Extreme Value Results for Scan Statistics

Abstract

Abstract:In the first part of this paper we focus on the classical scan and multiple scan statistic, defined on a sequence of independent and identically distributed (i.i.d.) binary trials and review a number of bounds and approximations for their distributions which have been developed by the aid of distance measures. Moreover, we discuss briefly a number of asymptotic results that have been established by setting up appropriate conditions guaranteeing the convergence (to zero) of the distance measures' upper bounds. In the second part, we study a multiple scan statistic enumerating variable by considering a general threshold-based framework, defined on i.i.d. continuous random variables. More specifically, we prove first a compound Poisson approximation for the total number of fixed length overlapping moving windows containing prespecified number of threshold exceedances. The classical scan and multiple scan statistic may be treated as a special case of this general model. Next we exploit the previous result to gain some new extreme value results for the scan enumerating statistic under the assumption that the continuous random variables belong to the maximum domain of attraction of one of the three extreme value distributions (Frechet, reversed Weibull, Gumbel). Finally, we elucidate how the general results can be applied in a number of classical continuous distributions (Pareto, Uniform, Exponential and Normal).

Approximating the Distribution of the Multiple Scan Statistic

“…Obtaining the exact distribution of the multiple scan statistic, defined on a sequence of Bernoulli trials, becomes unwieldy when the width of the scanning windows and/or the sought number of successes within each window is of moderate size, resulting frequently the whole procedure as non-feasible (see, e.g., Wu et al, 2013;Koutras and Alexandrou, 1995;Fu et al, 2012;Fu, 2001;Boutsikas et al, 2009). Therefore, the development of asymptotic results and approximations for the distribution of the multiple scan statistic is of special interest.…”

Asymptotic results for the multiple scan statistic

“…Glaz et al (1994) and Chen and Glaz (1999)). Since windows with high concentration of successes tend to occur in clumps (local groups), an approximation of the W n,k,r distribution by an appropriate compound Poisson distribution is expected to offer the most efficient approach; Boutsikas et al (2009) reviewed a number of asymptotic results of this kind. Boutsikas and Koutras (2006) considered the problem of establishing compound Poisson approximations for W n,k,r that yield upper bounds which are manageable under the conditions n, k, r → ∞, with the success probability remaining fixed.…”