IJPAM: Volume 69, No. 4 (2011)

A POINTWISE NEGATIVE BINOMIAL
APPROXIMATION BY $w$-FUNCTIONS

P. Malingam$^1$, K. Teerapabolarn$^2$
Department of Mathematics
Faculty of Science
Burapha University
Chonburi 20131, THAILAND
$^2$e-mail: kanint@buu.ac.th


Abstract. We use Stein's method and $w$-functions to determine a result in approximating the distribution of a non-negative integer-valued random variable $X$ by negative binomial distribution with parameters $r\in[1,\infty)$ and $p=1-q\in(0,1)$ in terms of the point metric between two such distributions together with its non uniform upper bound. In addition, when $r=1$, we also give a non-uniform upper bound on pointwise geometric approximation to the distribution of $X$. For applications, we use these results to approximate some discrete distributions such as Pólya, negative Pólya, hypergeometric and negative hypergeometric distributions.

Received: March 22, 2011

AMS Subject Classification: 60F05, 62E17

Key Words and Phrases: geometric approximation, negative binomial approximation, negative binomial distribution, non-uniform upper bound, point metric, Stein's method, $w$-functions

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 69
Issue: 4