Studies On Some High Dimensional Statistical Problems In Tobit And Relative Error Models

In practical application,high dimensional data often occurs when dimension of parameters in the studied model increases with sample size.Statistical methods based on linear regression models are widely studied,however,for Tobit model and relative error model,the researches with high dimensional data are rare.Tobit models are frequently used in many fields such as econometrics,and for the analysis of scale-dependent data,the relative errors are more of concern than the absolute error.Therefore,this paper mainly focuses on statistical inference on Tobit model and the relative error model when the dimension of parameters diverges.Firstly,we study the asymptotic properties of parameter estimation in Tobit model,which includes consistency,asymptotic normality,convergence rate and variable selec-tion.This research involves three difficulties:(1)With the increase of sample size,the parameter space and the design matrix are different from those with a fixed dimension of parameters;(2)the least absolute error criterion based on the Tobit model is non-convex and non-smooth,such that the previous research methods on the linear model are not suitable to the current model;(3)the maximum inequality in the traditional empirical process theory assumes that the dimension of parameters is fixed.Thus,We establish a maximal inequality when the dimension of parameters is di-vergent,the resulting inequality does not depend on the Tobit model and can be applied to other models.The least absolute deviation criterion gives an estimate of the parame-ters in the Tobit model,denoted by the least absolute deviation estimator.Under some regularization conditions,by using the proposed inequality,we obtain the asymptotic properties of the parameter estimates:weak consistency,convergence rate in probabil-ity and asymptotic normality.In order to avoid estimating nuisance parameters,such as the density function of the error term in the asymptotic normal distribution,we use the random weighted resampling method to approximate the asymptotic distribution of the parameter estimates.To the best of our knowledge,there are less researches on the strong consistency of parameter estimation in the literature when the dimension of pa-rameters is divergent,even though for the linear model they are also few.This paper proves that when the dimension of parameters and sample size meet certain conditions,the strong consistency and convergence rate of the parameter estimation of Tobit model are obtained.It shows that the rate depends on order of dimension of parameters with sample size.When the dimension of parameters more quickly increases with the sample size,the convergence rate becomes slower.The results also include the case when the dimension of parameters is fixed.The finite sample properties of the proposed method are illustrated by numerical simulation and analysis of real data.Variable selection is one of the most important issues in statistics.The existed re-searches on the variable selection method of Tobit model mainly focus on the case when the dimension of parameters is fixed.With the development of science and technology,more and more observational factors will be observed,especially with the increase of sample size.In order to improve the prediction accuracy of the model,it is necessary to select the important factors.Considering the divergence dimension of parameters in Tobit model,we give a non-concave penalty function based on the absolute error criterion.Under certain regularity conditions,for example,the coefficients of the ex-planatory variables are sparse.This paper proves that the proposed variable selection method has Oracle properties.Simulation results show that our estimation method can distinguish non-zero coefficients from zero coefficients,and give a good estimate of nonzero coefficients.Finally,we establish the statistical properties of the relative error model when the dimension of parameters diverges:the asymptotic properties of the parameter estimates and the properties of the linear hypothesis test.Due to the difference of measurement units and observation data,the relative error is more reasonable to analyze data than the absolute error criterion,such as stock data.There are many researches on the relative error,but they assumed dimension of parameters is constant number.Under the diver-gent dimension of parameters,we propose the least product relative error criterion and generalized relative error criterion of parameter estimation,respectively.Further,we consider the linear hypotheses test of parameter,and give a M-type test statistic and its limiting distribution.By using simulation data and actual data to compare performance of the least product relative error criterion,the least absolute relative error criterion and the least square criterion,it shows that the relative error method is more reasonable and effective.