of ~ in the case when parameters are estimated with consistent estimates, thus providing a major advantage over the empirical procedures based o n Um,p(~l ). In Section 3, we extend our procedure to the case where the data are the minima of m groups of k observations each. This is a case of indirect sampling and there is no procedures based on the empirical distributions known for it and this may be difficult to obtain. Our procedure, however, extends without much difficulty.

2. Results The problem we address in this section is to test H 0 : F = F 0 (completely known and continuous) versus Hi : F ~ Fo using the functional defined in (1.1). Note that (2.1) can be written as

Up(~/)=Ko(p,~,)-pU~,p(Foe-lO(Fo))+~_~(-1)r

U~,p(FoP-r 0(Fo)),

(2.1)

r=2

Ko(p,~,)=fFoP(x)~,(Fo(x))dFo(x) is a known constant (F(x))r(Fo(x))P-'O(Fo(x))dFo(x). We write Ko for Ko(p,O)

where

and for

r= 1,...,p, Ur,e(FoP-'¢(Fo))=f_~ for Ur,p(FoP-rO(Fo)) from now on.

and U~ For r = 2, 3 . . . . . p we estimate U~ by its empirical counterpart which reduces to

: U,,AFo

~P(Fo)) . . . . il =1

uP-'~,(u)du ir=l

:m-r~'~'"~-~qgr, fo(Xil . . . . . X/r), i1=1

F0(max(Xq,...,X,, ))

say.

(2.2)

i=l

For r = 1, we estimate (-/1 by ~rl,y :

~-71,7(F0P-II~(Fo)) : m -1

up-l~l(u)du=

C/,m(~ ) i=l

(X~)

m -1

Ci, m ~ p ( S i ) ,

say,

(2.3)

i=l m

where {Ci, m(7)}im__l, m >~ 1 is a triangular array of real numbers such that (l/m)~--~i=1 Ci,m(7)= l + O ( m -1/2) and satisfying m -1 ~-'~im__lC~m(7) --~ C 2 > 1 as m --~ 1 for all 0 < 7 ~< 1. Thus,

I.A. AhmadI Statistics & ProbabilityLetters 35 (1997) 261-268

263

we propose to estimate Up(0) in (2.1) by Op,y ( O ) = K o - p F J l , ~ + E ( - 1 )

~

(2.4)

0~.

r=2

The following result gives the asymptotic normality of Up(0) both under H0 and under HI. Theorem 1. As m--~oo, m~/2(@,r(O) - Up(O)) is asymptotically normal with mean 0 and variance ap, given in (2.8). Under H0, Up(0)= 0 and the null variance is given by

o 2 = p Z { C Z - 1 } { //Fo(min(x,y))FoP-l(x)FoP-l(y)O(Fo(x))O(Fo(y))dFo(x)dFo(y)

(2.5) In the special case 0 ( u ) = 1, a~ = pZ( CZ-1)/( p + l )2( 2 p + l ), while in the case 0 ( u ) = [u(1-u)]-l,ao 2 = 2 p 2 (C 2 - 1) y'~e~=o[(p+ ()(2p + f - 1)] -1.

Proof. Writing ~Ol,r(Sl)=E[q)r,Fo(Sl. . . . . Xr)IX1] and using the standard theory of U-statistics, cf. Serfling (1980), we see that for r = 2 , 3 .... ,p, (2.6) Thus, it follows that

Up(O)- Up(O) = - T I , m + E ( - - 1 ) r

Tr,m+Op(m-1/2),

(2.7)

r=2

with Tl,m = p{01,~ -/-/1}. Hence, ml/2(Up(O)- Up(O)) is asymptotically normal with mean 0 and variance 2 7 =mVar(Tl O'p,

m)+m E

Var(Tr, m ) - 2 m

r=2

(:)(:)

-2mEE(-llr+s r~s

E(-1) r=2

Cov(Tl,m, Tr,m)

r

(2.8)

Cov(Tr, m, Zs,m).

