Likelihood and statistical evidence in survey sampling

Abstract

A comparison is made of the two concepts, generalized likelihood (from Bjørnstad, 1996) and Royall's measure of empirical statistical evidence in prediction problems, here called evidential likelihood, applied to survey sampling when a population model is assumed. As shown by Bjørnstad (1996), the likelihood principle based on the generalized likelihood is implied by conditionality and sufficiency principles, generalizing the fundamental result from Birnbaum (1962). The main difference between the two likelihood concepts is that the generalized likelihood is a basis for statistical inference containing all available statistical information, while the evidential likelihood concentrates on the empirical evidence per se which does not contain all available statistical information about the variable to be predicted. One can regard the evidential likelihood as the sample evidence of the generalized likelihood, changing the prior distribution of the population total to the generalized likelihood after the data have been obtained.