Confidence intervals for negative binomial random variables of high dispersion.

Area of Science:

• Statistics
• Probability Theory
• Data Analysis

Background:

• Confidence intervals for the mean of a Negative Binomial distribution are crucial in various fields.
• Large sample sizes often lead to the use of Normal distribution approximations for interval construction.
• High dispersion in Negative Binomial data can challenge the accuracy of standard interval methods.

Purpose of the Study:

• To investigate the limitations of Normal approximation and bootstrap methods for Negative Binomial confidence intervals.
• To propose and evaluate alternative methods for constructing accurate confidence intervals.
• To provide guidelines for assessing the performance and applicability of different interval estimation techniques.

Main Methods:

• Analysis of the convergence rate of the sample mean of Negative Binomial distributions to the Normal distribution.
• Development of new confidence interval methods based on Bernstein's inequality, Gamma, and Chi Square distributions.
• Simulation experiments comparing proposed methods with standard Normal approximation and bootstrap techniques.

Main Results:

• Standard Normal approximation and bootstrap methods result in overly narrow confidence intervals and undercoverage for highly dispersed Negative Binomial data, especially at small sample sizes.
• Proposed methods utilizing Bernstein's inequality, Gamma, and Chi Square distributions demonstrate improved accuracy and coverage.
• A ratio statistic is proposed to guide the selection and evaluation of the Chi Square method and other interval types.

Conclusions:

• The Normal approximation is unreliable for Negative Binomial confidence intervals when dispersion is high.
• Alternative methods based on Bernstein's inequality and distribution theory offer more robust and accurate confidence intervals.
• The proposed ratio statistic aids in determining the suitability of methods for specific Negative Binomial datasets.