But m Var(Tl,m) ~ p2C2{ j j

-

F(min(x, y))FoP-l(x)FoP-~(y)O(Fo(x))O(Fo(y))dFo(x)dFo(y)

I/ F(x , op

(x)O(Fo(x)) dfo(x

,11 .

(2.9)

I.A. Ahmad I Statistics & Probability Letters 35 (1997) 261 268

264 Also for r = 2 ..... p,

m Var (Tr, m) ---+r 2 { / / F ( m i n ( x ,

--

y))F~-I(x)Fr-I(y)FoP-r(x)FoP-r(y)~(Fo(x))~,(Fo(y)) dFo(x) dFo(y)

Fr(x

(x)~O(Fo(x))dFo(x

,

(2.10)

while

f ff

m Cov (Tl,m, Tr,m) ~ rp I ] J F(min(x, y))Fr-'(x)FoP-r(x)FoP-I(y)~(Fo(x))O(Fo(y)) dFo(x) dFo(y)

- [f Fr(x)Fg-r(x)g4Fo(x))o(x)] x [fF(x)FoP-'(x)O(Fo(x))dFo(x)l },

(2.11)

and finally, for r ~ s,

mCov(Tr,m,T~,m)~rs {f/F(min(x,

y ) )Fr- l (x )FoP-r(x )FS- l ( y )FoP-S( y )~l(Fo(x ) )~l(Fo( y ) )

x dFo(x ) dFo(y ) - [ f Fr(x)FoP-r(x)~(Fo(x))dFo(x)]

x [f(x)FoP-%)q4Fo(x))dFo(x)] F }.

(2.12)

Now, under H0, all terms in (2.4)-(2.12) in the braces { } are equal and the total number of terms is

[~_~ p2(C2- 1)+

(;)]2

p_l

=p2(C2-1)+P2{r~=I(-1)r

r=l ( - 1 ) r r

(

/}2 rP-1-1

= p2(C2 - 1).

The special case ~9(u)= 1 is easily obtained while for ~ ( u ) = (u(1-u))-1 the result follows from the following: ao2 = p2(C2 - 1)/f[Fo(min(x, y))

-Fo(x)Fo(y)]FoP-l(x)FoP-~(y)[Fo(x)Po(x)Fo(Y)Po(y)]-~dFo(x) dFo(y),

(2.13)

where Fo = 1 -Fo. Thus, putting u=Fo(x),v=Fo(y) and changing variables we get 0"2 = 2p2(C 2 - 1)

/o'/o

uP-lvp-2(1

--

u) -1 dudv.

Expanding (1 - u) -1 and integrating gives the result. The theorem is now proved. []

(2.14)

I.A. Ahmad/ Statistics & ProbabilityLetters 35 (1997) 261-268

265

Next, let us detail how the statistic 0 p a ( 0 ) can be calculated in the special case ~k(u)= 1. In this case we shall take Ci,m= Ci, m(7) = 1 + 7 if i is odd and 1 - 7 if i is even. It is not difficult to see that

'

@ ( 1 ) = (p + 1)_1 ~ ( _ l ) r r=0

/ p +Fl

=(p+1)_1~(_1) p+l F

r=0

rn_r

h=l

... ~--~ [1 _ F0p-,+l(max(X~, ..... X#))] ir=l

)

u,.,p

In this case a 2 = pZy2/(p+ 1)(2p + 1). Next, we note that Watson (1961) showed how to modify the Cram6r-vonMises statistic to fit distributions defined on a circle by subtracting the center of the functional, namely he showed that for U2(1) = f[F(x)F0(x)] 2 dF0(x), the modification U2*(1 ) = f~{F(x)- Fo(x)- f[F(x)- Fl(x)]dFo(x)}EdFo(x)can be used for testing goodness of fit of a distribution defined on a circle. Similarly, we see that

Up*(O)=f {F(x)-Fo(x)- f [F(x)-Fo(x)]dFo(x)}PO(Fo(x))dFo(x),

(2.16)

is suitable for testing distributions defined on circles. Also note that Up*(0) =

Up(O)-

P

E(-lf

(p)

(f Up-r(O)

[F(x) - Fo(x)]dFo(x)

}r

.

