Hoi Wai-leung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 62-63). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Preminilaries --- p.8 / Chapter 2.1 --- Gauss and Jacobi Sums --- p.8 / Chapter 2.2 --- Basic Properties of Characters and Jacobi Sums --- p.10 / Chapter 2.3 --- Characters on Ok/Q --- p.11 / Chapter 2.4 --- Bernoulli Numbers and Bernoulli Polynomials --- p.15 / Chapter 2.5 --- Evaluation of Bp-1(k)-Bp-1 (mod p) --- p.16 / Chapter 2.5.1 --- "k = 2,3,4,6" --- p.16 / Chapter 2.5.2 --- "k = 5,8,10,12" --- p.17 / Chapter 3 --- The Main Result --- p.19 / Chapter 3.1 --- p-adic Gamma Functions and Gross-Koblitz Formula --- p.19 / Chapter 3.2 --- The Sum Σ - j(modp) --- p.22 / Chapter 3.3 --- The Main Theorem --- p.23 / Chapter 4 --- Evaluation of J(xk-r： Xk-s) (mod Q2) --- p.29 / Chapter 4.1 --- Basic Result --- p.29 / Chapter 4.2 --- Results for k = 8 --- p.30 / Chapter 4.3 --- Results for k = 12 --- p.32 / Chapter 4.4 --- "Results for k = 10,5" --- p.35 / Chapter 5 --- Evaluation of Binomial Coefficients --- p.41 / Chapter 5.1 --- Results for k = 8 --- p.41 / Chapter 5.2 --- Results for k = 12 --- p.44 / Chapter 5.3 --- Results for k = 10 --- p.49 / Chapter 5.4 --- Numerical Examples --- p.54 / Chapter 5.4.1 --- k = 8 --- p.54 / Chapter 5.4.2 --- k = 10 --- p.56 / Chapter 5.4.3 --- k = 12 --- p.58 / Bibliography --- p.61

The well known binomial coefficient is the building block of Pascal’s triangle. We explore the relationship between functions of the binomial coefficient and Pascal’s triangle, providing proofs of connections between Catalan numbers, determinants, non-intersecting paths, and Baxter permutations.

Thesis (Ph. D.)--University of Oregon, 2006. / Typescript. Includes vita and abstract. Includes bibliographical references (leaf 77). Also available for download via the World Wide Web; free to University of Oregon users.

In survey research, researchers often add up a finite number of binary responses to form an index of some political attitude or behavior, such as political knowledge and political participation. Indices of this sort are called grouped binary variables in political science research; they comprise finite and countable binary items that take on only integer values ranging from zero to the total number of survey items. Commonly-used distributions for modeling these kinds of indices are the binomial, beta binomial, and extended beta binomial distributions. But whether these distributions are appropriate depends on the assumptions that the binary responses are identically and independently distributed Bernoulli random variables. If these assumptions are violated, the binomial, beta binomial, and extended beta binomial models are rendered questionable, and it may be more useful to turn to other distributions of sums of Bernoulli variables, called generalized binomial distributions. To facilitate the use of generalized binomial distributions in political science research, this report is a review of the various probability distributions of grouped binary variables. This report clarifies the nature of the distributions of sums of Bernoulli variables in survey research by considering whether the Bernoulli variables are independently and/or identically distributed, whether there is heterogeneity across survey items and/or across respondents, and the consequences of these considerations for the relative dispersion of each generalized binomial distribution. / text

Two of the more well known congruences for binomial coefficients modulo p, due to Gauss and Jacobi, are related to the representation of an odd prime (or an integer multiple of the odd prime) p as a sum of two squares (or an integer linear combination of two squares). These two congruences, along with many others, have been extended to analogues modulo p^2 and are well documented. More recently, J. Cosgrave and K. Dilcher have extended the congruences of Gauss and Jacobi to analogues modulo p^3. In this thesis we discuss their methods as well as the potential of applying them to similar congruences found in the literature.

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

<p> This thesis is concerned with the properties of sequential binomial estimation. It illustrates the construction of optimal sequential binomial sampling plans for point estimation problems in which, according to custom, each loss function is taken to be a constant times the square of the error. The way such a constant affects the sizes of the constructed sampling plans is also within the scope of this thesis.</p> / Thesis / Master of Science (MSc)

Simularam-se 1000 ensaios para cada uma das 112 combinações de 4 populações (tratamentos), englobando casos de populações iguais e casos com diferenças em m, k ou ambos. Deve-se ter cuidado ao aplicar transformações de dados, particularmente se não há homogeneidade de k. A estatística C(&#945;) proposta por BARNWAL & PAUL (1988), mostrou alguma robustez para valores não homogêneos de k, conduzindo a resultados equivalentes àqueles obtidos usado dados não transformados. A análise de variância, usado o teste de mínimo qui-quadrado XU2 mostrou ser viesado, superestimando valores quando a matriz de variâncias e covariâncias é desconhecida. Se a matriz de variâncias e covariâncias é conhecida, os resultados são equivalentes a aqueles obtidos dos dados originais. Resultados similares foram obtidos para populações menores, n=10, quando um poder decrescente do teste foi detectado. Foram escolhidos 20 casos e simularam-se 1000 ensaios para cada caso. / The well-known negative binomial distribution is quite frequently used to interpret counting variables, through different techniques. In order to compare these techniques, four populations of size n=50 were computer-generated for different values of the parameters m and k, using NORMAN & CANNON (1972) procedure. Comparisons of the transformations of variables, as suggested by BARBOSA (1985), were used. For that, 1000 essays were simulated for each one of the 112 combinations of 4 populations. This covered equal and different populations with respect to the parameters m, k or both. In conclusion, for some values of the parameters m and k, there is no necessity of any data transformation, particularly if depending of k. Statistics like C(&#945;) proposed by BARNWAL & PAUL (1988) showed some robustness for non-homogeneous values of k, leading to equivalent results to that ones obtainned using untransformed data. The analysis of variance, using the minimum chi-square test U2 showed to be biased superestimating values when the variance-covariance matrix is unknown. If the variance-covariance matrix is knew the results are equivalent from those obtainned from original data. Similar results were obtainned for smaller populations, n=10, when a decreasing power of the tests was detected. In such a case 20 combinations and 1000 simulations for each combination were performed

I have developed a tiling interpretation of the q-binomial coefficients. The aim of this thesis is to apply this combinatorial interpretation to a variety of q-identities to provide straightforward combinatorial proofs. The range of identities I present include q-multinomial identities, alternating sum identities and congruences.

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