Fares Qeadan

Publication Date



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

Degree Name


Level of Degree


Department Name

Mathematics & Statistics

First Advisor

Christensen, Ronald

First Committee Member (Chair)

Ronald Christensen

Second Committee Member

Yan Lu

Third Committee Member

Michael D. Sonksen

Fourth Committee Member

Huining Kang




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

Document Type