Weighted ECF-type Test And In The Goodness Of Fit Of The Application

Goodness-of-fit test has a special status in statistics. It is not only a part of fundamental knowledge, but also has a close relationship with practical applica-tions. We often need to judge whether population distribution or data belongs to some a distribution family required by the corresponding statistic model. It’s well known that choosing test statistics is key to goodness-of-fit and different test statistics have different power. So it is very important to find a test which has good power and convenient calculation.This paper first reviews present research situation of the classical EDF and ECF tests, and shows that the ECF test works well, but it does not have the optimal choice for the weight function and parameter, what’more it has compli-cated calculation. In order to overcome these disadvantages, we need to find a improved method.A probability weighted ECF test which is proposed by Meintanis latest. It combines empirical characteristic function with the weighted function using probability distribution function of underlying distribution and its structure still comes from the old ECF test method, which does’t need to consider the weight function in the classical ECF test and can reflect the tail properties of distribu-tion. Motivated by this new method,we want to do more research and consider its application in the goodness-of-fit of scale-local distributions.The chapter2mainly discusses the probability weighted ECF testing method including the weak convergence properties of the probability weighted empirical characteristic function (PWECF)as a process and its application in the goodness-of-fit. We focus on the asymptotic distribution of the test under the null hypothesis according to the classical method and its consistency.In the chapter3we use Monte-Carlo simulation method to get the power of Cauchy distribution, Laplace distribution, Logistic distribution as typical distri-butions in local-scale distribution family and compare with other omnibus tests. Then we show that the new test is best mostly,especially in the case of small samples.In the chapter4we use the new weighted ECF test to do goodness-of-fit testing including Laplace distribution and Logistic distribution combined with some real data collections from relevant literatures.We also show that this test is workable and has simple calculation. In the end of the part, we discuss some deficiency of it and look forward to more research in the future.