Asymptotic analysis for point and interval estimation in some statistical models

尽管我的研究计划涉及统计学和概率论的许多不同方面，但它围绕着一个主题：渐近性(统计或分布的扩展)。我的研究涉及以下四个主要部分：基因表达的统计检验(pFDR和d-Risk方法)；相依结构；一些统计分布参数的点估计；以及两个二项比例之比的区间估计问题。 1.我认为基因表达的数据分析问题是在所谓的d-后验方法框架下的多重假设检验问题的特例。它是基于贝叶斯范式的，可以应用于各种统计实验的案例。每一次实验都会导致一个决定，必须保证错误率。我将把最优检验应用于识别导致疾病的过度活跃基因的问题，并将建立一个通用的贝叶斯模型来解决类似的问题，特别是低活性基因选择问题。 2.我对相依结构的兴趣源于它们在一些统计程序中的迷人应用，在这些程序中，观测的独立性假设被违反。一个经典的例子是依赖引导过程，其中重采样是在没有替换的情况下完成的。我多年来一直在研究这些问题，我的主要目标是获得依赖Bootstrap过程的重对数律。这将引导我对相依自举随机变量的渐近行为进行完整的描述。 在我对相依结构的研究中，另一个有趣的部分与一项关于弱形式和强形式的大数定律适用于负相关随机变量的假设的调查有关。有界负相协随机变量最大和的指数不等式的推导对于极限定理，特别是建立负相协随机变量的弱大数定律和强大数定律是至关重要的。这里的主要困难是要证明假设是必要的和充分的，也就是建立标准。 3.我提议的下一个部分是关于两个二项比例的可信区间的构造。到目前为止，这个统计问题只在两个样本中都有固定观测次数的抽样方案中得到了解决。我的目标是为不同抽样方案下的比例比找到一种通用的可信区间构造方法。 4.二项分布参数的矩估计方法存在问题。这些估计器甚至没有期望值，其值可能超出参数的自然范围。因此，需要对这些估计值进行修改。

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.