Order-restricted semiparametric inference for the power bias model

The power bias model, a generalization of length-biased sampling, is introduced and investigated in detail. In particular, attention is focused on order-restricted inference. We show that the power bias model is an example of the density ratio model, or in other words, it is a semiparametric model that is specified by assuming that the ratio of several unknown probability density functions has a parametric form. Estimation and testing procedures under constraints are developed in detail. It is shown that the power bias model can be used for testing for, or against, the likelihood ratio ordering among multiple populations without resorting to any parametric assumptions. Examples and real data analysis demonstrate the usefulness of this approach.