Deficiency bounds for the multivariate inverse hypergeometric distribution

TL;DR

This paper derives asymptotic bounds on the information loss when using the negative multinomial model instead of the more accurate multivariate inverse hypergeometric model in finite population sampling, supported by theoretical analysis.

Contribution

It provides the first asymptotic deficiency bounds between MIH and NM models, and between MIH and multivariate normal approximations, clarifying the impact of model choice in finite sampling.

Findings

Asymptotic deficiency bounds between MIH and NM models are established.

Bounds between MIH and multivariate normal models are derived.

Theoretical support includes local approximation and Hellinger distance analysis.

Abstract

The multivariate inverse hypergeometric (MIH) distribution is an extension of the negative multinomial (NM) model that accounts for sampling without replacement in a finite population. Even though most studies on longitudinal count data with a specific number of `failures' occur in a finite setting, the NM model is typically chosen over the more accurate MIH model. This raises the question: How much information is lost when inferring with the approximate NM model instead of the true MIH model? The loss is quantified by a measure called deficiency in statistics. In this paper, asymptotic bounds for the deficiencies between MIH and NM experiments are derived, as well as between MIH and the corresponding multivariate normal experiments with the same mean-covariance structure. The findings are supported by a local approximation for the log-ratio of the MIH and NM probability mass functions, and by Hellinger distance bounds.