A NOTE ON JOINT INCLUSION PROBABILITIES IN MAXIMUM ENTROPY SAMPLING

The expression of sampling variance of an estimator of a finite population total involves the first two orders inclusion probabilities pi(i) and pi(ij). The expression of the second order inclusion probabilities pi(ij) for most of the unequal probability sampling design is very complex. A number of researchers have paid their attention in approximating pi(ij) in terms of pi(i) and pi(j) and derived some useful results. In this paper we have tried to classify the methods into three groups and compare their performance by calculating their entropy values and obtained some important results. This gives easy to execute approximations of Horvitz-Thompson (1952) variance of population total and its estimate. The entropy of these new groups of approximations is compared empirically. The pi(ij) can be approximated using the first order inclusion probabilities of any sampling design. We will here apply Aftab and Hanif's (2012) maximum entropy sampling design to provide the maximum entropy pi(ij)'s. In Section 2 we will introduce some basic relationships relating to the first and second order inclusion probabilities to be derived in next three sections. In Sections 3, 4, and 5 designated as groups 1, 2 and 3 modified approximations will be derived. Numerical calculations, comparisons and concluding remarks will be discussed in the last section.