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SPEAKER: |
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TITLE: |
"A Bound for the Distribution of the Sum of Discrete Associated or NA Random Variables" |
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DAY: |
Wednesday, November 24, 1999 |
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TIME: |
3:30 p.m. [Coffee & cookies in BSB-202 at 3:00 p.m.] |
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PLACE: |
BSB-108 |
Let X1, X2, ..., Xn be a sequence of integer-valued random variables (r.v.'s). The need to approximate the distribution of the sum arises in various disciplines, such as probability theory, statistics, reliability, biology, etc.
The Poisson approximation of the distribution of the sum, known in the statistical literature as "Poisson law of small numbers" when the Xi's are independently distributed binary r.v.'s, has been the subject of continuing theoretical interest for more than one and a half centuries. Throughout statistical history several interesting generalizations/extensions of the classical Poisson law have been brought to light. Recently, the remarkable work by Chen (1975) led to the development of a group of flexible, powerful techniques that can be effectively used to estimate the error in the Poisson, Binomial and compound Poisson approximation of the sum of dependent indicator r.v.'s.
The purpose of this talk is to present simple tools, useful for approximating the distribution of the sum of integer valued (not necessarily binary) dependent r.v.'s by the distribution of the sum of independent variables with the same marginals as the original ones. This is accomplished at the expense of restricting the nature of the dependence to that of associated or negatively associated (NA) r.v.'s.
The organization of the talk is as follows: we first review preliminaries on a number of useful distance metrics (total variation distance, Kolmogorov distance, Wasserstein distance), association, negative association and several other types of positive or negative dependence. We present our general result for the approximation of the distribution of the sum of integer-valued associated or NA r.v.'s and specialize to the case of binary r.v.'s thereof obtaining a simple inequality for the mean of the product X1 X2 ... Xn and an upper bound for generalized binomial approximations. A compound Poisson approximation for the sum is examined and an application of the general result in a simple sampling scheme is presented. It is worth mentioning that our bounds require the computation of the first and second moments of Xi but not the higher ones.
Dr Markos Koutras is an Associate Professor in the Section of Statistics and O.R. of the Department of Mathematics at the University of Athens, Greece. He obtained his Ph.D. under the guidance of the legendary Professor Theofilos Cacoullos. His research interests include combinatorial models, discrete distributions, reliability theory and elliptical distributions.
The reference below, which Dr Koutras has suggested as useful background for his talk, has been placed on reserve at Thode Library (STATS 770: Statistics Seminar):
[1] Chen, LHY (1975), "Poisson Approximation for Dependent Trials," THE ANNALS OF PROBABILITY Vol. 3, No. 3, pp. 534-545.