Sequential tests for hypergeometric distribution

Xiaoping Xiong, Purdue University

Abstract

The problem of studying the truncated Generalized Probability Likelihood Ratio Sequential (GPLRS) test for the dependent case of sampling without replacement from a dichotomous population is considered. To test the composite hypothesis: $H\sb0 : p \le p\sp*, H\sb1 : p > p\sp*,$ by applying the criterion of generalized probability likelihood ratio sequential as well as truncation, with the practical consideration of convenience for application, a new test procedure is established. Also a new computation method is developed to give exact calculation of power function and expected sampling sizes for this procedure. This test procedure is based on a (log) ratio function $$\eqalign{&G(x,y;p\sp*) = p\sp*\log{1\over p\sp*} - x\log{x+y\over x} - (p\sp* - x)\log{1-x-y\over p\sp*-x}\ +\cr &\quad (1 - p\sp*)\log{1\over1-p\sp*} - y\log{x+y\over y} - (1-p\sp* - y)\log{1-x-y\over 1-p\sp*-y},\cr}$$which is well defined on and only on two-dimension rectangular region $R\sb{p\sp*} = \{(x,y) : 0 \le x \le p\sp*, 0 \le y \le 1 - p\sp*\}$ for given $p\sp*.$ Based on the same $G(x,y;p\sp*),$ a similar procedure is established to test the hypothesis: $H\sb0 : p = p\sp*,H\sb1 : p \ne p\sp*.$ By considering a random walk $\{U\sb{t},V\sb{t}) : t = 0, {1\over N\sp*},\cdots, 1\}$ on rectangular grids $$Rg\sb{p\sp*}(N\sp*) = \left\{\left({i\over N\sp*}, {j\over N\sp*}\right) : 0 \le i \le p\sp*N\sp*, 0 \le j \le (1 - p\sp*)N\sp*\right\},$$H$\sb0$ is rejected if $(U\sb{t}, V\sb{t})$ crosses curves $v = \varphi\sbsp{a}{h}(u)$ or $u = \varphi\sbsp{b}{l}(v),$ before $t=\gamma.$ Boundary function $v = \varphi\sbsp{a}{h}(u)$ is given by $G(u, \varphi\sbsp{a}{h}(u); p\sp*) = a$ and $u < p\sp*(u + \varphi\sbsp{a}{h}(u));$ Boundary function $u = \varphi\sbsp{b}{l}(v)$ is given by $G(\varphi\sbsp{b}{l}(v), v;p\sp*) = b$ and $v < (1$ - $p\sp*)(\varphi\sbsp{b}{l}(v) + v).$ An asymptotical approximation of type-I error for the truncated GPLRS procedure, as $N\sp*\to\infty,$ is derived. The asymptotical approximation of absorption probabilities on the rejection boundaries have the form of a simple function of population size $N\sp*$: $$\sqrt{N\sp*p\sp*(1 - p\sp*)\over2\pi}e\sp{-cN\sp*} \int\sbsp{0}{x\sb0} h(x)dx$$where $c, x\sb0$ and $h(x)$ (depend on $a, b, p\sp*, \gamma)$ do not depend on $N\sp*.$

Degree

Ph.D.

Advisors

Lalley, Purdue University.

Subject Area

Statistics|Mathematics

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