Parameter Estimation Of Statistics Model And Independence Of Quadratic Form

Gamma distribution,exponential distribution and weibull distribution are widely used in survival analysis and reliability analysis.They are very important distributions in statistics.Statistical inference about them has always been the hot topic of statistics.Statistical inference mainly includes parameter estimation and hypothesis testing.Parameter estimation includes point estimation and interval(domain)estimation.For the duality between interval(domain)estimation and hypothesis testing,we focus on parameter estimation in this thesis.Firstly,as for point estimation of Gamma distribution,methods such as moment estimation,maximum likelihood estimation,quasimaximum likelihood estimation,which mainly aim at correcting the bias and refer to little efficiency improvement.However,the excellence of point estimation directly affects the accuracy of interval estimation and the efficacy of hypothesis testing.Then it is important to study the efficiency of point estimation.Secondly,for interval(domain)estimation,the shortest confidence interval of a single parameter is mostly mature.However,there is no integrated method or conclusion for the optimal region estimation of multiple parameters especially in incomplete samples(such as censored data and record values).Therefore,we study the efficient estimation of Gamma distribution,investigate the optimal confidence regions for two-parameter exponential distribution based on censored samples,construct optimal confidence regions for two-parameter exponential distribution and weibull distribution based on record values,and study the independence of two quadratic form functions in normal variate.The main work of this thesis is as follows:1.The thesis concerns the efficient estimation of Gamma distribution.We propose a method of profile log-likelihood function and Brent's algorithm of univariate optimization without derivatives,which can greatly reduce the calculation difficulty of maximum likelihood estimation.Applying a linear function of the maximum likelihood estimation of shape parameter to estimate the shape parameter,we construct a linear regression model by minimizing the mean squared error.We obtain new estimations by Monte Carlo simulation and prove that new estimations have the large sample property.Simulation study is investigated to illustrate the effectiveness and feasibility of the proposed method,2.The thesis studies the optimal confidence region for two-parameter exponential distribution based on censored data.According to the sufficiency principle in statistics,we propose a method based on a sufficient statistic to construct the minimum area confidence region and study the invariance with the change of data scale.Applying the new method to two-parameter exponential distribution under progressively Type ?censored sample and Type ? doubly censored sample,we obtain the minimum area confidence regions respectively and compare them with classical confidence regions.Comparative study and real data analysis are investigated to illustrate the feasibility and optimality of the new method.3.The thesis considers the minimum area confidence regions for two-parameter exponential distribution and weibull distribution under record values.We obtain the minimum area confidence region for two-parameter exponential distribution by a method based on a sufficient statistic.Under certain conditions,we put forward a new method to establish the minimum area confidence region for weibull distribution,and prove that the optimality of confidence region does not change when the data scale changes.Then we compare the minimum area confidence regions with classical regions by estimation efficiency.Simulation study and real data analysis show the feasibility and optimality of the proposed method.4.The thesis investigates the independence of two quadratic form functions in normal variate.Based on research productions of the Craig-Sakamoto theorem that have been obtained,we generalize and obtain two unified theorems about the independence of two linear forms,a linear form and a quadratic form function and two quadratic form functions:one in multivariate standard normal variate and another in general multivariate normal variate.