Studies Of Some Statistical Issues In Censored Regression Model

Limited dependent variable (LDV) regression model is a virtual model and widely used in econometrics studies. Moreover, many important advances of econometrics studies are related to LDV model. Generally, range of response variable in LDV model is restricted to a subset (interval) of the real line. LDV model includes truncated dependent model and censored dependent model, where response variables of truncated dependent model are sampled from an incomplete population, and those of censored dependent model are generated from complete population but individuals' values are not specified when they are larger (smaller) than one preset value. Censored regression ( Tobit ) model studied in this paper is a special LDV model for which response variable has a nonnegative limitation, where only segment of response variable being not less than 0 can be measured. For censored regression model, this paper mainly presents 3 aspects of research work: LASSO-type approach to variables selection and estimation of regression parameters, randomly weighted approximation method for linear hypothesis test statistic of regression coefficients and change-point estimation in censored regression model and its convergence rate.First, this paper establishes a LASSO-type approach to variables selection and estimation of regression coefficients in censored regression model, where explanation variables are known design points. model (variables) selection is an essential part in model construction. For censored regression model, studies of variables selection are few at present references. We propose a LASSO-type method for variables selection and estimation: diverse penalty L₁x constraint method (DPLC). DPLC method can select variables whose coefficients are significantly nonzero, while giving an estimator of the corresponding coefficients. Under some conditions, we get consistent property of this estimator and its asymptotic distribution. In simulation studies, ability of selecting variables and estimation of DPLC method is compared with that of generally best subset selection method (BSSM). Extensive simulation studies show that DPLC almost possesses the same performance as BSSM. When number of variables is huge, however, BSSM method is difficultly implemented.Second, we propose a randomly weighted approximation method for linear hypothesis test statistic in censored regression model, where explanation variables are known design points. Linear hypothesis test problem has been widely studied in censored regression model, however, asymptotic distribution of the common test statistic includes a nuisance parameter, density of error distribution. Therefore critical value which is utilized to whether reject the null hypothesis or not is difficultly determined. Especially with small sample size, estimation of error's density is greatly rough. In this paper we use randomly weighted method to construct a randomly weighted test statistic for linear hypothesis test. And utilize conditional distribution of the weighted test statistic to approximate to the null distribution of the test statistic. Given observation samples, under some assumptions we prove that the conditional limiting distribution of the weighted test statistic, no matter the null hypothesis or local alternative is true, is same as the null limiting distribution of the test statistic. Consequently, we do not need to estimate the nuisance parameter and use randomly weighted method to determine the critical value. For a preset nominal significance level, iteratively make randomly weighted version of the weighted test statistic by generating randomly weighted variables, and take (1-nominal significance level) quantile of these randomly weighted versions as our critical value. When the test statistic is larger than this critical value we reject the null hypothesis. It is easy to show that, given nominal significant level, the test with critical value determined by the weighted test statistic has the same asymptotic level and asymptotic power under the alternative as the test with critical value obtained by estimating nuisance parameter. Simulation studies show that the conditional distribution of the weighted test statistic is rather close to the null distribution of the test statistic.Finally this paper investigates change-point problem in censored regression model. Studies of change point problem are always popular issue of statistics and derive from the automatic control of industry. As sociality developing, change point model has more and more application in many fields, such as econometric, finance, compute and so on. This paper studies estimation problem of change point in the context of existing at most one change point in censored regression model, where we consider randomly explanation variables. A nonparametric estimator of change point is presented and its strong consistent property is obtained under some conditions. Furthermore, we achieve the almost sure convergent rate of change-point estimator, O(n^-1/2 log n). Simulation studies show that our results are reasonable.