Hybrid Poissonextreme value distribution

Let us consider the estimation problem for extreme value time series in a more general manner. In principle, the distribution function of Xout(i) may change with time, T(i). The sample is used to estimate properties of the time-dependent PDF. Fitting a stationary distribution (GEV or GP; Section 6.2) corresponds to using an "estimation area" (Fig. 6.11) with dxout arbitrarily small and ¿t (sample level) equal to the whole observation interval, [t(n) — t(1)]. Fitting an inhomogeneous Poisson process (Section 6.3.2) corresponds to using an estimation area with ¿t small (in the order of the bandwidth, h) and dxout arbitrarily large (interval from u to to). Fitting a GEV or GP model with time-dependent parameters (Section 6.3.1) means using an estimation area with dxout arbitrarily small and ¿t in principle also small. By writing "in principle" we acknowledge that here the comparison is flawed and the

Figure 6.11. Estimation area for extreme value time series. The area (dark shading) is given by dxout • St. In the GP case, where POT values (circles) are analysed, St = [t(n) —1(1)]; in the inhomogeneous Poisson case, where event times are analysed, dxout ^ to. u, threshold; xout(i) = x(i) — u.

Figure 6.11. Estimation area for extreme value time series. The area (dark shading) is given by dxout • St. In the GP case, where POT values (circles) are analysed, St = [t(n) —1(1)]; in the inhomogeneous Poisson case, where event times are analysed, dxout ^ to. u, threshold; xout(i) = x(i) — u.

time-dependence is actually modelled parametrically and not estimated nonparametrically.

The hybrid model (Smith 1989, 2004) is a mixture between a non-parametric description of the time-dependence via the inhomogeneous Poisson process and a parametric extreme value distribution model such as the GP. The probability that an event occurs is multiplied with the probability that the extreme has a size within a certain interval. The Poisson-GP hybrid model corresponds to a combined rate measure,

Analogously, the Poisson-GEV hybrid uses the GEV density function,

The Monte Carlo experiment (Section 6.3.2.11) employed a hybrid model to generate the data. Conditional on the existence of an event at Tout(j), which is described by the occurrence rate, we drew xout(j) from a chi-squared distribution with v = 1.

Fitting the hybrid model corresponds to using an estimation area with both ¿t and dxout small (Fig. 6.11). Davison and Ramesh (2000) developed an estimation method for the hybrid model based on kernel smoothing and maximum likelihood fitting (Section 6.5). We remark that such "two-dimensional" (T, X) estimations may require a large sample size to achieve acceptably small error bars.

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