Asymptotic analysis for point and interval estimation in some statistical models

某些统计模型中点估计和区间估计的渐近分析

• 批准号: RGPIN-2017-06304
• 负责人: Volodin, Andrei
• 金额: $ 1.17万
• 依托单位: University of Regina
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=667870
• 关键词: Asymptotic; analysis; point; interval; estimation

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.

尽管我的研究计划涉及统计学和概率论的许多不同方面，但它围绕一个主题：渐近学（统计或分布的扩展）。我的研究涉及以下四个主要部分：基因表达的统计测试（pFDR 和 d-risk 方法）；依赖结构；一些统计分布参数的点估计；以及两个二项式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所谓的 d 后验方法框架中多重假设检验问题的一个特例。它基于贝叶斯范式，可以应用于统计实验的各种情况。每次实验都会产生一个决定，并且必须保证错误率。我将应用最佳测试来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，特别是低活跃基因选择的问题。******2。我对依赖结构的兴趣源于它们在一些统计程序中的令人着迷的应用，在这些程序中，观察的独立性假设被违反了。一个典型的例子是相关引导过程，其中重采样是在不进行替换的情况下完成的。我多年来一直致力于解决这些问题，我的主要目标是获得相关自举过程的迭代对数定律。这将使我对相关自举随机变量的渐近行为进行完整的描述。********我对相关结构的研究的另一个有趣的组成部分与对大数定律适用性假设的研究有关以弱和强的形式与负相关的随机变量。有界负相关随机变量最大和的指数不等式的推导对于极限定理至关重要，特别是为负相关随机变量建立弱和强大数定律。这里的主要困难是证明当下假设是必要且充分的，即建立标准。******3．我的提案的下一个组成部分与两个二项式比例的比率的置信区间构造有关。迄今为止，这个统计问题仅针对两个样本中具有固定观测值的采样方案得到解决。我的目标是找到一种通用方法来构建不同抽样方案比例的置信区间。******4。二项式分布参数的矩估计方法存在问题。这些估计量甚至没有期望，并且可能具有超出参数自然范围的值。因此，需要修改这些估计器。