Asymptotic factorial powers expansions for binomial and negative binomial reciprocals

Rempała, Grzegorz

doi:10.1090/S0002-9939-03-07254-X

Asymptotic factorial powers expansions for binomial and negative binomial reciprocals
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by Grzegorz A. Rempała PDF

Proc. Amer. Math. Soc. 132 (2004), 261-272 Request permission

Abstract:

By considering the variance formula for a shifted reciprocal of a binomial proportion, the asymptotic expansions of any order for first negative moments of binomial and negative binomial distributions truncated at zero are obtained. The expansions are given in terms of the factorial powers of the number of trials $n$. The obtained formulae are more accurate than those of Marciniak and Wesołowski (1999) and simpler, as they do not involve the Eulerian polynomials.

References

R. R. Bahadur, Some approximations to the binomial distribution function, Ann. Math. Statist. 31 (1960), 43–54. MR 120675, DOI 10.1214/aoms/1177705986
William G. Cochran, Sampling techniques, 3rd ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York-London-Sydney, 1977. MR 0474575
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Edward W. Frees, Infinite order $U$-statistics, Scand. J. Statist. 16 (1989), no. 1, 29–45. MR 1003967
Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete mathematics, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994. A foundation for computer science. MR 1397498
M. C. Jones, Inverse factorial moments, Statist. Probab. Lett. 6 (1987), no. 1, 37–42. MR 907257, DOI 10.1016/0167-7152(87)90056-3
Ewa Marciniak and Jacek Wesołowski, Asymptotic Eulerian expansions for binomial and negative binomial reciprocals, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3329–3338. MR 1637400, DOI 10.1090/S0002-9939-99-05105-9
Grzegorz Rempala and Gábor Székely, On estimation with elementary symmetric polynomials, Random Oper. Stochastic Equations 6 (1998), no. 1, 77–88. MR 1610209, DOI 10.1515/rose.1998.6.1.77
Edward B. Rockower, Integral identities for random variables, Amer. Statist. 42 (1988), no. 1, 68–72. MR 936685, DOI 10.2307/2685265
Robert J. Serfling, Approximation theorems of mathematical statistics, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1980. MR 595165