Semiparametric regression analysis under imputation for missing response data

We develop inference tools in a semiparametric regression model with missing response data. A semiparametric regression imputation estimator and an empirical likelihood based one for the mean of the response variable are defined. Both the estimators are proved to be asymptotically normal, with asymptotic variances estimated with Jackknife method. The empirical likelihood method is developed. It is shown that when missing responses are imputed using the semiparametric regression method the empirical log-likelihood is asymptotically a scaled chi-square variable or a weighted sum of chi-square variables with unknown weights in the absence of auxiliary information or in the presence of auxiliary information. An adjusted empirical log-likelihood ratio, which is asymptotically standard chi-square, is obtained. Also, a bootstrap empirical log-likelihood ratio is also derived and its distribution is used to approximate that of the imputed empirical log-likelihood ratio. A simulation study is conducted to compare the imputed, adjusted and bootstrap empirical likelihood with the normal approximation based methods in terms of coverage accuracies and average lengths of confidence intervals. Based on biases and standard errors, a comparison is also made by simulation between the proposed two estimators. The simulation indicates that the empirical likelihood methods developed perform competitively and the use of auxiliary information provides improved inference.