A Three-Parameter Binomial Approximation

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.

A Pólya Approximation to the Poisson-Binomial Law

“…Example 1. In this example we intend to convey the typical behaviour of the various approximations when applied to the Bayesian hierarchical modelling problem described in [24]. Table 1: Approximating distributions for the Poisson-binomial law.…”

“…The Poisson-binomial distribution is the law of the number of successes in a sequence of independent, nonidentically distributed trials and, as such, has found utility in several modelling applications, including Bayesian heirarchical models [24], generalized linear models [15], and noisy threshold models [21]. What is more, as the convolution product of nonidentical two-point distributions, the Poisson-binomial distribution is also intimately linked to several combinatorial and occupancy problems [26], including the 'birthday paradox' [32].…”

Approximate models for aggregate data when individual‐level data sets are very large or unavailable

“…Thus, it would be best to use only common distributions such as the binomial, Poisson or normal. In the article by Peköz et al [17], several different approximations to the distribution of the sum of independent Bernoulli random variables are considered and evaluated. A Poisson approximation performs well if the Bernoulli probabilities are all very small.…”

“…This then allows three parameters to be adjusted and, in most cases, three moments or three derivatives could be matched. The article by Peköz et al [17] evaluates all the approximations discussed here (and derives error bounds) over a range of different settings and finds that a translated binomial approximation performs the best. As we show later, our analysis demonstrates that a binomial approximation with two fitted parameters performs much better than the usual binomial approximation with one fitted parameter, and the three parameter binomial approximation does not perform much better than the two parameter binomial approximation-though it requires additional computations and data to be transmitted and stored.…”

A Pólya Approximation to the Poisson-Binomial Law

“…The earliest work focused on the Poisson approximation, of which a detailed account can be found in [6]. Subsequently, variations of the Poisson law were used: Poisson-Charlier signed measures (see [6] and the references therein), (signed) compound Poisson [5], [9], [22], shifted Poisson [4], [13]; as were the binomial law and its variations: binomial [13], [17], [30], 'almost' binomial [33], signed binomial-Krawtchouk [28], (signed) compound binomial [10], [11], [12], shifted binomial [24], [27]. Among more recent work, a class of signed measures was introduced in [7] that yielded notably impressive approximations in the special case of counting records in an independent and identically distributed sequence, 746 M. SKIPPER a two-parameter polynomial birth-death distribution was proposed in [8], and an arbitrary Gibbs measure approximation was developed in [18].…”