Utilizing Skew Normal And Inverse Scale Skew Normal Distribution For Regression Model

The Tobit model, which is also called the sample selection model or the limited dependent variable model, was first introduced by James Tobin in 1958 in order to model a specific type of data in which the interpreted variable is limited. It is a specific case of a censored regression model and assumes that the continuous component of the data is normally distributed. In later research, it has shown that even small departures from the normal assumption may lead to inconsistent estimators. Arabmazar and Schmidt(1982) explored the robustness of the Tobit estimator when estimating a population mean when the assumption of normality is violated. They concluded that the bias can be quite large and that the bias is dependent on the proportion of censoring. One technique that is often utilized in an attempt to compensate for this weakness in the case of long-tailed distributions is to apply a log or square root transformation to the data. Another way is to consider semi-parametric approaches. What we plan to do is to use more flexible parametric models, such as the inverse scale skew normal distribution and skew normal distribution which include the norm ones as a special case to extend to the original Tobit model. Here the inverse scale skew normal distributionwas introduced by C.Fernandez and M.F.L Steer(1998) when they tried to transfer the symmetrical distributions to the unsymmetrical ones.In this paper, we assume that the underlying distribution is inverse scale skew normal, and get some new properties and results. The maximum likelihood functions of model parameters with their asymptotic properties are derived. For illustrating our results, a simulation study for the inverse scale skew normal Tobit regression model is discussed. At last, the comparison between our new model and the original one is proposed and the result showed that when γ=1, they both performed equally well. There was no substantial penalty for using the inverse scale skew normal Tobit model when Tobit model assumptions held. But when γ>1, the Tobit model did not perform as well as the inverse scale skew normal Tobit model.