Exact values and sharp estimates for the total variation distance between binomial and Poisson distributions. (English) Zbl 1157.62005
Summary: We present a method to obtain both exact values and sharp estimates for the total variation distance between binomial and Poisson distributions with the same mean \(\lambda \). We give a simple efficient algorithm, whose complexity order is \(\sqrt\lambda\), to compute exact values. Such an algorithm can be further simplified for moderate sample sizes \(n\), provided that \(\lambda \) is neither close to \(l + \sqrt l, l = 1, 2, \dots \), from the left nor close to \(m -\sqrt m, m = 2, 3, \dots \), from the right. Sharp estimates, better than other known estimates in the literature, are also provided. The 0s of the second Krawtchouk and Charlier polynomials play a fundamental role.
MSC:
| 62E17 | Approximations to statistical distributions (nonasymptotic) |
| 65C60 | Computational problems in statistics (MSC2010) |
| 60E15 | Inequalities; stochastic orderings |
Keywords:
Poisson approximation; binomial distribution; total variation distance; Krawtchouk polynomial; Charlier polynomialReferences:
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