Further improved recursions for a class of compound Poisson distributions

propose a spatial scan statistic using a compound Poisson distribution to detect spatial cluster of cases that may have multiple (repeated) events over the study region. Their method is motivated using emergency department visits data set, where the same individual can make multiple visits to a hospital. As noticed by Chang and Rosychuk, 2 the recursive formula 1 becomes computationally unfeasible for large data sets; hence, with this in mind, they propose the negative binomial approach to overcome this problem. In this letter, we introduce an alternative way to calculate the likelihood, and prove that it is equivalent to the recursive formula for the compound Poisson scan. 1 We apply this alternative way to calculate the likelihood of events for the large and extra large data sets.Let the population size of the subregion S i is denoted by n i for i = 1, … , I and also C i ∼ Poisson( i n i ), where i > 0. Consider C i is the total number of individuals with at least one event in subregion S i . Assume that within a period, say a fiscal year, individuals only have event(s) within the subregion of their residence. For individuals with at least one event, let the random variable X il denote the number of event(s) occurred by the lth individual in subregion S i , l = 1, … , C i . The probability mass function of the random variable X il can be arbitrary depended on the context; however, Rosychuk and Chang 1 assume that X il is discrete and follows a zero truncated Poisson distribution with parameter i . The total number of events from the population of the subregion i can be written as U i = ∑ C i l=1 X il , where C i is independent of X il for l = 1, … , C i and i = 1, … , I. With this formulation, U i is a compound Poisson random variable.In general, the recursive formula of Panjer 3 can be used to obtain likelihoods of the compound Poisson distribution. Hence, Rosychuk and Chang 1 used this fact to calculate probabilities of U i recursively as follows:Pr(U i = 0) = e − i n i , Pr(U i = u i ) = i n i u i u i

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“…Hence, Rosychuk and Chang used this fact to calculate probabilities of U i recursively as follows: