ON TIlE BOUNDED-INFLUENCE REGRESSION ... - CiteSeerX

ON TIlE BOUNDED-INFLUENCE REGRESSION ESTIMATOR OF KRASKER AND WELSGf

David Ruppert 1

Inavid Ruppert is Associate Professor, Department of Statistics, University of North Carolina, Chapel Hill, NC 27514. This research was supported by National Science Foundation Grant MCS-8l00748.

ABSTRACT

• Recently, Krasker and Welsch (1982) considered a class of bounded-inflllence regression estimators.

They showed that within this class the

so-called Krasker-Welsch estimator is the only solution to a first-order necessary condition for strong optimality, i.e., for minimizing, in the sense of positive definiteness, the asymptotic covariance matrix.

However, whether

any strongly optimal estimator in fact exists remained an open question. this note, an example is given where no strongly optimal estimator exists.

In

1.

I NfRODUCfION

In a recent article, Krasker and Welsch (1982) consider robust estimators for the linear regression model y. = x. B + I

I

€.

1

where x.1 is a p-dimensiona1 row vector, (y.I ,x.), i=l, ... ,n, are independent. I 2 and identically distributed, and E. is distributed N(O,a ) independently of I

x·. I

Their interest is in bounding the influence of outliers in both xl' and

y..

They consider M-estimators of the form

1

n

(1.1)

. o = i=l I w(y.1 ,x.;B ) (y. -x. B )x~ I n I I n I

where w is a scalar-valued weighting ftmction. S11bstitute an estimator a for a. n this estimator is ~?(y,x;w)

If w depends on a, then we

The influence ftmction (Hampel, 1974) for

= B-w1 w(y,x;B) (y-xB)x t

where (1. 2)

The asymptotic covariance matrix is V = B-lA B- 1 w w ww where (1. 3)

Kr3sker-Welsch discuss several measures of sensitivity. ,Ul

They choose to use

invariant (and quite reasonable) measure y w defined to be

-2-

==

• ==

sup y,x

[ ~? t

(y,x;w ) V-1 ~(y,x;w )]J,.~ w

sup [(x A-1 Xt )2.!-1 y-xS I W(y,x;S)] w y,x

Because an estimator's influence function is normed by the asymptotic covariance matrix of the estimator itself, Stahel (1981) calls y w the self-standardized sensitivity. Now let a bound a > 0 be given, and consider the class of all weighting functions w such that y

and

oj

w

:,; a

depends on y only through ly-xSI •

We say that w is strongly optimal

within this class of (V . w-V) w is positive semidefinite for all w in the class . Krasker and Welsch show that if a strongly optimal w exists, then (up to a scalar multiple) it must be (letting

E

= y-xS)

w(y,x;S)

(Thi sis an implicit definition since A appears on the right -hand side.) w

They conjecture that this w is strongly optimal.

In this note we show by

example that in general no strongly optimal estimators exists. The Krasker-Welsch estimator is invariant to nonsingular reparametrization.

Invariance is not necessarily desirable when the parameters have

physical meanings.

To show the practical significance of the lack of a strong-

ly optimal estimator, we discuss how one might choose alternatives to the

Krasker-Welsch estimator when there are nuisance parameters.

2•

AN EXAMPLE

We will take

x.

= 1.

Suppose p

(l-D.)Z.

= (1

1

where U1, is

0

1

=3

and

D.Z.)

1

1 1

distributedBemoulli(~), 2 ,

is distributed N(O,l), and

1

E., 1

D., 1

and Z. are mutually independent. 1

For h. > 0 let w(y,x;~,I'::.)

=

I'::.

min {I, a/(1'::. IEI (x if x(Z)

=

-:f

-1 t

1-

X )2)}

AI'::.

°

min {l,a/(I€I(xA~lxt)~)} if x(Z)

=

°,

where x(Z) is the second coordinate of x and A" u

(1. 3) f\f':.,

.

By theorem 1 of Maronna (1976)

is a solution to equation

wI'::.

is unique.

AI'::.

This also proves that

is diagonal, since the syrronetry of Z implies that the matrix obtained by

l~lltiplying

Say

=A

AI'::.

(2.1)

=

the off-diagonal elements of

diag

(~ 1'::.'

Al , I'::.

,

1

=

2

+

2

1

AZ,1'::.' •

A

3

AI'::.

by -1 is also a solution to (1.3).

,1'::.)· Now, equation (1.3) can be rewritten as Z

Z -1

E mln{(I'::.E) , a (AI ,I'::. • Z Z -1 E mln{E , a (AI , I'::.

+

+

-1 Z-I

Az,I'::.Z)

}

-1 Z-l A3 ,I'::.Z) }

(2. Z)

(2.3)

We will take a > Z.

If we divide (Z.l) by

and add the resulting equations, we obtain

~,I'::.'

then divide (Z.Z) by

AZ,I'::.'

-4(2.4)

1

+ 2

Now, choose a sequence 6.

m

A. 1,00

= lim Ai 6. + m

00

+

2 -1 -1 2 -1 -1 a (A_ -1., ~ + A3, ~Z) }Al , ~

,

00

= 1,Z,3

such that for i

exist (but are possibly +00).

6. ,

Z

•

E mln{E

m

If

the limits ~