Asymptotic properties of weighted M-estimators for variable probability samples

I provide a systematic treatment of the asymptotic properties of weighted M-estimators under variable probability stratified sampling. The characterization of the sampling scheme and representation of the objective function allow for a straightforward analysis. Simple, consistent asymptotic variance matrix estimators are proposed for a large class of problems. When stratification is based on exogenous variables, I show that the unweighted M-estimator is more efficient than the weighted estimator under a generalized conditional information matrix equality. When population frequencies are known, a more efficient weighting is possible. I also show how the results carry over to multinomial sampling.