Time series and state space model with generalized extreme value distributed marginals and α-stable distributed errors

Abstract

This thesis is mainly focused on the estimation and filtering of extreme events time series and models with generalized extreme value distributed marginals and the multiplicative errors from α-stable distribution. First a non-linear time series with Fréchet distributed marginals and α-stable distributed errors is considered. To estimate the stability parameter, three recursive procedures are proposed. The first is based on the Hill estimation, the second is a modified Fan’s estimation that uses the property of the α-stable distribution, and the last is an application of Kantorovich-Wasserstain metric. For the state space model with generalized extreme value distributed marginals and α-stable distributed errors, the estimation is more complex, especially when the stability parameters are small. In the model with Gumbel distributed marginals, if one of the stability parameter is known, a procedure that generates an ensemble from the known error distribution by Monte Carlo followed by estimation is proposed. For a model with generalized extreme value distributed marginals and unknown stability parameters, first a recursive regression estimation is applied to obtain the generalized extreme valued parameters, then the Yule-Walker estimation or generalized least square regression model is used to estimate the stability parameters. Regarding filtering, the estimation of unobserved states and their empirical conditional densities are our interests. The estimation of states is obtained numerically via Monte Carlo, based on the model structure. This procedure outperforms Kalman filter. As to the empirical conditional density, sequential importance sampling with different importance functions, particle filter with discrete sample space, auxiliary particle filter and plain linearization are used and compared. The asymptotic properties and rates of convergence of the proposed estimations are studied analytically and through simulation. The methods and procedures developed in this thesis have been applied to analyze the air pollution data in New York city.