A comparison of likelihood-based methods for size-biased sampling

Three likelihood approaches to estimation under informative sampling are compared using a special case for which analytic expressions are possible to derive. An independent and identically distributed population of values of a variable of interest is drawn from a gamma distribution, with the shape parameter and the population size both assumed to be known. The sampling method is selection with probability proportional to a power of the variable with replacement, so that duplicate sample units are possible. Estimators of the unknown parameter, variance estimators and asymptotic variances of the estimators are derived for maximum likelihood, sample likelihood and pseudo-likelihood estimation. Theoretical derivations and simulation results show that the efficiency of the sample likelihood approaches that of full maximum likelihood estimation when the sample size $n$ tends to infinity and the sampling fraction $f$ tends to zero. However, when $n$ tends to infinity and $f$ is not negligible, the maximum likelihood estimator is more efficient than the other methods because it takes the possibility of duplicate sample units into account. Pseudo-likelihood can perform much more poorly than the other methods in some cases. For the special case when the superpopulation is exponential and the selection is probability proportional to size, the anticipated variance of the pseudo-likelihood estimate is infinite.