Asymptotic analysis for point and interval estimation in some statistical models

虽然我的研究计划涉及统计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组成部分：基因表达的统计检验（pFDR和d风险方法）;相关结构;一些统计分布参数的点估计;以及两个二项比例的区间估计问题。 1.我认为基因表达的数据分析问题作为一个特殊的情况下，在所谓的d-后验方法的框架中的多个假设检验的问题。它基于贝叶斯范式，可以应用于统计实验的各种情况。每个实验都导致一个决定，并且必须保证虚假率。我将最优检验应用于识别负责疾病的高活性基因的问题，并将建立一个通用的贝叶斯模型来解决类似的问题，特别是低活性基因的选择问题。 2.我对依赖结构的兴趣来自于它们在一些统计过程中的迷人应用，在这些过程中，观测的独立性假设被违反了。一个经典的例子是依赖的引导过程，其中不进行替换而进行重新排序。多年来，我一直致力于这些问题，我的主要目标是获得法律的迭代对数的依赖自助程序。这将导致我一个完整的描述的渐近行为的相依自助随机变量。 在我对相依结构的研究中，另一个有趣的组成部分是关于弱形式和强形式的大数定律对负相关随机变量的适用性假设的研究。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.这里的主要困难是证明矩假设是必要的和充分的，也就是说，建立标准。 3.我的建议的下一个组成部分与两个二项比例的置信区间构造有关。到目前为止，这个统计问题已经解决了只有一个固定数量的观察在两个样本的抽样方案。我的目标是找到一个通用的方法来构造不同抽样方案下比例比的置信区间。 4.二项分布参数的矩估计方法存在一个问题。这些估计量甚至没有期望值，并且可能具有超出参数自然范围的值。因此，需要对这些估计量进行修改。

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.