Multiple inverse sampling in post-stratification with subpopulation sizes unknown: a solution for quota sampling

We extend traditional inverse sampling to multiple case. We then modify the multiple inverse sampling design to a version with taking a simple random sample at the beginning similar to Chang et al (J. Statist. Plan. Inference 69 (1998) 209) and a truncated version similar to Chang et al (J. Statist. Plan. Inference 76 (1999) 215). Using Murthy (Sankhya 18 (1957) 379) we develop their unbiased estimators and their unbiased variance estimators. These unbiased estimators can also be applied to a frequently used sampling scheme called quota sampling by practitioners. The multiple inverse sampling may be viewed as an improved version of quota sampling in some sense. We show that our estimators for estimating the proportions (weights) of subpopulations are more efficient and robust than available estimators using a small simulation study.