Approach To Simultaneous Testing Of Continuous Populations Parameters And Convolution Of Negative Binomial Random Variables

Simultaneous testing of multiple continuous populations parameters by order statistics and exact distribution of the convolution of negative binomial random variables by probability generating function and BELL polynomials are two main contents of this thesis.It is important to test homogeneity of parameters of multiple continuous populations in statistics.For example,testing for homogeneity of mean-values of multiple normal populations is a classical problem in the statistical analysis problems,and it has been applied in many fields,such as experimental design,venture capital and quality management,etc.So,it is great significance and necessary to carry out the research of simultaneous testing of multiple continuous populations parameters.In addition,the case of order statistics of independent and identical random variables sequence discussed more than the order statistics of independent but not identical random variables sequence in existing literatures.So,it is novelty and uniqueness that testing homogeneity of parameters of multiple continuous populations by order statistics of independent but not identical random variables sequence.Definition and nature of order statistics of independent but not identical random variable sequence are given in chapter 3.Then,definition of maximum-ratio statistic is given according to the order statistics.After that,testing homogeneity of scale parameters of multiple uniform populations,testing homogeneity of scale parameters of multiple one-parameter exponential populations,testing homogeneity of scale parameters of multiple two-parameter exponential populations,testing homogeneity of variances of multiple normal populations by maximum-ratio statistic are discussed in chapter 3.Finally,some figures of probability density functions of maximum-ratio statistic and some tables of quantiles of maximum-ratio statistic are presented in chapter 3.Different problems can be solved by the same method,and pivot variable is consistent in the latter three problems mentioned above,so the pivot variable need to be more discussed and can be widely used in the further.Related properties about order statistics,record values,quantiles,limit theorems,convolutions etc of discrete random variables sequence are less discussed than continuous random variables sequence,because the properties mentioned above of discrete random variables sequence are discussed by tools and methods in combinatorics which are very complicated and tedious to be calculated.So,compared with classical result about continuous random variables sequence,it is very meaningful to profoundly study probabilistic properties of discrete random variables sequence in detail.Negative binomial random variables are common in discrete random variables which have practical significance in reliability experiment and insurance claim etc.Geometric distribution is a special negative binomial distribution and a important discrete distribution.Geometric distribution and exponential distribution have many similar properties.The exact distribution and the recursion of the convolution of independent negative binomial random variables is derived by using the method of generating function and BELL polynomials in chapter 4.Moreover,being a particular case,the exact distribution and the recursion of the convolution of independent geometric random variables is given in chapter 4 too.Finally,some figures of probability mass function of convolution and a new equation about different theorems of convolution of geometric random variables are presented in chapter 4.