Multivariate Goodness Of Fit Tests And Statistical Analysis Of Recurrent Event Data

Goodness-of fit and parameter estimation are the eternal topic in statistical inference.They have various contents in different models.In this thesis,three problems are studied.First,two approaches are proposed in order to modify Pearson's chi-squared test.These modified tests remove the weakness that Pearson's chi-squared test is not stable and partition of sample space is not unique. Second,some applications of vertical density representation in goodness of fit tests of multivariate distribution are considered.Finally,several regression models for recurrent event data are proposed.And the unknown parameters in these models are estimated.The resulting estimators are proven to be consistent and asymp-totically normal.In Chapter 2,two methods are proposed to modify Pearson's chi-squared test. On the one hand,maximized chi-squared test is proposed.A construction of the maximized chi-squared test statistic is obtained.And the maximized chi-squared test is applied to test whether the vectorial data come from the uniformity defined on the hypersphere.Tests include the maximized chi-squared test,Rayleigh,Ajne, Giné,and Bingham tests,are compared the empirical power against the hypothesis of a Von Mises-Fisher distribution or a Watson distribution in some cases. The simulation results show that the maximized chi-squared test is stable against different alternative.On the other hand,based on the value of probability density function,a new principle of partition of classes is proposed.Furthermore,a formula to calculate division points is represented.The new principle removes the weakness that the traditional partition of classes is not unique for the same sample. The simulation studies demonstrate that Pearson's chi-squared test based on new principle is more powerful than that based on traditional partition in abscissa.In Chapter 3,based on the results of vertical density representation and center-similar distribution,a statistical model of center-similar multivariate distribution is proposed.The proposed model includes multivariate normal distribution as a special case.Firstly,the unknown parameters of the proposed model are estimated by method of moment.The asymptotic properties of the resulting estimators are established.Some examples are presented to illustrate the application of the proposed model.Secondly,by maximum likelihood method,the estimators of unknown parameters in the proposed model are obtained.The system of estimation equations is presented.Finally,results of vertical density representation are used to goodness-of-fit tests of multivariate distribution,including goodness-of-fit tests of spherically symmetric distribution and center-similar distribution, partition ofχ~2 test through vertical density representation.In Chapter 4,for recurrent event data,several regression models are proposed. Firstly,a class of general additive-multiplicative rates models for single type recurrent event is proposed.The proposed models include the additive rates and multiplicative rates models as special cases.For the inference on the model parameters,estimating equation approaches are developed,and asymptotic properties of the proposed estimators are established through modern empirical process theory.In addition,the proposed models are applied to multiple-infection data from a clinic study on chronic granulomatous disease(CGD).Secondly,general additive-multiplicative rates models for multiple type recurrent event data also are considered.We formulate estimating equations for unknown parameters and nonparametric function of proposed models.Under some regularity conditions,the consistency and asymptotic normality of resulting estimators are shown.Finally, we present a flexible additive-multiplicative rates model for multiple type recurrent event data.Procedures for making inference about the model parameters are provided.Asymptotic properties of the proposed estimators are established.