#### Degree Type

Dissertation

#### Date of Award

1987

#### Degree Name

Doctor of Philosophy

#### Department

Statistics

#### First Advisor

David A. Harville

#### Abstract

For an n x 1 observable random vector y, consider the mixed linear model y = X[alpha] + Z[beta] + e, where X is an n 4x a constant matrix, Z an n x b constant matrix, [alpha] an a x 1 unknown fixed vector, [beta] a b x 1 unobservable random vector, and e an n x 1 unobservable random vector. It is assumed that [beta] [superscript]~ MVN (O,[sigma][subscript]sp[beta]2I), e [superscript]~ MVN (O,[sigma][subscript]sp e2I), and [beta] is independent of e. In many applications, it is of interest to construct a confidence set for [lambda] = [sigma][subscript]sp[beta]2/[sigma][subscript]sp e2;Various tests of the null hypothesis H[subscript] o: [lambda] = [lambda][subscript] o are to be discussed, including the most powerful translation and scale invariant test (against a specified alternative), the locally most powerful translation and scale invariant test, and the likelihood ratio translation and scale invariant test. The most powerful translation and scale invariant test, the locally most powerful translation and scale invariant test, and the likelihood ratio translation and scale invariant test are also "optimal" in the class of translation-invariant similar tests;Corresponding to each family of tests is a confidence set for [lambda]. The properties of these tests and confidence sets are to be discussed and compared with those of Wald's test and confidence set. Computing the critical points and power functions of various tests are considered;Numerical comparisons on the performance of various confidence sets, in terms of the probability of covering a false value, are given for each of five data patterns. For constructing a confidence lower limit, Wald's procedure performs well when the true value of the parameter is large. For other occasions, the confidence sets corresponding to the most powerful invariant tests are very attractive.

#### DOI

https://doi.org/10.31274/rtd-180813-11358

#### Publisher

Digital Repository @ Iowa State University, http://lib.dr.iastate.edu/

#### Copyright Owner

Tsung-Hua Lin

#### Copyright Date

1987

#### Language

en

#### Proquest ID

AAI8805111

#### File Format

application/pdf

#### File Size

140 pages

#### Recommended Citation

Lin, Tsung-Hua, "Confidence sets for the ratio of variance components in a mixed linear model with two variance components " (1987). *Retrospective Theses and Dissertations*. 9276.

https://lib.dr.iastate.edu/rtd/9276