The Estimation Of Parametric Tobit Model When The Covariates Are Measured With Errors

In practical application,due to the limitation of actual condition,when the response variable is greater or less than a threshold value,we can not observe the true value of the response variable.Under this situation,Tobit regression models are usually applied to analyze this kind of data.Parametric models are valuable tools for exploring the relationship between a response and a set of covariates.It has many advantages,such as high precision,good interpretation and more accurate prediction,so this paper studies the parametric Tobit model.Besides,measurement errors are often encountered due to the limitation of measurement equipments or methods.For these reasons,it is meaningful to study the parametric Tobit model with measurement errors in covariates.Parametric estimation for regression models is an important problem in statistics.This thesis studies the estimation of parametric Tobit model with measurement errors in covariates.For measurement errors,we do not assume the known form of model structure or variance.Furthermore,repeated measurement data is not necessary.This thesis obtains an estimator of the unavailable true variable with the help of the instrument variable.Firstly,the estimation of unobserved covariates is estimated by using the nonparametric kernel smoothing method,and then the measurement errors are processed by using this estimation instead of the unobserved variables.In this way,the regression parameter can be estimated by the corrected least square method under the assumption that the distribution of the model error is symmetric.We prove the consistency and the asymptotic normality of the proposed estimator.And a specific algorithm is given.Finally,we make use of Monte Carlo simulation to investigate the performance of the proposed method.The proposed least squares estimation and existed maximum likelihood estimation are considered compared with or without considering measurement error adjustment.All of the results show that the proposed method better than the naive method(i.e.don’t take measurement errors into consideration).In addition,the maximum likelihood estimation result is better when the model errors follow the normal distribution,and the least squares result is better when the model errors follow the non-normal symmetric distribution.Meanwhile,the real data analyses also show that the proposed method is better.