The sparsity index in Poisson size-biased sampling: Algorithms for the optimal unbiased estimation from small samples

If the probability that a statistical unit is sampled is proportional to a size variable, then size bias occurs. As an example, when sampling individuals from a population, larger households are overrepresented. With size-biased sampling, caution must be applied in estimation. We propose two exact algorithms for the calculation of the uniformly minimum variance unbiased estimator for the sparsity index in size-biased Poisson sampling. The algorithms are computationally burdensome even for small sample sizes, which is our setting of interest. As an alternative, a third, approximate algorithm based on the inverse Fourier transform is presented. We provide ready-to-use tables for the value of the optimal estimator. An exact confidence interval based on the optimal estimator is also proposed, and the performance of the estimation procedure is compared to classical maximum likelihood inference, both in terms of mean squared error and average coverage probability and width of the confidence intervals.