"Bootstrap, confidence sets, and asymptotic analysis for some procedures of statistical inference"

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Although my research program concerns many different aspects of statistics and probability theory, it revolves around a single theme: asymptotics (expansion of a statistic and/or of a distribution). The research involves the following four main components: 1. A general statement of the interval estimation problem for the ratio of two binomial proportions arising from data coming from two independent samples will be considered. Each sample may be obtained in the framework of direct or inverse binomial sampling. Asymptotic confidence intervals are constructed in accordance with different types of sampling schemes and unbiased estimators of success probabilities and their logarithms will be obtained. The ratio of two binomial proportions arises in prospective studies, in biological experiments, and in the comparison of manufacturing processes for quality control in industry. Statistical procedures for the ratio of binomial proportions (often called the relative risk) are also quite common in clinical trials and epidemiological studies, or more generally in a pharmaceutical setting. Hence we expect that this part of my (mostly theoretical) research will have many significant applications. 2. Asymptotic probability for the deviations of some nonclassical bootstrap U-statistics from the sample U-statistic will be investigated. I am mostly interested in such limit theorems for dependent bootstrap random variables as the law of the single logarithm, the central limit theorem, and the law of large numbers. 3. The construction of a confidence region based on the shrinkage estimator with constant confidence level will be investigated. This will require rectification of formulae for coverage probabilities by confidence sets centered at both James-Stein estimators. 4. Construction of stochastic models for crack development (models related to the Birnbaum-Saunders and Inverse Gaussian distributions) will be investigated. Solutions to the statistical problems of parameter estimation and hypothesis testing for these new models will be provided.