Asymptotic Expansions for Studentized M-Estimators With Applications to Multiple Comparisons

Ringland, James Thomas

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https://hdl.handle.net/2142/68179

Description

Title

Asymptotic Expansions for Studentized M-Estimators With Applications to Multiple Comparisons

Author(s)

Ringland, James Thomas

Issue Date

1980

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Language

eng

Abstract

Consider the one-way layout with Y(,ij) being the i-th observation from group j, j = 1(1)p, i = 1(1)n(,j). Let N denote the total number of observations. Assume Y(,ij) = (theta)(,j) + e(,ij), where the e's are independent random disturbances with a common symmetric, but not necessarily normal,(, ) distribution. M-estimates of location and scale, (theta) and (sigma), are defined as(, )^roots of the equations (SIGMA)(,i)(psi)((Y(,ij)-(theta)(,j))/(sigma)) = 0, j = 1(1)(,p) and (SIGMA)(,j)(SIGMA)(,i)(chi)((Y(,ij)-(theta)(,j))/(sigma)) = C. The Studentized location statistics T(,j) = ((theta)(,j)-(theta)(,j))/(tau)(,j), where (tau)(,j) estimates the standard error of (theta)(,j), have a limiting N(,p)(O,I) distribution, but the small sample behavior is generally unknown.

This thesis derives the Edgeworth expansion for the joint density of T under suitable regularity conditions. Probabilities obtained through integration are shown to be valid with an error of order O(p('3)/N) provided p('3)/N (--->) 0, which allows for, but does not require, p (--->) (INFIN). Explicit computation of the O(l/N) correction is given in terms of the moments of the (psi) and (chi) functions and their derivatives.

Multiple comparisons of the (theta)(,j)'s may be based on (i) the marginal distributions of the T(,j)'s using the Bonferroni inequality, (ii) Max(T(,j))-Min(T(,j)) using the Tukey method, or (iii) (SIGMA)(,j)T(,j)('2) using the Scheffe method. Expansions for each of these, akin to the t, q, and F distributions, are then derived from the joint expansion.

General examination of these expansions and explicit evaluation for a variety of estimators and heavy-tailed distributions for the e's suggest the following. The t, q, and F distributions provide reasonable approximations if N/p > 10. For smaller samples, the Bonferroni method, which uses the extreme tail of the marginal distribution, can be highly sensitive to changes in the distribution of the e's. The Tukey method is generally well served by the q distribution, even for N/p > 5, unless p is quite large (p = 10). The Scheffe method is less stable for small values of p but is excellently served by the F distribution for large values of p. For p = 2, each of the three distributions in question tend to be less spread as the distribution of the e's becomes heavier in the tails; thus, the procedure of using t, q, or F critical points tends to give conservative results. For p = 5, the opposite is true of the Bonferroni method. For p = 10, only the Scheffe method tends to be conservative. These variations are strongest for the sample means and standard deviation. Highly trimmed estimators, such as Huber (1), tend to be relatively stable, but generally give more liberal results when used with t, q, or F critical points.

Monte Carlo simulation was used to check the accuracy of the expansions in a selection of situations. Excellent agreement between simulation and expansion was found for moderately trimmed estimators such as Huber (1.5). Somewhat less accuracy was found for untrimmed (least squares) and highly trimmed estimators.

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