(2.17 /

r=| We estimate Up(O) by Op,r(O) and estimate Up-r(O)by f[F(x) - Fo(x)] dFo(x) by f[Fm(x)- Fo(x)] dFo(x). Hence,

Op_r(O)

(the empirical counterpart) and estimate

m|/z{o;(O)- U;(O)}=ml/Z{Op(O)- Up(O)}- E" (-1) r (r') {ml/2(Op-Z(O)-Up-r(O))} r=l

P ×{I[Fm(x)-Fo(x)IdFo(x)}-~_,= (-1)r(P~)U,-.(O)

x {ml/2 [(f[Fm(x)-Fo(x)]dFo(x)) r]

where the right-hand side of (2.18) has the same limiting distribution as

ml/2{Up(O) - UP(O)} - E" ( - 1)r

(r'){I

}"

[r(x) - Fo(x)] dFo(x) {ml/2[Op_r(O) - Vp--r(O)]}

- Pr~=I(-I)'(P)up-r(O){mI/E[f Fm(x)dFo(x)-/F(x)dFo(I)}}{f[F(I)-Fo(x)]dFo(x)} r-'. (2.19)

1.A. Ahmad / Statistics & Probability Letters 35 (1997) 261 268

266

Hence, the asymptotic normality of m l / 2 { U ; ( f f ) - Up(if)} can be established and its variance evaluated. Under Ho, all terms in (2.17) vanish except the first term which as in Theorem 1, is asymptotically normal with variance a0z given above. We also note that the test based on Up,r(~k) is indeed consistent, this follows from Theorem 1 above. Let us indicate briefly how Theorem 1 may be altered to handle the case when parameters may be estimated. Here we want to test H0 : F ( p + ax) = Fo(x),F0 is completely known. Suppose that we have a consistent m estimate of the vector of parameters 2 = (p, a)', say J. = (/~, 6) I such that 2 = 2 + m -1 ~ i = l ~(X/)+ Op(n-1/2), where E~(X/) = 0 and E~(X/)~(Xj) in finite. Assume that F has a bounded density and set

P (P) Op,7(~I,2)=Ko(p,O)- pOI,p,~(2)+ Z (-1)~ Ur,p,7(~), r=2

where 01,p,y(,~ )

=

up-l~(u)du

Ci, m(]) )

m -1

for r = 2 . . . . . p,

(2.20)

,JFo((Xi--I~)/d)

i=1 m

Ur,p,r(J.) m-~ ~ ' "

uP-r~(u)du.

=

i1=1

#=1

(2.21)

F0((max(Xil..... X~r)--~)/~)

Since it is not difficult to show that all conditions of Theorem 2.13 of Randles (1982) are met, then one sees that 0p,~(~k, J.) has same distribution as 0p,~(~,2) which by Theorem 1 above is normal and under H0(2 = 0) the null variance remains unaffected. Note that DeWet and Randles (1987, Theorem 2.16) showed similar result for Urn,2(1). Extending this to the general Um,p(~k) seems difficult to do. Hence our procedure has a major advantage.

3. Extension to sampled minima In some reliability studies, and maybe others, the following sampling plan may be used: Put on test a set of m groups of observations each of size k. Observe only the minimum life of each group. Thus, if Z11 . . . . . Zlk . . . . . Zm(k-1)+l . . . . . Zmk are the original data, we observe only X1..... Xm where X / = minl~
£

oo

[Pk(x) -

P~(x)]P~(Fo(x))dF0(x),

(3.1)

where P = 1 - F. Note that as above, L~,p(~9) may be written as

Uk,p(~ll) = go(k, p, ~ll) - pUl,k,p(Fk(p-l)O(Fo)) -k Z

P

(--1)P

(P)

Ur'k'p(Fk(p-r)O(Fo))'

(3.2)

i=2

where

Ko(k, p, 0)= fPkoP(x)~(Fo(x))dFo(x) is

a known constant and

~Jr,k, p( Fk(p-r)~J( Fo ) ) = f Fkr (x )Fko(P-r) (x )~J(Fo(x ) ) dF0(x),

(3.3)

