Abstract

When the probability that a statistical unit is sampled is proportional to a size variable, then size bias occurs. For example, when sampling individuals from a population, larger households may be overrepresented. Indeed, as a motivating problem we briefly discuss a recent analysis of a plague dataset from 1630 in Carmagnola, Italy, in which data on the health outcome after admission to the plague ward ("lazzaretto") was analyzed.
With size-biased sampling, caution must be applied in estimation. We obtain the uniformly minimum variance unbiased estimator for the sparsity index, that is the inverse of the parameter, for a size-biased sample obtained from a Poisson distribution.
We propose two exact algorithms for the calculation of the optimal estimator. The algorithms are computationally burdensome even for small sample sizes, which is the setting of interest for the estimator. As an alternative, a third, approximate algorithm based on the inverse Fourier transform is presented.In the associated manuscript, we provide ready-to-use tables for the value of the estimator.
An exact confidence interval is also proposed, and the performance of the inferential procedures is compared to classical maximum likelihood inference, both in terms of mean squared error and average coverage probability and width of the confidence intervals.

References

Alfani G, Bonetti M and Fochesato M (2023). Pandemics and socio-economic status. Evidence from the plague of 1630 in Northern Italy. Population Studies (Camb) Vol. 78(1): 21-42. doi: 10.1080/00324728.2023.2197412.

Bondi L, Pagano M and Bonetti M (2024). The sparsity index in Poisson size-biased sampling: Algorithms for the optimal unbiased estimation from small samples. Statistics and Probability Letters 214, 110217.