Efficient algorithms for calculating the probability distribution of the sum of hypergeometric-distributed random...

Area of Science:

• Probability Theory and Statistics
• Computational Statistics

Background:

• The sum of independent and identically distributed (i.i.d.) random variables typically follows a convolution of their individual distributions.
• Unlike binomial, geometric, or Poisson distributions, the sum of i.i.d. hypergeometric random variables lacks a closed-form probability mass function (p.m.f.) or cumulative distribution function (c.d.f.).
• This limitation poses challenges in various statistical applications requiring the exact distribution.

Purpose of the Study:

• To propose a novel approximation for the distribution of the sum of i.i.d. hypergeometric random variables.
• To compare the proposed approximation with existing numerical methods, namely direct convolution and the De Pril recursive algorithm.
• To provide efficient MATLAB implementations for these methods and demonstrate their utility in Statistical Process Monitoring (SPM).

Main Methods:

• Development of a new approximation for the distribution of the sum of i.i.d. hypergeometric random variables.
• Implementation of direct convolution and the De Pril recursive algorithm for comparative analysis.
• Utilizing MATLAB for coding and efficient computation of probability distributions.

Main Results:

• The proposed approximation demonstrates remarkable accuracy and efficiency in calculating the distribution of the sum of i.i.d. hypergeometric random variables.
• Comparison with direct convolution and the De Pril algorithm shows the approximation is simpler to apply.
• The approximation significantly reduces computational time while maintaining high accuracy, as evidenced by an SPM application.

Conclusions:

• The proposed approximation offers a practical and effective solution for dealing with the sum of i.i.d. hypergeometric random variables.
• This method is well-suited to replace established practices where closed-form solutions are unavailable.
• The findings are applicable across diverse fields encountering challenges with hypergeometric distribution convolutions.