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Robust Bootstrap for Location and Scale

Assume the location model (2) with unknown scale, and let Tn and Sn be defined by (4) and (5). Under mild assumptions (see Maronna and Yohai, 1981) we have

 (32)

where N2 denotes a normal distribution in , and the asymptotic variance-covariance matrix is

 (33)

where

 (34)

 (35)

and . Let the weights and be defined as

Similarly define and for . We have

To simplify the notation call

and

 (36)

The motivation for theorem 5 below is as follows. We have the identity

 (37)

Rearranging the terms after a Taylor expansion around we get

 (38)

where

and is a point between and . Equation (37) motivates our bootstrap, and (38) suggests the correction needed to account for the use of fixed weights. As before, we can complete our argument by proving that the sequence of matrices converges almost surely to a non-singular matrix. The necessary regularity conditions for this proof are very restrictive. The next theorem is the location-scale extension of theorem 4.

Theorem 5   Assume that (32) and (33) to (35) hold, and that , with .
1.
Let be defined by (36). Then

where

 (39)

and .
2.
There exists a matrix such that

 (40)

where is given by (33) to (35) and is given by (39).
3.
If the functions , , , and are uniformly continuous, then there exists a sequence of matrices such that

where A satisfies (40).

Proof of theorem 5:
1.
The following argument will show that there exists a function , and n independent and identically distributed random vectors , with mean , such that

This representation will allow us to calculate the asymptotic variance-covariance matrix of . By Slutzky's Theorem it will be given by

where denotes the matrix of first derivatives of g at , and is the matrix of variance-covariance of the random vectors . Note that

and

For each define the random vectors as

Their sample mean is

Define by

and note that

Now let . We have , , and . This implies

We have then expressed as a smooth function of means:

It is easy to check that for any with we have

so that

where . From here, part 1 of the theorem follows.
2.
Consider the matrix A such that

where . Note that A is non-singular if and only if D in equation (35) is non-singular. It is easy to see that A satisfies (40).
3.
From Lemma A in page 253 of Serfling (1980) and the assumed regularity conditions we get the result with

where . This finishes the proof of the theorem.
The questions that remain to be answered are:
• Can we bootstrap ?
• When we replace by , is the bootstrap distribution close to the one we want to estimate? This question can be seen as a matter of continuity'' or qualitative robustness of the bootstrap (see Cuevas, et. al., 1993).
The representation

we get from the proof of Part 1 in Theorem 5 together with Theorem 3 positively answer the first question. We are still working on the second question. Empirical results obtained are very encouraging.

Next: Challenges Up: Robust Bootstrap Previous: Robust Bootstrap for Location
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2000-05-29