Asymptotic analysis for point and interval estimation in some statistical models

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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 or of a distribution). My research involves the following four main components: statistical testing for expression of genes (pFDR and d-risk approaches); dependent structures; point estimation for parameters of some statistical distributions; and the interval estimation problem for the ratio of two binomial proportions.******1. I consider the problem of data analysis of gene expression as a special case of the problem of multiple hypothesis testing in the framework of the so-called d-posterior approach. It is based on the Bayesian paradigm and can be applied to the various cases of statistical experiments. Each experiment leads to a decision and the falsity rate must be guaranteed. I will apply the optimal test to the problem of identifying of hyperactive genes responsible for a disease and will establish a general Bayesian model for solving similar problems, in particular problems of hypoactive genes selection.******2. My interest in dependent structures arose from their fascinating applications to some statistical procedures where the assumption of independency of observations is violated. A classical example would be the dependent bootstrap procedure where resampling is done without replacement. I have been working on these problems for many years, and my main goal is to obtain the Law of the Iterated Logarithm for the dependent bootstrap procedure. This will lead me to a complete description of the asymptotic behaviour of the dependent bootstrap random variables.******Another interesting component of my investigation on dependent structures is connected with an investigation on the assumptions of applicability of the law of large numbers in weak and strong forms to negatively associated random variables. A derivation of exponential inequalities for maximum sums of bounded negatively associated random variables is crucial for limit theorems, especially establishing weak and strong laws of large numbers for negatively associated random variables. The main difficulty here is to show that the moment assumptions are necessary and sufficient, that is, to establish criteria.******3. My next component of the proposal is connected with a confidence interval construction for a ratio of two binomial proportions. To date, this statistical problem has been solved only for sampling schemes with a fixed number of observations in both samples. My goal is to find a universal approach for confidence interval construction for the ratio of proportions with different sampling schemes.******4. There is a problem with the method of moments estimation of parameters of the binomial distribution. These estimators do not even have expectation and can have values which are out of the natural range of the parameters. Hence, modifications of these estimators are required.