A Growth Estimator for the Mean of Negative Binomial Random Variables with Unknown Dispersion

• Activity Number: 670
• Type: Contributed
• Date/Time: Thursday, August 4, 2011 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): David Shilane and Derek Bean*+
• Companies: Stanford University and University of California at Berkeley
• Keywords: Negative Binomial distribution ; confidence intervals ; coverage probability ; small sample inference ; growth estimator

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 expected value in this setting. In previous work, we have examined alternative methods of generating confidence intervals for the mean when the dispersion parameter is unknown. These methods were based upon Gamma and Chi Square approximations or tail probability bounds such as Bernstein's Inequality. We now propose a growth estimator of the Negative Binomial mean. Under high dispersion, zero values are likely to be overrepresented in the data. The growth estimator constructs a Normal-style confidence interval by effectively removing a small, pre-determined number of zeros from the data. We will demonstrate that the growth estimator's confidence interval provides improved coverage over a wide range of parameter values. Asymptotically, the procedure becomes equivalent to the Normal approximation. Interestingly, the proposed growth estimator succeeds despite adding both bias and variance to the Normal approximation.