tailieunhanh - Báo cáo hóa học: "EXACT KOLMOGOROV AND TOTAL VARIATION DISTANCES BETWEEN SOME FAMILIAR DISCRETE DISTRIBUTIONS"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: EXACT KOLMOGOROV AND TOTAL VARIATION DISTANCES BETWEEN SOME FAMILIAR DISCRETE DISTRIBUTIONS | EXACT KOLMOGOROV AND TOTAL VARIATION DISTANCES BETWEEN SOME FAMILIAR DISCRETE DISTRIBUTIONS JOSÉ A. ADELL AND P JODRA Received 9 June 2005 Accepted 24 August 2005 We give exact closed-form expressions for the Kolmogorov and the total variation distances between Poisson binomial and negative binomial distributions with different parameters. In the Poisson case such expressions are related with the Lambert w function. Copyright 2006 J. A. Adell and P. Jodra. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Estimates of the closeness between probability distributions measured in terms of certain distances particularly the Kolmogorov and the total variation distances are very common in theoretical and applied probability. Usually the results refer to upper estimates of those distances even sharp upper bounds in some sense. As far as we know only a few exceptions deal with exact formulae see . Kennedy and Quine 5 where the exact total variation distance between binomial and Poisson distributions is given for small values of the success parameter of the binomial . Although numerical computations seem to be unavoidable exact expressions are only useful if they are easy to handle. The aim of this note is to provide exact closed-form expressions for the Kolmogorov and the total variation distances between Poisson binomial and negative binomial distributions with different parameters. In many occasions these distances appear as ingredients to estimate other distances in more complex situations see . Ruzankin 8 . On the other hand it is interesting to observe that in the Poisson case such exact formulae involve the Lambert w function. This function for which efficient numerical procedures of evaluation are known has many applications in pure and applied mathematics for more details see Corless .