Zero Biasing and a Discrete Central Limit Theorem
نویسندگان
چکیده
We introduce a new family of distributions to approximate IP(W ∈ A) for A ⊂ {· · · ,−2,−1, 0, 1, 2, · · · } and W a sum of independent integer-valued random variables ξ1, ξ2, · · · , ξn with finite second moments, where with large probability W is not concentrated on a lattice of span greater than 1. The well-known Berry–Esseen theorem states that for Z a normal random variable with mean IE(W ) and variance Var(W ), IP(Z ∈ A) provides a good approximation to IP(W ∈ A) for A of the form (−∞, x]. However, for more general A such as the set of all even numbers, the normal approximation becomes unsatisfactory and it is desirable to have an appropriate discrete, nonnormal, distribution which approximates W in total variation, and a discrete version of the Berry–Esseen theorem to bound the error. In this paper, using the concept of zero biasing for discrete random variables [cf Goldstein and Reinert (2005)], we introduce a new family of discrete distributions and provide a discrete version of the Berry–Esseen theorem showing how members of the family approximate the distribution of a sum W of integer valued variables in total variation.
منابع مشابه
Zero Biasing and a Discrete Central Limit Theorem by Larry Goldstein
University of Southern California and University of Melbourne We introduce a new family of distributions to approximate P(W ∈A) for A ⊂ {. . . ,−2,−1,0,1,2, . . .} and W a sum of independent integer-valued random variables ξ1, ξ2, . . . , ξn with finite second moments, where, with large probability, W is not concentrated on a lattice of span greater than 1. The well-known Berry–Esseen theorem s...
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