## Mathematics & Statistics ETDs

1-28-2015

#### Abstract

I study the equivalence between the Likelihood Ratio Test (LRT), Restricted Likelihood Ratio Test (RLRT) and the F-test when testing variance components within the class of generalized split-plot (GSP) models. In this work, I derive explicit expressions for both the maximum likelihood estimates (MLEs) and restricted maximum likelihood estimates (RMLEs) for the variance components of the GSP model and show the equivalence between the F-test, the LRT or the F-test and the RLRT when the level of the test, $\\alpha$, is less or equal to one minus the probability, $p$, that the LRT or the RLRT statistic is zero. However, when $\\alpha> 1-p$, I show that the F-test has a larger power than either the LRT or RLRT. Further, we derive the statistical distribution of these tests under both the null and alternative hypotheses $H_0$ and $H_1$ where $H_0$ is the hypothesis that the whole plot variance is zero. To establish the power inequality for the case $\\alpha> 1-p$, I developed a new stochastic inequality involving a class of distributions that includes, for example, the F and Gamma distributions. I call random variables (r.v.s.) that inherit this inequality to be quantile-stochastic. The stochastic representation of the new inequality involves $\\alpha, p\\in (0,1)$ such that if $p>\\alpha$ and $k>1$ with $W$ being a random variable with an $F(\ u_1,\ u_2)$ or $Gamma(\ au,\ heta)$ distribution then it's always true that \$\\frac{1}{p}P\\left(W<\\frac{W_p}{k}\\right)>\\frac{1}{\\alpha}P\\left(W<\\frac{W_{\\alpha}}{k}\\right), \$ where \$\\gamma=P(W

Statistics

Doctoral

#### Department Name

Mathematics & Statistics

Christensen, Ronald

Lu, Yan

Sonksen, Michael

Kang, Huining

English

#### Keywords

MLEs, REMLs, LRT, RLRT, F-Test, mixed model, variance components, stochastic inequality, quantile-stochastic

Dissertation

COinS