Growth Estimators and Confidence Intervals for the Mean of Negative Binomial Random Variables with Unknown Dispersion

TL;DR

This paper introduces growth estimators for the mean of Negative Binomial variables that improve confidence interval coverage, especially under high dispersion, without needing to estimate the dispersion parameter.

Contribution

It proposes novel growth estimators that construct confidence intervals by removing zeros or adjusting the mean, enhancing inference accuracy in highly dispersed Negative Binomial data.

Findings

Growth estimators improve confidence interval coverage.

Methods do not require dispersion parameter estimation.

Intervals asymptotically converge to the sample mean.

Abstract

The Negative Binomial distribution becomes highly skewed under extreme dispersion. Even at moderately large sample sizes, the sample mean exhibits a heavy right tail. The standard Normal approximation often does not provide adequate inferences about the data's mean in this setting. In previous work, we have examined alternative methods of generating confidence intervals for the expected value. These methods were based upon Gamma and Chi Square approximations or tail probability bounds such as Bernstein's Inequality. We now propose growth estimators of the Negative Binomial mean. Under high dispersion, zero values are likely to be overrepresented in the data. A growth estimator constructs a Normal-style confidence interval by effectively removing a small, pre--determined number of zeros from the data. We propose growth estimators based upon multiplicative adjustments of the sample mean and direct removal of zeros from the sample. These methods do not require estimating the nuisance dispersion parameter. We will demonstrate that the growth estimators' confidence intervals provide improved coverage over a wide range of parameter values and asymptotically converge to the sample mean. Interestingly, the proposed methods succeed despite adding both bias and variance to the Normal approximation.