Fisher Information in Weighted Distributions

Standard inference procedures assume a random sample from a population with density fθ(x) for estimating the parameter θ. However, there are many applications in which the available data are a biased sample instead. Fisher modeled biased sampling using a weight function w(x) ≥ 0, and constructed a weighted distribution with a density f θ (x) that is proportional to w(x)fθ(x). In this paper, we assume that fθ(x) belongs to an exponential family, and study the Fisher information about θ in observations obtained from some commonly arising weighted distributions: (i) the kth order statistic of a random sample of size m, (ii) observations from the stationary distribution of the residual lifetime of a renewal process, and (iii) truncated distributions. We give general conditions under which the weighted distribution has greater Fisher information than the original distribution, and specialize to the normal, gamma, and Weibull distributions. These conditions involve the distributions’ hazard rate and the reversed hazard rate functions.