Likelihood-based estimators for endogenous or truncated samples in standard stratified sampling

Standard stratified sampling (SSS) is a popular non-random sampling scheme. Maximum likelihood estimator (MLE) is inconsistent if some sampled strata depend on the response variable Y ('endogenous samples') or if some Y-dependent strata are not sampled at all ('truncated sample' - A missing data problem). Various versions of MLE have appeared in the literature, and this paper reviews practical likelihood-based estimators for endogenous or truncated samples in SSS. Also a new estimator 'Estimated- E_X MLE' is introduced using an extra random sample on X (not on Y) to estimate the distribution E_X of X. As information on Y may be hard to get, this estimator's data demand is weaker than an extra random sample on Y in some other estimators. The estimator can greatly improve the efficiency of 'Fixed-X MLE' which conditions on X, even if the extra sample size is small. In fact, Estimated-E_XMLE does not estimate the full FX as it needs only a sample average using the extra sample. Estimated-E_X MLE can be almost as efficient as the 'Known-F _XMLE'. A small-scale simulation study is provided to illustrate these points.