Semiparametric efficient estimation for partially linear single-index models with responses missing at random

In this paper, we establish the semiparametric efficient bound for the heteroscedastic partially linear single-index model with responses missing at random, and develop an efficient estimating equation method. By solving the estimating equation, we obtain estimators for the parameter vectors in the linear part and the single index part simultaneously. The estimators are asymptotically semiparametrically efficient when the propensity score function is specified correctly. It should be noted that the inverse probability weighted efficient estimating equation cannot be obtained directly from the full data efficient estimating equation by the inverse probability weighted approach. We establish the estimating equation by deriving the observed data efficient score function. Some simulation studies and a real data application were conducted to evaluate and illustrate the proposed methods.