The Parameter Estimation And Hypothesis Testing Of A Special Kind Of Two-parameter Exponential Distribution

Two-parameter exponential distribution, as one of the most basic distributions in probability theory, is used in many areas such as data analysis on survival and longevity, research of reliability theory and so on. As a result, the parameter estimation and hypothesis testing of two-parameter exponential distribution is an interesting subject to many statisticians from all the time, and it has been researched deeply so far. Certaintly, it has been widely used in practice. In this article, however, we discuss the parameter estimation and hypothesis testing on a special kind of two-parameter exponential distribution. Its density function is f(x)=λXe-λ(x-λ),x>λ>0.Firstly, we introduce the practical background of this function. Secondly, we deduce and get the conclusion of the moment estimation and the maximum likelihood estimation on parameter A, and analyze the unbiasedness and effectiveness of the estimation results. Besides, this article provides the test statistic and its distribution, which is2λn(X-λ)-X2(2n). For different hypothesis testing problems, we give the corresponding region of rejection. In this process, we discuss four methods of hypothesis testing:probability ratio test, monotone likelihood ratio and the side test, likelihood ratio test and the p value test. In addition, we consider the Bayes test problem of this special two-parameter exponential distribution, and derive the two expressions of posterior probability. By comparing the posterior probability, we can easily decide to accept which one of the two hypotheses.