Semiparametric model averaging prediction for dichotomous response

Model averaging has attracted abundant attentions in the past decades as it emerges as an impressive forecasting device in econometrics, social sciences and medicine. So far most developed model averaging methods focus only on either parametric models or nonparametric models with a continuous response. In this paper, we propose a semiparametric model averaging prediction (SMAP) method for a dichotomous response. The idea is to approximate the unknown score function by a linear combination of one-dimensional marginal score functions. The weight parameters involved in the approximation are obtained by first smoothing the nonparametric marginal scores and then applying the parametric model averaging via a maximum likelihood estimation. The proposed SMAP provides greater flexibility than parametric models while being more stable than a fully nonparametric approach. Theoretical properties are investigated in two practical scenarios: (i) covariates are conditionally independent given the response; and (ii) the conditional independence assumption does not hold. In the first scenario, we show that SMAP puts weight one to the true model and hence the model averaging estimators are consistent. In the second scenario in which a “true” model may not exist, SMAP is shown to be asymptotically optimal in the sense that its Kullback–Leibler loss is asymptotically identical to that of the best – but infeasible – model averaging estimator. Empirical evidences from simulation studies and a real data analysis are presented to support and illustrate our methods.