Weighted distributions and size-biased sampling with applications to wildlife populations and human families

Abstract

When an investigator records an observation by nature according to a certain stochastic model, the recorded observation will not have the original distribution unless every observation is given an equal chance of being recorded. A number of papers have appeared during the last ten years implicitly using the concepts of weighted and size-biased sampling distributions. In this paper, we examine some general models leading to weighted distributions with weight functions not necessarily bounded by unity. The examples include: probability sampling in sample surveys, additive damage models, visibility bias dependent on the nature of data collection and two-stage sampling. Several important distributions and their size-biased forms are recorded. A few theorems are given on the inequalities between the mean values of two weighted distributions. The results are applied to the analysis of data relating to human populations and wildlife management. For human populations, the following is raised and discussed: Let us ascertain from each male student in a class the number of brothers, including himself, and sisters he has and denote by k the number of students and by B and S the total numbers of brothers and sisters. What would be the approximate values of B/(B+S), the ratio of brothers to the total number of children, and (B+S)/k, the average number of children per family? It is shown that B/(B+S) will be an overestimate of the proportion of boys among the children per family in the general population which is about half, and similarly (B+S)/k is biased upwards as an estimate of the average number of children per family in the general population. Some suggestions are offered for the estimation of these population parameters. Lastly, for the purpose of estimating wildlife population density, certain results are formulated within the framework of quadrat sampling involving visibility bias.