A Study On Estimating And Coparing The Variances Under A Semiparametric Density Ratio Model

Estimating and hypothesis testing of the variance are basic statistical problems. The traditional variance estimation and hypothesis testing focus on the parametric or nonparametric method. Parametric method is often under the normal distribution assumption.There are the individual variance of Chi-square test and two side comparison of F testing, etc. Nonparametric method does not require the normal distribution assumption. We can construct the order statistics or the asymptotic distribution of estimator for the statistical inference. The latest nonparametric method aims at the empirical likelihood theory. The asymptotic relative efficiency of the nonparametric method is worse than the parametric method when data are normal, conversely nonparametric method is better. In this paper, a new method is proposed for the variance estimation and testing. For the statistic inference of the variance, we put forward a semiparametric method which is under a semiparametric density ratio model. Some theoretical results as well as simulation results are given in this paper, also presented are the analysis of the real data sets. The simulation results show that the semiparametric method is slightly superior to some commonly used parametric and nonparametric methods when data are normal, and is significantly better than them when data are no longer normal. Instance analysis, we apply our method to the data set collected from the literature, and we obtain the same conclusion from the nonparametric test statistic.In chapter 1, we introduce the background and significance of the statistic inference about the variances, then summarized the research status and describe the related knowledge of this paper briefly.In chapter 2, under the semiparametric density ratio model, the first we give the semiparametric estimation of the single population variance and the Bootstrap description of the variance. Then we derive the asympotic distributions of the two population variances’estimator and show that is more efficient than traditional nonparametric method. At last, the Wald semiparametric test statistic is constructed.In chapter3, In the diffident distribution assumption, we have a comparison during a parametric F test method, a nonparametric method and the proposed semiparametric method. Then real examples are analysed by the method we proposed.At last, we make a summary of the paper and give a prospect.