# STOHASTIÄŒKI MODELI PREKORAÄŒENJA VISOKOG NIVOA I PROBLEMI ÄŒEKANJA

Title: | STOHASTIÄŒKI MODELI PREKORAÄŒENJA VISOKOG NIVOA I PROBLEMI ÄŒEKANJA |

Author: | JockoviÄ‡, Jelena |

Abstract: | Statistical methodology for dealing with extremes depend on how extreme values are defined. One way to extract extremes from a given sample x1, x2, ..., xN is to consider maxima (minima). The other way is to consider values y1 = x1 âˆ’ u, y2 = x2 âˆ’ u, . . . , yn = xn âˆ’ u, where y1, y2, . . . , yn are sample members above (below) a given predetermined threshold u. These two methods lead to two different approaches in extreme value theory. This doctoral dissertation has two main goals. One of them is to apply the techniques from extreme value framework to certain type of combinatorial problems. The other goal is to contribute to the field of statistical modeling of extremes. The dissertation consists of three chapters. In the first chapter, we introduce generalized extreme value distributions and generalized Pareto distributions (GPD). These two families play key roles in the two approaches to modeling extremes. We set out the theoretical background for both approaches. In the second chapter, we apply the extremal techniques to combinatorial waiting time problems. Precisely, we consider Coupon collectorâ€™s problem, defined as follows: elements are sampled with replacement from the set Nn = {1, 2, . . . , n} under assumption that each element has probability 1/n of being drawn. The subject of interest is the waiting time Mn until all elements of Nn or some other pattern are sampled. We focus our attention to the following two cases: 1. Mn is the waiting time until all elements of Nn are sampled at least r times, where r is a positive integer; 2. Mn is the waiting time until all pairs of elements jj, j âˆˆ Nn are sampled. We present new results related to the asymptotic behavior of the waiting time Mn, if it is known that a large number of trials was performed and the experiment is not over. For both cases, we determine the limiting distribution of exceedances of Mn over high thresholds, and answer some related questions: how to choose a suitable high threshold (depending on n) in order to obtain a limiting distribution; under what conditions the limit does not depend on the threshold; are the generalized Pareto distributions the only possible limits. We also estimate the speed of convergence in both cases. The third chapter of the dissertation is devoted to estimation of parameters and quantiles of the generalized Pareto distributions. We restrict the attention to the two-parameter version of GPD, defined as: WÎ³,Ïƒ(x) = ï£±ï£´ï£´ï£² ï£´ï£´ï£³ 1 âˆ’ eâˆ’x , x â‰¥ 0, Î³ = 0 1 âˆ’ 1 + Î³ Ïƒx âˆ’1 , x â‰¥ 0, Î³ > 0 1 âˆ’ 1 + Î³ Ïƒx âˆ’1 , x âˆˆ h 0,âˆ’Ïƒ Î³ i , Î³ < 0. Well known problem with this model is inconsistency with the sample data, which is that one or more sample observations exceed the estimated upper bound in case when Î³ < 0. We propose a new, general technique to overcome the inconsistency problem and improve performance of the existing GPD estimation methods. We apply the proposed technique to methodof- moments and method-of-probability-weighted-moments estimates, investigate its performance through computer simulation and provide some real data examples. Finally, we address the problem of estimating high GPD quantiles. We evaluate the robustness of some estimation methods through simulation study and present a case study from finance (value-at-risk estimation), with special emphasis to certain difficulties related to this field of application. |

URI: | http://hdl.handle.net/123456789/4271 |

Date: | 2012 |

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