The Minimum Nonparametric Likelihood Ratio Test For Goodness Of Fit

In real data analysis, data are always sampled from an unknown population. We want to inferthe population distribution or its numeric characters. There are two kinds of questions:estimationand test. How to solve the questions based on the data from the known distribution family withunknown parameters. It is one of the important topic in Statistics.The distribution family with unknown parameters are investigated in this thesis. Firstly, weshould estimate the unknown parameters. Then, using the estimated parameters, the test problemis discussed. The minimum distance method is one of the important method to solve the abovequestions. It is to construct a function between the empirical distribution function and its theoreticone in the discussed distribution family.Then,we estimate the parameter by minimize the distancefunction.Next,we can obtain the test statistics by substituting the minimum estimator for the un-known parameter. The estimator and the test depend on the data, and can make full use of theinformation.Specially,when the true distribution does not belong to the specified distribution fam-ily,it may give a distribution which is closest to the true distribution. Thus the methods can avoidto misuse the data information.From this point,the minimum distance method is a good idea.It hasmore important theoretic and applied values.Two distance methods are investigated in this thesis,one is the weighted KS distance,the otheris the nonparametric likelihood ratio distance.The former is an extension of traditional KS,thelatter is a further study of the weighted KS distance.Therefore the weighted KS is the basic,and thenonparametric likelihood ratio is a further development.Firstly,the estimator of minimum weighted KS is given.That isThen the corresponding prosperities of T(Fn),such as consistence,robustness and the limit distri-bution.At the same time, the limit distribution of the corresponding test statistics under the nullhypothesis is derived. Next,the estimator of minimum nonparametric likelihood ratio are given.That isSimilar to the weighted KS, the corresponding prosperities of T(Fn)is discussed.Finally,in order to compare our tests with the traditional test,we make a lot of simula-tions.Simulations results show that,under certain circumstances and criterions, the proposed es-timators are more robust than the MLE, and the powers of the new test are higher for some casewe conducted.The new idea of the thesis is as follows:1. As a new distance,the new nonparametric likelihood ratio is investigated under the com-posite hypothesis.2. The new minimum distance estimators are more robust than MLE, and the new tests arepowerful than the ordinary tests.3. Our new mathematical tools for our proof are complex.4. Our results can make up the theory of goodness of fit and nonparametric statistics.