Statistical estimation of the composition of a finite set

A survey of recent advances in the problem of estimating the structure of a finite population is given. We include into the survey the results closely connected with the allocation problem and consider both homogeneous and stratified populations. A comparative analysis of the results obtained by using sufficiency and likelihood principles is carried out for a general sampling scheme. The asymptotical approach is also presented. The optimal (asymptotically optimal) inferences for both point and confidential estimation are the main object of the article. Possible extensions of the investigations in this area are also discussed.