From a comment I made in an email exchange:
My work on survey adjustments has very much been inspired by the ideas of Rod Little. Much of my efforts have gone toward the goal of integrating hierarchical modeling (which is so helpful for small-area estimation) with post stratification (which adjusts for known differences between sample and population). In the surveys I’ve dealt with, nonresponse/nonavailability can be a big issue, and I’ve always tried to emphasize that (a) the probability of a person being included in the sample is just about never known, and (b) even if this probability were known, I’d rather know the empirical n/N than the probability p (which is only valid in expectation). Regarding nonparametric modeling: I haven’t done much of that (although I hope to at some point) but Rod and his students have.
As I wrote in the first sentence of the above-linked paper, I do think the current theory and practice of survey weighting is a mess, in that much depends on somewhat arbitrary decisions about which variables to include, which margins to weight on, and how to trim extreme weights. And that’s just for estimating population totals. Once we move to regressions, weighting becomes even messier. This is not to say that weighting should not be done—I construct survey weights myself sometimes—but I think it’s important to recognize that the theory has been holes, it’s not a simple matter of clean unbiased estimates as is sometimes presented in introductory presentations and even to users.
P.S. In response to a comment, I elaborated:
It’s hard for me to see how anyone who has actually constructed survey weights can disagree with the statement that survey weighting is a mess. But I suppose one could also say that regression modeling is a mess. In both cases there is so much awkwardness surrounding the steps of selecting or rejecting variables for the adjustment, and there are so many potential interactions while at the same time big practical limitations on how much can be included using the least-squares-like estimation and adjustment procedures that are standard. To me, survey weighting seems even messier than regression modeling in that survey weighting involves two additional complications: (1) the numbers that we want to adjust to (census totals or sampling probabilities) are commonly not themselves known as precisely as we would like, and (2) there’s that whole business of trimming weights.