MOn the efficiency of some semi- and nonparametric kernel estimators in the modeling of binary response data

Abstract

The semiparametric estimation procedure of binary response data is composed of two steps. In the first step, various semiparametric estimators are used for estimating the model parameters, whereas nonparametric regression type estimators are required in the second step to obtain the probability estimates dependent on the estimated model parameters in the first step. In this study, we have investigated the efficiency level of the Klein and Spady estimator which is widely used in the current literature in first step of the semiparametric modelling and the classical Nadaraya-Watson kernel estimator used in the second step of the estimation procedure of binary response data when the parametric model assumptions are satisfied. We have also wanted to see the variation in the estimates when the adaptive Nadaraya-Watson kernel estimator has been used instead of the classical estimator. So far, there has neither been any simulation study nor a study comparing those methods, analytically in Statistics literature. Therefore, a comprehensive simulation study has been conducted and data sets from the logistic distribution have been generated to display that success in practice. Four different sample sizes have been considered to see the differences along with the variation in the sample sizes. All findings have been assessed in terms of both the mean averaged square error and the correct classification rate criteria for ordinary, Pearson and deviance residuals, respectively. Additionally, a real data set has been used to demonstrate the effectiveness of the simulation results with the results in practice. The simulation results indicate that the semiparametric Klein and Spady estimations give considerable close results to the parametric counterparts when parametric model assumptions are satisfied. This obviously means that there is no significance difference between the parametric and the semiparametric approaches in case of satisfaction of the parametric model assumptions. Another considerable finding is that there is no clear superiority between classical and adaptive Nadaraya-Watson estimators. Key words: Nonparametric regression, kernel estimation, semiparametric estimation, binary data.