Approximations for multidimensional discrete scan statistics

In this thesis, we derive accurate approximations and error bounds for the probability distribution of the multidimensional discrete scan statistics. We start by improving some existing results concerning the estimation of the distribution of extremes of 1-dependent stationary sequences of random variables, both in terms of range of applicability and sharpness of the error bound. These estimates play the key role in the approximation process of the multidimensional discrete scan statistics distribution. The presented methodology has two main advantages over the existing ones found in the literature: first, beside the approximation formula, an error bound is also established and second, the approximation does not depend on the common distribution of the observations. For the underlying random field under which the scan process is evaluated, we consider two models: the classical model, of independent and identically distributed observations and a dependent framework, where the observations are generated by a block-factor. In the i.i.d. case, in order to illustrate the accuracy of our results, we consider the particular settings of one, two and three dimensions. A simulation study is conducted where we compare our estimate with other approximations and inequalities derived in the literature. The numerical values are efficiently obtained via an importance sampling algorithm discussed in detail in the text. Finally, we consider a block-factor model for the underlying random field, which consists of dependent data and we show how to extend the approximation methodology to this case. Several examples in one and two dimensions are investigated. The numerical applications accompanying these examples show the accuracy of our approximation. All the methods presented in this thesis leaded to a Graphical User Interface (GUI) software, implemented in Matlab®.