Several large-sample confidence intervals for the ratio of independent binomial proportions are compared in terms of exact coverage probability and width. A non-iterative approximate Bayesian interval is derived and its frequency properties are superior to all of the non-iterative confidence intervals considered. The approximate Bayesian interval, which is very easy to compute, has performance characteristics that are very similar to the computationally intensive score method. Two sample size determination formulas are presented, one for desired absolute precision and the other for desired relative precision.

This thesis is an extension of the development of orthogonal statistics which can be used to investigate sampling properties of moment estimators. This work is particularized for estimators of parameters of the negative binomial distribution. / Master of Science

Ph. D.

Ph. D.

This dissertation deals primarily with the development of the technique of orthogonal statistics and the use of this technique to investigate sampling properties of moment estimators of parameters of the negative binomial distribution. The general technique of orthogonal statistics which is based on the existence of an infinite set {q_r(x)} of orthogonal polynomials associated with a particular distribution, enables one to obtain expansions of sampling moments of statistics which are functions of say, the first k sample moments m₁, m₂,…, m_k. The thesis describes the technique in general, and gives tables which facilitate the expansion through terms in n⁻⁵ of sampling moments of statistics which are functions of any four sample moments. The need for the development of this technique resulted from an interest in the problem of investigating sampling properties of certain moment estimators for the case of the negative binomial distribution. Thus further work was done on the technique for this particular case. Tables are given in the thesis which simplify the procedure for moment statistics which result from a sample taken from this particular distribution. Sampling properties of moment estimators for the negative binomial distribution were investigated. The distribution forms considered in depth were due to Anscombe [Biometrika, 37 (1950}, pp. 358-362] with parameters λ and α, Evans [Biometrika, 40 (1953), pp. 186-211] with parameters m and a, and Fisher [Annals of Eugenics, 11 (1941), pp. 182-187] with parameters p and k. The purpose of this study was to obtain an insight into the behavior of expansions through high powers of 1/n (e.g., terms in n⁻⁴) of the bias, variance, and higher moments for these estimators. It was felt that the usual asymptotic properties described by the first term approximations might be misleading for practical cases (i.e., ordinary sample sizes). The results verified what was suspected. For the moment estimators of Ansaombe's form, when α > λ the sample sizes needed to make high order terms negligible for the expansion of the bias and variance were extremely large. (For one particular case, in order to use the usual asymptotic variance safely one would need an n of 2 million.) This then reveals the hazardous practice of using the first term approximation and resulting in a very serious under-assessment of the true variance of the estimate of α. Since for Fisher's form k̂ = α̂, the same applies. For Evans' form, the situation was in marked contrast. Higher order terms were "damped off" with much smaller sample sizes, and in most cases one is justified in using first term approximations. Studies for Evans' estimators were confined to the range λ > 1 and α > 1. The results for the estimators of Anscombe's form were compared with similar results for the maximum likelihood estimator of α, in order to ascertain the effect on efficiency of the chaotic nature of the n⁻³ term in the expansion of the covariance determinant of α̂. The maximum likelihood results were taken from Bowman [Thesis submitted for Ph.D. degree, Virginia Polytechnic Institute, Moments to Higher Orders for Maximum Likelihood Estimators with an Application to the Negative Binomial Distribution]. This study revealed that there is a striking similarity in the n⁻³ term in the covariance determinant for the two estimators. This made the "true" efficiency almost identical to the asymptotic efficiency in cases when sufficiently large sample sizes are used to "sink" terms beyond n⁻³. This statement cannot be generalized, however, to include any sample size, since for α > λ only relatively large sample sizes "damp off' further terms in the covariance determinants for both estimators. Hence one cannot be sure of the behavior of these determinants beyond n⁻³ unless these large sample sizes are used. Tables and charts are given which display the nature of the expansions given in the text. In particular, charts are given of minimum sample size needed in order that the expansions given can safely be used as approximations. / Ph. D.

This dissertation deals with the properties of two modified moment estimators for parameters of the negative binomial distribution (NBD). Several parametric forms have been suggested for the NBD. The estimation problems vary according to the form which is used. In particular, the form proposed by Anscombe [Biometrika, 37 (1950), pp. 358-382), with parameters λ and α, has received wide attention and was selected for study in this investigation. In Anscombe's parametric form, the mean of the NBD is λ and the variance is λ + λ²/α. While the parameter λ is universally estimated by the sample mean, many different methods of estimation for α have been attempted. Among these, the maximum likelihood estimator α* and the simple moment estimator â are most often used. However, α* is quite difficult to obtain numerically and often this computation requires the use of an electronic computer. In addition, â, while not difficult to compute, is often inefficient. For these reasons, it was felt that a study of the two modified moment estimators â₁ and â₂, suggested by Shenton and Wallington [Moment Estimators and Modified Moment Estimators with Special Reference to the Negative Binomial Distribution (unpublished)], was needed. In the text, the method of obtaining modified moment estimators in general is given in detail. The application of this method to the NBD is discussed and, in particular, the derivations of â₁ and â₂ are presented. Since orthogonal statistics play an important part in this work, their definition and applications are reviewed. In order to evaluate the small sample properties of â₁ and â₂, asymptotic expansions, in powers of 1/n, of their biases, variances, covariance determinants, and higher moments were determined numerically in the parameter space (1 ≤ α ≤ 100, 1 ≤ λ ≤ 100), through terms to n⁻⁴. The computational method for this work is described in detail. Tables and charts which display the nature of the expansions are given in the text. The results show that the behavior patterns of the moment expansions for â₁ and â₂ are somewhat similar to those for â and α*. For both â₁ and â₂, the n⁻⁴ term contributes heavily in all the expansions when α > λ. Thus, as with the other estimators, a first term approximation would not suffice for the properties of â₁ and â₂. Further, the results give evidence that â₁ and â₂ are highly efficient for most α and λ, and, in some regions of the parameter space, have less bias than α* and â. Some experimental data was fitted to the NBD using the estimators â₁, â₂, â, and α*. In all of the examples given, the modified moment estimators provided a better fit of the data than did the simple moment estimator and, in one instance, a better fit than was obtained by the maximum likelihood estimator. / Ph. D.

