Contribution to fixed sample and sequential testing of point null hypothesis

Hyun Sook Oh, Purdue University

Abstract

In Part I of the thesis, we revisit the problem of testing a sharp null hypothesis and the issue of conflict and reconciliation between the Bayes and frequentist approaches. Specifically, let $X\sb1, X\sb2,\...,X\sb{n}$ be i.i.d. observations from a distribution $P\sb{\theta}$ in $R\sp{k}.$ For testing $H\sb0:\theta = \theta\sb0$ vs. $H\sb1:\theta\ne\theta\sb0,$ let g be a prior density on the alternative, where $g\in\Gamma,$ some specified class of priors. We consider the prior probability $\pi\sb0$ on $H\sb0$ required for inf $P(H\sb0\vert X\sb1,\...,X\sb{n})=Q,$ where $Q=Q(X\sb1,\...,X\sb{n})$ denotes the P-value of a common classical test; here the infimum is over the prior family $\Gamma$. In Berger and Sellke (1987), Cassella and Berger (1987), and others, a fixed prior probability of 0.5 was used. Clearly, $\pi\sb0$ is formally a function of the data $X\sb1,\...,X\sb{n}.$ We derive the null distribution of $\pi\sb0$ and some particular features of this distribution, such as the median and the mean. In some cases, the distribution is in fact also unimodal and in many cases, the median of $\pi\sb0$ is exactly 0.5, even though the distribution of $\pi\sb0$ is not at all symmetric about 0.5. Also, we consider the null distribution of the ratio and the difference between the P-value and inf $P(H\sb0\vert X\sb1,\...,X\sb{n})$ and the Wolfowitz distances of distributions of the P-value and inf $P(H\sb0\vert X\sb1,\...,X\sb{n})$ are evaluated. Special cases that are considered include $P\sb{\theta} = N(\theta,I),$ and $\Gamma$ = Normal priors/Spherically symmetric unimodal priors/Symmetric Scale-Parameter with M.L.R. priors. In Part II of the work, two problems in sequential analysis are considered. In the first, we go back to the well known Stein two stage confidence interval and find global upper bounds on the ratio of the expected sample size for appropriate nonnormal data and normal data. The first stage sample size is taken to be 2 and the nonnormal distributions considered are the bounded uniform, t with 3 and 5 degree of freedom, the Double Exponential, the Logistic, and the von Mises extreme value density. In the second problem, we consider the question of efficiency of the Bayes sequential procedure with respect to the optimal fixed sample size Bayes procedure in a simple vs. simple testing problem for data coming from a general nonregular density $b(\theta)h(x)I(x<\theta).$ Efficiency is defined in two different ways in these calculations.

Degree

Ph.D.

Advisors

DasGupta, Purdue University.

Subject Area

Statistics

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