Date of Award

4-1987

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Dr. Joseph W. McKean

Second Advisor

Dr. Gerald Sievers

Third Advisor

Dr. Michael Stoline

Abstract

Three robust rank procedures are considered to test the multivariate linear hypothesis HBK = 0. In the least squares setting, the Lawley-Hotelling trace criterion provides a UMP when the covariance structure is known. Our work reflects the theoretical structure of the Lawley-Hotelling test with robust rank analogues.

The Lawley-Hotelling test has an asymptotic chi-square distribution when the covariance structure is estimated. The trace criterion can be written in three algebraically equivalent forms: a quadratic test based on full model estimates, a quadratic test based on reduced model residuals and a test based on the drop in the least squares dispersion function. Our work considers robust rank analogues to each of the three forms. These rank tests are based on the component-wise extension of the R-estimates which Jaekel (1972) proposed for the univariate linear model. Each has an asymptotic chi-square distribution. The three tests are asymptotically equivalent when testing HB = 0.

The procedure that is based on the residuals from an R-fit of the reduced model is generally called an aligned rank test. Our test is asymptotically equivalent to the one developed by Puri and Sen (1985). The test that is based on a drop in dispersion when passing from the reduced to the full model is an extension of the univariate procedure proposed by McKean and Hettmansperger (1976).

We extended the work of Heiler and Willers (1979) to the multivariate setting. They showed that the asymptotic linearity result of Jureckova (1969) remains true when her "concordance conditions" are replaced with Huber's condition. Our asymptotic distribution theory assumes that Huber's condition holds and that the centered design matrix is of full rank. We also developed the theory under a sequence of contiguous alternatives, from which asymptotic relative efficiency properties are obtained. Estimates of covariance and scale parameters are considered.

Access Setting

Dissertation-Open Access

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