Weighted Distributions and Size-Biased Sampling with Applications to Wildlife Populations and Human Families

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Abstract
Some general models leading to weighted distributions with weight functions not necessarily bounded by unity are examined. Examples include; probability sampling in sample surveys, additive damage models, visibility bias dependent on the nature of data collection and 2-stage sampling. Several important distributions and their size-biased forms are recorded. A few theorems are given on inequalities between mean values of 2 weighted distributions. Results are applied to analysis of data relating to human populations and wildlife management. For human populations, the following is 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? B/(B + S) apparently 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 average number of children per family in the general population. Some suggestions are offered for 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.

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