Statistical Inference And Model Selection For The Processes Of Ornstein-Uhlenbeck Type

The process of Ornstein-Uhlenbeck (OU) type is an important moving average process. In recent years, it not only has very good application value in theory, and also has been widely used in the field of finance and econometrics. In this paper, we mainly focus on the following problems:we provide a new parameter estimation method and a model selection method for the process of OU type; we also provide a test statistics to test the equality of the distributions for two processes of OU type; then, we investigate a variable selection method for the linear model, in which the error terms are from the process of OU type; last, we study the ruin probability of a class of renewal risk model. In this paper, the main work and conclusions are as follows:(1)、We develop a new method of parameters estimation for the process of OU type. This proposed method is based on GMM with continuum of moment conditions (CGMM for short). We show that this estimator has consistency and asymptotical nor-mality. We also show that this estimator enjoys good finite sample performance through simulation study.(2)、We study the least absolute shrinkage and selection operator-type (LASSO-type) CGMM estimator for the process of OU type. This LASSO-type objective func-tion is formed by the CGMM objective function with the addition of a penalty term. The exponent of the penalty term in the LASSO-type estimator is less than one in the analysis here, and this is reduced to avoid the asymptotic bias. This estimator selects the correct model and estimates it simultaneously. The asymptotic theory for our esti-mator is nonstandard. We conduct a simulation study show that LASSO-type CGMM correctly selects the true model much more than the traditional methods based on the CGMM objective function.(3)、We study the problem of testing the equality of distributions for two processes of OU type. We propose a two step method, the first step is testing the equality of correlation coefficient of the two processes by LS estimation, and then by bootstrap method to testing the deviation of the two LS estimator, we reject the original testing if the deviation is not zero, we turn to the second step if the deviation is zero; the second step is testing the equality of marginal distributions of the two processes. We construct the test statistic by a weighted modulus of the difference between the two empirical characteristic functions, and then testing the statistic by sequential bootstrap method. We conduct a simulation study to evaluate the performance of the proposed approach.(4)、We propose composite quantile regression (CQR) for linear model, in which the errors are from the process of OU type and theoretical properties are given. We also conduct simulation study to evaluate that the performance of the proposed approach.(5)、We consider a renewal risk model with upper-tail independent heavy-tail claims, which belongs to the class D ∩ L, and with constant interest force. We focus on the asymptotic relationship of the finite time ruin probability.The innovations of the methodologies in this dissertation are described as follow-ing. Firstly, using CGMM method to estimate the parameters of the process of OU type, we not only avoids to deal with the complex or nonexistent of the density function for the process of OU type, but also avoids to deal with the number of moment conditions. This method provide an effective and convenient method for the parameter estimation. Secondly, we provide a model selection method for multidimentional processes of OU type, and theoretical properties are given. Thirdly, we put forward a type of hypothesis testing method to the two sample problems in the processes of OU type. Fourthly, we propose a variable selection method for linear model, in which the errors are from the process of OU type, and provides the theoretical properties. At last, we study the tail asymptotic probability for a class of more general renewal process.