Bounds for the Distribution of Two-Dimensional Binary Scan Statistics

Abstract

In the present article we develop some efficient bounds for the distribution function of a two dimensional scan statistic defined on a (double) sequence of iid binary trials. The methodology employed here takes advantage of the connection between the scan statistic problem and a equivalent reliability structure and exploits appropriate techniques of reliability theory to establish tractable bounds for the distribution of the statistic of interest. An asymptotic result is established and a numerical study is carried out to investigate the efficiency of the suggested bounds.

Scan Statistics Viewed as Maximum of 1-Dependent Random Variables

“…We compare numerically our results with results obtained using the product approximation, the Poisson approximation and Bonferroni inequality techniques as presented in Glaz et al (2001). For binary Y i,j 's we compare our values to bounds obtained in Boutsikas and Koutras (2003).…”

Evaluating methods for the reliability of a large 2‐dimensional rectangular k‐within‐consecutive‐(r, s)‐out‐of‐(m, n):F system

“…Many studies on 2-dimensional discrete scan statistics can be found in Chapter 16 of Glaz, Naus, and Wallenstein [11]. Recently, Boutsikas and Koutras [7] had proposed some bounds on 2-dimensional discrete scan statistics and reported asymptotic results on them (an anonymous referee pointed out this for us).…”

Approximations for two-dimensional discrete scan statistics in some block-factor type dependent models

“…Since there are no exact formulas for P(S ≤ n), various methods of approximation and bounds have been proposed by several authors. An overview of these methods as well as a complete bibliography on the subject can be found in Chen and Glaz [1996], Glaz, Naus and Wallenstein [2001, Chapter 16], Boutsikas and Koutras [2003], Haiman and Preda [2006] and the references therein.…”