On regression analysis with incomplete covariate data

Approximate likelihoods are considered for regression analysis with covariate data missing at random. The methods are based on discrete approximations to the covariate distribution. In the first approximation, the covariate distribution is assumed to be concentrated on the observed complete ca,ses, with kernel averaging used for the likelihood contribution of partially complete cases. The second method places the mass of the covariate distribution on a discrete grid of points, and uses a penalty function to regularize the covariate distribution estimator. Simulations indicate the methods give reasonably good performance for estimated parameters, but that inferences based on large sample approximations may not be reliable.