Nesta dissertação estudou-se o modelo GARMA para modelar séries temporais de dados de contagem com as distribuições condicionais de Poisson, binomial e binomial negativa. A principal finalidade foi analisar no contexto clássico e bayesiano, o desempenho e a qualidade do ajuste dos modelos de interesse, bem como o desempenho dos percentis de cobertura dos intervalos de confiança dos parâmetros para os modelos adotados. Para atingir tal finalidade considerou-se a análise dos estimadores pontuais bayesianos e foram analisados intervalos de credibilidade. Neste estudo é proposta uma distribuição a priori conjugada para os parâmetros dos modelos e busca-se a distribuição a posteriori, a qual associada a certas funções de perda permite encontrar estimativas bayesianas para os parâmetros. Na abordagem clássica foram calculados estimadores de máxima verossimilhança, usandose o método de score de Fisher e verificou-se por meio de simulação a consistência dos mesmos. Com os estudos desenvolvidos pode-se observar que, tanto a inferência clássica quanto a inferência bayesiana para os parâmetros dos modelos em questão, apresentou boas propriedades analisadas por meio das propriedades dos estimadores pontuais. A última etapa do trabalho consiste na análise de um conjunto de dados reais, sendo uma série real correspondente ao número de internações por causa da dengue em Campina Grande. Estes resultados mostram que tanto o estudo clássico, quanto o bayesiano, são capazes de descrever bem o comportamento da série / In this work, it was studied the GARMA model to model time series count data with Poisson, binomial and negative binomial discrete conditional distributions. The main goal is to analyze, in the bayesian and classic context, the performance and the quality of fit of the corresponding models, as well as the coverage percentages performance to these models. To achieve this purpose we considered the analysis of Bayesian estimators and credible intervals were analyzed. To the Bayesian study it was proposed a priori distribution joined to the models parameters and sought a posteriori distribution, which one associate with to certain loss functions allows finding out Bayesian estimates to the parameters. In the classical approach, it was calculated the maximum likelihood estimators using the method of Fisher scoring, whose interest was to verify, by simulation, the consistence. With the studies developed we can notice that, both classical and inference Bayesian inference for the parameters of those models, presented good properties analysed through the properties of the punctual estimators. The last stage of the work consisted of the analysis of one real data set, being a real serie corresponding to the admission number because of dengue in the city of Campina Grande. These results show that both the classic and the Bayesian studies are able to describe well the behavior of the serie

Nesta dissertação estudou-se o modelo GARMA para modelar séries temporais de dados de contagem com as distribuições condicionais de Poisson, binomial e binomial negativa. A principal finalidade foi analisar no contexto clássico e bayesiano, o desempenho e a qualidade do ajuste dos modelos de interesse, bem como o desempenho dos percentis de cobertura dos intervalos de confiança dos parâmetros para os modelos adotados. Para atingir tal finalidade considerou-se a análise dos estimadores pontuais bayesianos e foram analisados intervalos de credibilidade. Neste estudo é proposta uma distribuição a priori conjugada para os parâmetros dos modelos e busca-se a distribuição a posteriori, a qual associada a certas funções de perda permite encontrar estimativas bayesianas para os parâmetros. Na abordagem clássica foram calculados estimadores de máxima verossimilhança, usandose o método de score de Fisher e verificou-se por meio de simulação a consistência dos mesmos. Com os estudos desenvolvidos pode-se observar que, tanto a inferência clássica quanto a inferência bayesiana para os parâmetros dos modelos em questão, apresentou boas propriedades analisadas por meio das propriedades dos estimadores pontuais. A última etapa do trabalho consiste na análise de um conjunto de dados reais, sendo uma série real correspondente ao número de internações por causa da dengue em Campina Grande. Estes resultados mostram que tanto o estudo clássico, quanto o bayesiano, são capazes de descrever bem o comportamento da série / In this work, it was studied the GARMA model to model time series count data with Poisson, binomial and negative binomial discrete conditional distributions. The main goal is to analyze, in the bayesian and classic context, the performance and the quality of fit of the corresponding models, as well as the coverage percentages performance to these models. To achieve this purpose we considered the analysis of Bayesian estimators and credible intervals were analyzed. To the Bayesian study it was proposed a priori distribution joined to the models parameters and sought a posteriori distribution, which one associate with to certain loss functions allows finding out Bayesian estimates to the parameters. In the classical approach, it was calculated the maximum likelihood estimators using the method of Fisher scoring, whose interest was to verify, by simulation, the consistence. With the studies developed we can notice that, both classical and inference Bayesian inference for the parameters of those models, presented good properties analysed through the properties of the punctual estimators. The last stage of the work consisted of the analysis of one real data set, being a real serie corresponding to the admission number because of dengue in the city of Campina Grande. These results show that both the classic and the Bayesian studies are able to describe well the behavior of the serie

