Regression Analysis with Covariates Missing at Random: A Piece-wise Nonparametric Model for Missing Covariates

Statistical analysis for the regression model f(y|x, z) with missing values in the covariate vector X requires modeling of the covariate distribution g(x|z). Likelihood methods, including Ibrahim (1990), Chen (2004), and Zhao (2005), need either X or Z to be discrete. This article considers extending the likelihood methods to deal with cases where both X and Z may be continuous. We propose modeling the covariate distribution g(x|z) using a piece-wise nonparametric model, then a maximum likelihood estimate (MLE) of can be computed following the maximum likelihood estimating procedure of Chen (2004) or Zhao (2005). The resulting estimation method is easy to implement and the asymptotic properties of the MLE follow under certain conditions. Extensive simulation studies for different models indicate that the proposed method is acceptable for practical implementation. A real data example is used to illustrate the method.