Estimating the Distributions of Scan Statistics with High Precision

In many statistical applications one is concerned with the estimation of the distribution of the maximum or minimum number of points in a moving window of fixed length, called scan statistics. Scan statistics are also extremes of 1-dependent sequences. A result of Haiman (1999) provides approximations of these distributions together with sharp bounds for the corresponding errors. Applications concern the maximum cluster of points on a line or on a circle and multiple coverage by subintervals or subarcs of fixed size. We compare our method with some existing empirical and non empirical methods and show how it can be applied to multidimensional scanning