Approximations for Discrete Scan Statistics on i.i.d and Markov-Dependent Bernoulli Trials

In this short note, we examine some approximations for the distribution of the discrete scan statistic defined on i.i.d. and Markov-dependent Bernoulli trials. The approximations are developed using the finite Markov chain imbedding technique of Fu and Koutras (J Am Stat Assoc 89(427):1050–1058, 1994) and the methods in Fu and Johnson (Adv Appl Probab 41(1):292–308, 2009) and Koutras and Milienos (J Stat Plann Inference 142(6):1464–1479, 2012). The approximations perform well for the cases considered and, in most cases, outperform the commonly used product approximation developed in Chen and Glaz (Approximations for the distribution and the moments of discrete scan statistics. In: Glaz J, Balakrishnan N (eds) Scan statistics and applications. Statistics for industry and technology. Birkhäuser, Boston, pp 27–66, 1999).