A note on exponential dispersion models which are invariant under length-biased sampling

Length-biased sampling situations may occur in clinical trials, reliability, queueing models, survival analysis and population studies where a proper sampling frame is absent.In such situations items are sampled at rate proportional to their length so that larger values of the quantity being measured are sampled with higher probabilities.More specifically, if f(x) is a p.d.f. presenting a parent population composed of nonnegative valued items then the sample is practically drawn from a distribution with p.d.f. g(x) = xf(x)/E(X) describing the lengthbiased population.In this case the distribution associated with g is termed a length-biased distribution.In this note we present a unified approach for characterizing exponential dispersion models which are invariant, up to translations, under various types of length-biased sampling.The approach is rather simple as it reduces such invariance problems into differential equations in terms of the derivatives of the associated variance functions.