Adaptive cluster sampling for negatively correlated data

Adaptive cluster sampling is a design specifically developed for rare and clustered populations. Using this sampling design, we consider the case when an auxiliary variable is available in addition to the variable of interest. The use of auxiliary information has been shown to improve the efficiency of estimators although this results in asymptotically design-unbiased estimators. Consider wildlife population in a protected area. Its distribution and abundance can partly be influenced by such factors as disease and pollution where the presence of wildlife diseases or higher environmental pollution decreases population totals and the distribution of wildlife. This paper proposes two product estimators and their associated variance estimators for the adaptive cluster sampling design to be used when the study and auxiliary variables are negatively correlated. The exact expression of the bias together with the mean square error to the first degree of approximation has been obtained. We derived the conditions under which the proposed estimators provided a more accurate estimation than the Horvitz–Thompson and Hansen–Hurwitz estimators with adaptive cluster sampling and the product estimator with simple random sampling. A simulation study was carried out to show the performance of the proposed estimators. Moreover, theoretical findings were supported by a numerical example using real data.