Consider $N$ times to repair, $T\sb1,T\sb2\cdots,T\sb{N}$, from a repair time distribution function $F(\cdot)$. Let $p\sb{0~1},p\sb{0~2},\cdots,p\sb{0~K}$ be $K$ proportions with $\sum\sbsp{\nu =1}{K}p\sb{0~\nu}$ $<$ 1. We wish to have at least 100 ($\sum\sbsp{\nu =1}{K}p\sb{0~\nu}$)% of items repaired by time $L\sb{i}$, $1 \le i \le K$, $K \ge 2$. Denote the unknown quantity $F(L\sb{i}$) - $F(L\sb{i-1})$ as $p\sb{i}$, $1 \le i \le K$. Thus we wish to test the hypothesis(UNFORMATTED TABLE OR EQUATION FOLLOWS) / A simple procedure is to test this hypothesis with the $K$ statistics $N\sb1$, $\sum\sbsp{\nu=1}{2}N\sb{\nu},\cdots,\sum\sbsp{\nu=a}{K}N\sb{\nu}$, where $\sum\sbsp{\nu=1}{i}N\sb{\nu}$ = the number of repairs that takes place on or before $l\sb{i}$, $1 \le i \le K$. Each $\sum\sbsp{\nu=n}{i}N\sb{\nu}$ is a binomial random variable with unknown parameter $\sum\sbsp{\nu=1}{i}p\sb{\nu}$. The hypothesis H$\sb0$ is rejected if any of the $\sum\sbsp{\nu=1}{i}N\sb{\nu}$ $\le$ $n\sbsp{i}{0}$, where the $n\sbsp{i}{0}$ are chosen from binomial tables. This test is shown to have several deficiencies. We construct an alternative procedure with which to test this hypothesis. / The Generalized Likelihood Ratio Statistic (GLRT) is based on the multinomial random variable ($N\sb1,N\sb2,\cdots,N\sb{K}$), with parameter ${(p\sb1,}$ $p\sb2,\cdots,$ $p\sb{K}$). The parameter space is(UNFORMATTED TABLE OR EQUATION FOLLOWS) / An algorithm is constructed and computer code supplied to calculate $\lambda(N)$ efficiently for any finite $N$. / For small samples computer code is given to calculate exactly $\delta$ or a p-value for an observed value of $\lambda(N(K))$, 2 $\le$ $K$ $\le$ 5, and $K\ \le\ N\ \le\ N(K)$. / For large $N$, we apply a theorem by Feder(1968) to evaluate the asymptotic critical values and power. / The GLRT statistic, $\lambda(N)$, is shown to be approximately a union-intersection test and thus is approximated by a collection of uniformly most powerful unbiased tests of binomial parameters. The GLRT is shown empirically in the case of $K$ = 3 to have higher power than competing union-intersection tests. / Two power estimation techniques are described and compared empirically. / References. Feder, Paul J. (1968), "On the distribution of the loglikelihood ratio test statistic when the true parameter is 'near' the boundaries of the hypothesis region," Annals of Mathematical Statistics, 39, 2044-2055. / Source: Dissertation Abstracts International, Volume: 49-08, Section: B, page: 3283. / Major Professor: Duane A. Meeter. / Thesis (Ph.D.)--The Florida State University, 1988.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_77837 |
| Contributors | Clair, James Hunter., Florida State University |
| Source Sets | Florida State University |
| Language | English |
| Detected Language | English |
| Type | Text |
| Format | 199 p. |
| Rights | On campus use only. |
| Relation | Dissertation Abstracts International |
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