I.A. Ahmad I Statistics & Probability Letters 35 (1997) 261-268

267

for r = 1..... p. Thus, as in Section 2, we propose to estimate ~,p(qJ) by

Ok,p,?(~l) = Ko(k, p, O) - pUI,k,p,?(Fk(p-1)~I(Fo)) -~ Zp (--1)

(p)

Ur,k,p(ff'k(p-r)~l(Fo)),

(3.4)

r=2 1

m

where Pmk,~(x)= m ~-]i=1 Ci,m('y)l( Xi

Fff(x) = ~

Z

> X) and for r = 2,3 ..... p,

I(min(X/, ..... X/r) > x),

where the sum is taken over 1 ~
As m ~ oo, ml/2(Uk,p,~(~t) - Uk,p(~k)) is asymptotically normal with mean 0 and variance

, V(f'r,k,p)--2mZ(--1)r i=2

z = m---~oo lim [,mV(f'l,k,p,r)+m Zr=2 ¢7k'P

--2m~gs(--1)r+s (P)

(p)

^ ^ k,p) Cov(TI,k,p,?, Tr,

(P)cov(Tr, k,p,~,k,p)) ,

(3.5)

where

lirnmV( f'l,k,p,7 ) : p2C2(7) {

ff

Fk(min( x, Y))/~0k(P-1)(x )Fk(p-ll(y)~l(Fo( x ))~b(Fo(y))dFo(x )dFo(y)

,

}

and for r -- 2 ..... p,

limmmV(f'r,k,p) = r 2 {

ff

Fk(min(x, y))pk(r-1)(x)pk(r-ll(y)F~(P-r)(x)Pko(P-rl(y)~b(Fo(x ))lp(FO(y))

while limmm cov(/~l,k, p,,, f'r,k,p)= 2p{

ff

Fk(min(x, y))ff'k(r-1)(x)P~(P-r'(x)Fko(P-l'(y)~(Fo(x ))O(FO(y)

xdFo(x)dFo(y)-[/Fkr(x)Fko(P-r)(x)~(Fo(x))dFo(x) ]

268

I.A. Ahmad / Statistics & Probability Letters 35 (1997) 261 268

and for r # s,

limmcov(Tr, k,p, Ts,k,p)=rs m

Fk(mm(x,y))Fk(r 1)(x

(p

r)(x)Fk(S 1)(y

(p S)(y)•(Fo(x) )

tJJ

[l Under Ho, the variance reduces to o'S,k,p,~, = p2(CX(,)-1

){

if

F°k(min(x' Y))F~(P-1)(x)F~(P-1)(Y) (3.6)

In the special case ~O(u) = 1, tro, ~ k,p, 2 reduces to p2k(C2(7 ) - 1)/(kp + 1 ) ( 2 k p - k + 2). While in the special case Ip(u) = [u(1 - u ) ] - i ,tro,2 k,p,~ reduces to k--1

2pZk(C2(7) - 1) Z j=o

[(kp+ 1)(2kp+( + j - k ) ] -l. t=o

Acknowledgements The author is grateful to the reviewer and the editor for their many suggestions which resulted in improving exposition and contents.

References Ahmad, I.A., 1993. Modification of some goodness of fit statistics to yield asymptotically normal null distributions. Biometrika 80, 466-472. Csorgo, M., Horvath, L., 1988. On the distribution of Lp-norms of weighted uniform empirical and quantile processes. Ann. Probab. 16, 142-161. Csorgo, M., Horvath, L., 1993. Weighted Approximations in Probability and Statistics. Wiley, New York. Dewet, T., Randles, R.H., 1987. On the effect of substituting parameters estimators in limiting X 2, U and V statistics. Ann. Statist. 15, 398-412. Randles, R.H., 1982. On the asymptotic normality of statistics with estimated parameters. Ann. Statist. 10, 462-474. Sertling, R.J. 1980. Approximation Theorems of Mathematical Statistics. Wiley, New York. Shorack, G.R., Wellner, J.A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York. Watson, G.S., 1967. Goodness of fit tests on a circle. Biometrika 48, 109-114.