Master of Science / Department of Statistics / Leigh Murray / This case study was motivated by the need for effective statistical analysis for a series of poultry feeding experiments conducted in 2006 by Kansas State University researchers in the department of Animal Science. Some of these experiments involved an automated auger feed line system commonly used in commercial broiler houses and continuous, proportion response data. Two of the feed line experiments are considered in this case study to determine if a statistical model using a non-normal response offers a better fit for this data than a model utilizing a normal approximation. The two experiments involve fixed as well as multiple random effects. In this case study, the data from these experiments is analyzed using a linear mixed model and Generalized Linear Mixed Models (GLMM’s) with the SAS Glimmix procedure. Comparisons are made between a linear mixed model and GLMM’s using the beta and binomial responses. Since the response data is not count data a quasi-binomial approximation to the binomial is used to convert continuous proportions to the ratio of successes over total number of trials, N, for a variety of possible N values. Results from these analyses are compared on the basis of point estimates, confidence intervals and confidence interval widths, as well as p-values for tests of fixed effects. The investigation concludes that a GLMM may offer a better fit than models using a normal approximation for this data when sample sizes are small or response values are close to zero. This investigation discovers that these same instances can cause GLMM’s utilizing the beta response to behave poorly in the Glimmix procedure because lack of convergence issues prevent the obtainment of valid results. In such a case, a GLMM using a quasi-binomial response distribution with a high value of N can offer a reasonable and well behaved alternative to the beta distribution.

Neste trabalho, alguns resultados, tais como, função geradora de momentos, relações de recorrência para os momentos e alguns teoremas da classe de distribuições em séries de potencias modificadas (MPSD) proposta por Gupta (1974) e da classe de distribuições em séries de potências modificadas inflacionadas (IMPSD) tanto em um ponto diferente de zero como no ponto zero são apresentados. Uma aplicação do Modelo Poisson padrão, do modelo binomial negativo padrão e dos modelos inflacionados de zeros para dados de contagem, ZIP e ZINB, utilizando-se as técnicas dos MLGs, foi realizada para dois conjuntos de dados reais juntamente com o gráfico normal de probabilidade com envelopes simulados. Também foi proposta a distribuição Weibull binomial negativa (WNB) que é bastante flexível em análise de dados positivos e foram estudadas algumas de suas propriedades matemáticas. Esta é uma importante alternativa para os modelos Weibull e Weibull geométrica, sub-modelos da WNB. A demostração de que a densidade da distribuição Weibull binomial negativa pode ser expressa como uma mistura de densidades Weibull é apresentada. Fornecem-se, também, seus momentos, função geradora de momentos, gráficos da assimetria e curtose, expressoes expl´citas para os desvios médios, curvas de Bonferroni e Lorenz, função quantílica, confiabilidade e entropia, a densidade da estat´stica de ordem e expressões explícita para os momentos da estatística de ordem. O método de máxima verossimilhança é usado para estimar os parametros do modelo. A matriz de informação esperada ´e derivada. A utilidade da distribuição WNB está ilustrada na an´alise de dois conjuntos de dados reais. / In this paper, some result such as moments generating function, recurrence relations for moments and some theorems of the class of modified power series distributions (MPSD) proposed by Gupta (1974) and of the class of inflated modified power series distributions (IMPSD) both at a different point of zero as the zero point are presented. The standard Poisson model, the standard negative binomial model and zero inflated models for count data, ZIP and ZINB, using the techniques of the GLMs, were used to analyse two real data sets together with the normal plot with simulated envelopes. The new distribution Weibull negative binomial (WNB) was proposed. Some mathematical properties of the WNB distribution which is quite flexible in analyzing positive data were studied. It is an important alternative model to the Weibull, and Weibull geometric distributions as they are sub-models of WNB. We demonstrate that the WNB density can be expressed as a mixture of Weibull densities. We provide their moments, moment generating function, plots of the skewness and kurtosis, explicit expressions for the mean deviations, Bonferroni and Lorenz curves, quantile function, reliability and entropy, the density of order statistics and explicit expressions for the moments of order statistics. The method of maximum likelihood is used for estimating the model parameters. The expected information matrix is derived. The usefulness of the new distribution is illustrated in two analysis of real data sets.