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

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 expected value 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, predetermined 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.

Applications of information measures to assess convergence in the central limit theorem

“…Shilane et al (2010) established that the normal confidence interval significantly under-covers the mean at moderate sample sizes and suggested alternative estimators based upon gamma and chi square approximations along with tail probability bounds such as Bernstein's inequality. Shilane and Bean (2013) proposed another method, namely the growth estimator, which provides improved confidence intervals for the mean of negative binomial random variables with unknown dispersion. They observed that their growth estimator produces intervals that are longer and more variable than the normal approximation.…”