A Goodness-of-fit Test Based On Empirical Euclidean Likelihood

In theoretical study and practical applications, goodness of fit test is one of the important topics in statistics. It got a great development in60s and70s last century. Besides the known chi-square type test, Empirical distribution function (EDF) type goodness of fit tests are encoun-tered frequently in literatures. These tests include Cramer-von Mises(CV) tes、Kolmogorov-Smirnov(KS) test、and Anderson-Darling (AD) test and so on. On the other hand, empirical likelihood method is a non-parametric statistical method proposed by Owen(1988,1990), and de-veloped by Qin and Lawless(1994) et al. Later, the method is discussed and applied to the practical problem by many statisticians. Empirical Euclidean likelihood is also a non-parametric method, where Euclidean distance is used to substitute the likelihood distance in empirical likelihood. Em-pirical Euclidean likelihood has the same large sample properties as empirical likelihood. It is an extension of Empirical likelihood. Therefore, study on goodness of fit test with empirical Eu-clidean likelihood is valuable in both theory and application.Goodness of fit test method based on empirical Euclidean likelihood and its main prosperi-ties are discussed in this thesis. Firstly, a new test statistics is constructed by means of empirical Euclidean likelihood, and then the asymptotic distribution of the test statistics is derived under the simple null hypothesis. Secondly, the parameters are estimated by maximum likelihood method (MLE) under the composite null hypothesis. Subsequently, the corresponding test statistics is con-structed and its limit distribution is given. Moreover, an improved principle of partition of classes using vertical density representation theory is proposed to overcome the weakness that the tradi-tional partition of classes is not unique for the same sample. The new packet principle based on the value of probability density function removes the packet arbitrariness, and improves the reliability of test results. Finally, the new test is compared with chi-square test and KS test. Simulation results show that the new test has higher power and smaller computational complexity and the same large sample properties as empirical likelihood. Therefore, the new test shows favorable application and potential prospect.The new idea of the thesis is as follows:1. The goodness of fit test problems based on the concept of empirical Euclidean likelihood and its corresponding prosperities are firstly discussed in this thesis. 2. A new principal of partition of classes based on vertical density representation theory is proposed, which not only solves the non-unique problem of traditional partition of classes, and also shows a higher power.3. Empirical Euclidean likelihood method has a smaller computational complexity and our results can make up the theory of goodness of fit and non-parametric statistics.