A Three-Parameter Binomial Approximation

TL;DR

This paper introduces a three-parameter shifted binomial distribution to accurately approximate sums of Bernoulli variables, providing bounds on the error and demonstrating improved accuracy over traditional methods.

Contribution

It presents a novel three-parameter binomial approximation matching the first three moments and offers error bounds, enhancing accuracy in probabilistic modeling.

Findings

Shifted binomial approximation outperforms Poisson and standard binomial methods.

Error bounds are established using Stein's method.

Numerical studies confirm improved approximation accuracy.

Abstract

We approximate the distribution of the sum of independent but not necessarily identically distributed Bernoulli random variables using a shifted binomial distribution where the three parameters (the number of trials, the probability of success, and the shift amount) are chosen to match up the first three moments of the two distributions. We give a bound on the approximation error in terms of the total variation metric using Stein's method. A numerical study is discussed that shows shifted binomial approximations typically are more accurate than Poisson or standard binomial approximations. The application of the approximation to solving a problem arising in Bayesian hierarchical modeling is also discussed.