Optimal estimating functions in incomplete data and length biased sampling data problems

Abstract

AbstractIt is well known that the score function is the optimal estimating function among all regular unbiased estimating functions (Godambe, 1960). In the presence of incomplete data such as missing data or length biased sampling data, Horvitz and Thompson's (1952) method is an effective way of eliminating the possible bias induced by using complete data only. In this article, we show that the inverse weighted Horvitz and Thompson score estimating function is not optimal in the presence of incomplete data. By using Godambe's estimating function theory, we can identify the optimal estimating function in this situation. In the case of the accelerated failure time model with length bias sampling data, the optimal estimating function can produce an unbiased estimator for the slope parameter even when the underlying density function is misspecified. Simulation studies show that the estimate derived from the optimal estimating function can be substantially better than the estimate derived from the inverse weighted score estimating function. The Canadian Journal of Statistics 39: 510–518; 2011 © 2011 Statistical Society of Canada