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Nevertheless, a Bayesian point of view may be considered in order to estimate the rate λ of the homogeneous Poisson process. In that direction, P´erez-Mu˜noz [62] considers λ an unknown quantity and assigns to it a Gamma prior distribution with hyperparameters α and β that are considered to be known and are given by the researcher. Hence, the parameter to be estimated is θ = λ . The observed data D are composed of the number of ozone exceedances of the threshold of interest that occurred in the period of 1 year.

K) Therefore, the posterior distribution of θ is proportional to a Gamma distribution with parameters (α + k) and (β + 1). 2) setting s = 1. The hyperparameters of the prior distribution of θ are obtained by taking advantage of the fact that the mean and variance of a Gamma(α , β ) density are given, respectively, by μ = α /β and σ 2 = α /β 2 . Hence, we can calculate the empirical mean and variance of the number of surpassings per year of the threshold of interest during the observed years.

For instance, if we have T years of measurements, then for ni the number of exceedances of the threshold of interest in the ith year (i = 1, 2, . . , T ), the empirical mean is μ = (1/T ) ∑Ti=1 ni . 28 3 Poisson Models and Their Application to Ozone Data A similar calculation is given for the empirical variance (see for instance [65]). Then, using the relation between μ and σ 2 and α and β we solve the system of equations to obtain the hyperparameters α and β of the Gamma prior distribution. In [62] data from the Metropolitan Area of Mexico City were used.