esearch is to find out the Performance of the bootstrap confidence region of a binomial distribution with unknown parameters.

Abstract

The goal of this research is to find out the Performance of the bootstrap confidence region of a binomial distribution with unknown parameters. The research is designed as follow:

The first step is to estimate unknown parameters m and p from binomial distribu- tion. In my case, I focus on method of moment to estimate. Since these estimators do not have moments of all orders, I cannot obtain the mean, variance, and covariance for these estimators. Thus, the Delta Method is used to derive the asymptotic nor- mality of the joint distribution of the Method of Moments estimators. After finding the estimators pˆ, and mˆ , I will work on the Asymptotic Normality of the Estimators. For Asymptotic Normality of the Estimators by method of moment, I will find the sampling from the binomial distribution with sample mean X¯ − mp and sample variance S2 − mp(1 − p) are asymptotically normal with zero mean vector and co-variance matrix. I will apply the Delta method consists of expansion of pˆ, and mˆinto two-dimensional Taylor series expansion, then using partial derivatives of these functions, so I should have the covariance matrix.

Next section, I find out that if random vector X is normally distributed with the mean vector E and covariance matrix Σ is distributed as chi-square with 2 degrees of freedom. Therefore, the 100(1 − α)% confidence region should be X2(p, m) ≤ X2(α) with 2 degree of freedom.

Some general steps of using the independent and dependent bootstrap sampling will be the next. I will give example of creating both independent and dependent bootstrap samples with different k, where k is the number of copies of original sample. I also will talk about the coverage probability of confidence regions and the areas of confidence regions.

Introduction

1.1 Research Objectives

The estimation of binomial distribution parameters m and p achieve in four common inference methods: Bayesian Method, the Maximum Likelihood, Method of Moments, Method of Least Squares. Method of Moments is the only method for this research.The independent bootstrap sampling method will be applied for resampling the estimators mˆand pˆ, in order to construct the 100(1 − α)% Confidence regions and calculate the converage probabilities, as well as the dependent bootstrap sampling method with different k. The asymptotic normality of the joint distribution of the method of moment estimators mˆ and pˆ is proved by the delta method. The confidence region of estimators mˆ and pˆ can be written as form which follows the chi-square distribution with two degrees of freedom.

1.2 Research Hypothesis

The goal is to see whether the bootstrap procedure can influence the confidence regions for Bin(m,p), such as changing the coverage probability, and the areas of confidence regions. Furthermore, I will compare the results for both independent bootstrap samples and dependent bootstrap samples, in order to test the performance of bootstrap sampling. In my case, I focus on method of moment to estimate. Delta Method is used to derive the asymptotic normality of the joint distribution of the Method of Moments estimators. After finding the estimators pˆ, and mˆ , I will work on the Asymptotic Normality of the Estimators.For the simulation part, I plan to generate N binomial samples with fixed m and p with each sample size n. Then I will generate the estimators for all m and p. Applying the independent and dependent bootstrap for re-sampling propose, I will have B sets of bootstrap samples of the estimator's form and p. A summary of all B bootstrap samples of coverage probabilities can give some results of this research. Moreover, I can change the value of true m, p, B, and k to compare all the results by plots and tables.

1.3 Research Statement

Using chi-square test, to define the rules for Confidence regions of Estimation of m and p from binomial distribution by method of moment on independent bootstrap method and dependent bootstrap method with different k. The coverage probability of confidence regions are not evidently different, but the Areas of confidence regions.

1.4 The Significance of This Research

The method of Sampling is the first step of statistical analysis. However, the samples which selected from the population are not perfect. Because the dataset of population must include every entry of the selected topic, it is impossible to find all of them. Therefore, constructing samples in statistical way is one of the major goal for statisticians.

A good statistical analysis relying on precise samples. The statisticians and data analyst spend time, patience, and financial support ceaselessly, in order to construct accurate samples. The classical sampling method, such as simple random sampling is often costly and inefficient due to all elements with same probability of being selected (Yates, David, and Daren, 2008). This research will focus on using simple and accurate sampling method: the bootstrap sampling.