As a classic parameter originated from the binomial distribution, the binomial pro- portion has been well studied in the literature due to its wide range of applications. In contrast, the reciprocal of the binomial proportion, also known as the inverse proportion, is often overlooked, although it plays an important role in sampling designs and clinical studies. To estimate the inverse proportion, a simple method is to apply the maximum likelihood estimation (MLE). This estimator is, however, not a valid estimator because it suffers from the zero-event problem, which occurs when there is no successful event in the trials. At first, we review a number of methods proposed to overcome the zero-event problem and discuss whether they are feasible to estimate the inverse proportion. Inspired by the Wilson (1927) and Agresti and Coull (1998), in this thesis, we focus on a family of shrinkage estimators of the inverse proportion and propose to derive the optimal estimator within this family. The shrinkage estimator overcomes the zero-event problem by including a positive shrinkage parameter, which is intrinsically related to the expected value of the resulting estimator. To find the best shrinkage parameter, the relationship between the shrinkage parameter and the estimation bias of the shrinkage estimator is investigated systematically. Note that the explicit expression of the expected value function of the estimator and the best shrinkage parameter are quite complicated to compute when the number of trials is large. Hence, we review three methods in the literature which were proposed to approximate the expected value function. And after being inspired, we propose a new approximate formula for the expected value function and derive an approximate solution of the optimal shrinkage parameter by the Taylor expansion. Because there still exist an unknown binomial proportion in the optimal shrinkage parameter, we suggest a plug-in estimator for the unknown proportion with an adaptive threshold. Finally, simulation studies are conducted to evaluate the performance of our new estimator. As baselines for comparison, we also include the Fattorini estimator, the Haldane estimator and a piecewise estimator in the simulations. According to the simulation results, the new estimator is able to achieve a better or equally good performance compared with the Fattorini estimators in most settings. Hence, our new estimator can be a reliable estimator for the inverse proportion in most practical cases

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Thesis (Ph. D.)--Virginia Polytechnic Institute, 1963. / Vita. Abstract. Includes bibliographical references (leaves 124-126). Also available via the Internet.

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This thesis is an extensive review of the major literature dealing with the negative binomial distribution. An account of the forms of the distribution is presented. This includes compound, limiting and truncated forms and examples of their occurrence. Parameter estimation is treated in a few specific cases. In particular, the inverse binomial sampling form, its analytic properties, occurrence in biological as well as industrial situations, estimation of the parameter p, and hypothesis testing are discussed in detail. Many examples are included. In the discussion of the inverse binomial sampling form several extremely useful, little known relations are presented. Among these is a discussion of the use of the Tables of the Incomplete Beta Function for the evaluation of the cumulative distribution function. Also discussed is the method of adapting Biometrika Table 41, which gives confidence limits on p in the positive binomial case, to the inverse binomial sampling case. / Master of Science

by Ho Yat Fan. / Bibliography: leaf 59 / Thesis (M.Phil.) -- Chinese University of Hong Kong, 1983

Typescript (photocopy). / Digitized by Kansas Correctional Industries

In this thesis, first we replace the condition X ¡Ø y in Huang and Su (2000) by X ¡Ù y and give necessary and sufficient conditions such that there exists a random variable X satisfying that E(g(X)| X ¡Ø y)=h(y) f(y )/ F(y), " y Î CX, where CX is the support of X.Next, we investigate necessary and sufficient conditions such that h(y)=E(g(X) | X ¡Ø y ), for a given function h and extend these results to bivariate case.

Resumo: O estudo da distribuição espacial de pragas é fundamental para elaboração de planos de amostragem para uso no manejo integrado de pragas. Para Toxoptera citricida foi estudada a distribuição espacial em talhões de pomares de citros comerciais de laranja (Citrus sinensis) da variedade Pêra, com 5, 9 e 15 anos de idade, durante o período de setembro de 2004 a abril de 2005. Foram realizadas 14 amostragens de número de pulgões-preto em intervalos aproximados de 15 dias entre as mesmas, utilizando-se armadilhas adesivas de cor amarela (0,11 x 0,11 m) fixadas à planta a 1,5 m de altura aproximadamente. As armadilhas foram distribuídas na área, a cada cinco plantas na linha, em linhas alternadas, totalizando 137 armadilhas no talhão com 5 anos, 140 no talhão com 9 anos e 80 no talhão com 15 anos. A lei de Taylor e a distribuição binomial negativa foram os modelos que melhor representaram a distribuição da população. Foram com construídos planos de amostragens para levantamento desta praga com base na lei de Taylor e na distribuição binomial negativa. / Abstract: The study of spatial distribution of insects is fundamental to elaborate sampling plans with potential to use in integrated pest management. The spatial distribution of Toxoptera citricida was studied in plots of commercial orchards of orange (Citrus sinensis) of the variety 'Pêra' with 5, 9 and 15 years of age, during the period of September of 2004 and April of 2005. Fourteen samples of the number of Toxoptera citricida was performed each 15d approximately, using yellow adhesive traps fixed at 1,5 m of height each 5 plants in alternated lines, summing 137 traps in the 5-years plot, 140 traps in the 9-years plot and 80 traps in the 15-years plot. The best models fitted the distribution of population were the Taylor Law and negative binomial distribution, which were used to elaborate the sampling plans. / Orientador: José Carlos Barbosa / Coorientador: Pedro Takao Yamamoto / Banca: Antonio Carlos Busoli / Banca: Wilson Itamar Maruyama / Mestre