On Size-Biased Sampling and Related Form-Invariant Weighted Distributions*

Consider a finite or infinite population of units carrying values of a non negative random variable X (6) having distribution with probability (density) function (pdf) f (x; 6) where x? > 0 and deQ, , the parameter space defined for /. In the usual random sampling onI = X (d), the probability of selection of each unit is the same, regardless of the value of x it carries, so that the pdf at the observation x is f (x; 6). In the size-biased sampling on X = X (d), the probability of selection of a unit is proportional to a predetermined weight function such as w (x)? x corresponding to the value of x that the unit carries, implying the pdf at the observation x to be/* (a;; 6) = w (x) f (x; d) (x) f (x; 6) or w (x) f (x; 6) | J w (x) f (x) 6) dx depending on whether X is discrete or continuous. In this situation, we shall say that X is size-biased and we shall denote the size-biased version by X*, which we will write sometimes as (X (6) ) *, when the parameter 6 needs special attention. It is clear that the pdf of X* is /*, the pdf of the weighted distribution defined b}^ the original distribution having pdf / together with the identity weight-function w. We may also emphasize here that the x-value of the unit is not the well-known ancillary variable of the pps sampling, but is itself the variable observed and recorded.