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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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##### References:
 [1] Barbour, A.D. (1989), Personal communication. [2] Barbour, A.D.; Eagleson, G.K., Poisson approximation for some statistics based on exchangeable trials, Adv. in appl. probab., 15, 585-600, (1983) · Zbl 0511.60025 [3] Barbour, A.D.; Hall, P., On the rate of Poisson convergence, Math. proc. Cambridge philos. soc., 95, 473-480, (1984) · Zbl 0544.60029 [4] Barbour, A.D.; Holst, L., Some applications of the stein—chen method for proving Poisson convergence, Adv. in appl. probab., 21, 74-90, (1989) · Zbl 0673.60023 [5] Chen, L.H.Y., On the convergence of Poisson binomial to Poisson distributions, Ann. probab., 2, 178-180, (1974) · Zbl 0276.60024 [6] Chen, L.H.Y., Poisson approximation for dependent trials, Ann. probab., 3, 534-545, (1975) · Zbl 0335.60016 [7] Freedman, D., A remark on the difference between sampling with or without replacement, J. amer. statist. assoc., 72, 681, (1977) [8] Hardy, G.H.; Littlewood, J.E.; Pólya, G., Inequalities, (1934), Cambridge Univ. Press Cambridge [9] Keilson, J.; Gerber, H., Some results for discrete unimodality, J. amer. statist. assoc., 66, 386-389, (1971) · Zbl 0236.60017 [10] LeCam, L., An approximation theorem for the Poisson binomial distribution, Pacific. J. math., 10, 1181-1197, (1960) · Zbl 0118.33601 [11] Stein, C., A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, Proc. sixth Berkeley symp. math. statist. probab., 2, 583-602, (1970) [12] Stein, C., Approximate computation of expectations, 7, (1986), Inst. Math. Statist Hayward, CA, IMS Monograph Series · Zbl 0721.60016 [13] Vatutin, V.A.; Mikhailov, V.G., Limit theorems for the number of empty cells in an equiprobable scheme for group allocation of particles, Theory probab. appl., 27, 734-743, (1982) · Zbl 0536.60017
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