Binomial approximation to the Poisson binomial distribution: The Krawtchouk expansion (Q2752952)

Let \(X_i\), \(1\leq i\leq n\), be independent Bernoulli distributed random variables with means \(p_i\). If the \(p_i\) are small, starting from a Poisson approximation, there are a number of progressively more accurate approximations to the distribution of \(S_n:= \sum^n_{j=1}X_j\), for instance using Poisson-Charlier expansions [\textit{P. Deheuvels} and \textit{D. Pfeifer}, Ann. Inst. Stat. Math. 40, No. 4, 671-681 (1988; Zbl 0675.60027)].NEWLINENEWLINENEWLINEIn this paper, expansions are developed around a binomial approximation, using Krawtchouk polynomials, which can be accurate in any range of values of the \(p_i\). Explicit bounds for the errors in these approximations are given in terms of total variation and point metrics.