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Binomial approximation to the Poisson binomial distribution. (English) Zbl 0724.60021
Let $$X_ 1,...,X_ n$$ be independent Bernoulli random variables with $$P(X_ i=1)=p_ i=1-q_ i.$$ Let $$S=X_ 1+...+X_ n,$$ $$P(k)=P(S=k)$$ and $$Q(k)=\left( \begin{matrix} n\\ k\end{matrix} \right)p^ kq^{n-k}=P(Z=k)$$ with $$p=n^{-1}(p_ 1+...+p_ n),$$ $$q=1-p.$$ The authors derive a lower and an upper bound for the variation distance $$\sum | P(k)- Q(k)|,$$ both of which are, roughly, proportional to the nonnegative quantity 1-(Var S/Var Z). Proofs follow the method of C. Stein [Approximate computation of expectations (Hayward, CA, 1986)]. Logconcavity of P(k)/Q(k) leads to similar bounds for the sup-distance between distribution functions. The hypergeometric distribution has probabilities P(k) for suitable $$p_ 1,...,p_ n$$ [see V. A. Vatutin and V. G. Mikhailov, Theory Probab. Appl. 27, 734-743 (1982); translation from Teor. Veroyatn. Primen. 27, No.4, 684-692 (1982; Zbl 0517.60008)]. From this result bounds for the variation distance between the multivariate hypergeometric and the multinomial distribution are derived.
Reviewer: A.J.Stam (Winsum)

##### MSC:
 60F05 Central limit and other weak theorems 62E17 Approximations to statistical distributions (nonasymptotic)
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