Extreme Value Theory for Tail-Related Risk Measures

Many fields of modern science and engineering have to deal with events which are rare but have significant consequences. Extreme value theory is considered to provide the basis for the statistical modeling of such extremes. The potential of extreme value theory applied to financial problems has only been recognized recently. This paper aims at introducing the fundamentals of extreme value theory as well as practical aspects for estimating and assessing statistical models for tail-related risk measures.Extreme Value Theory; Generalized Pareto Distribution, Generalized Extreme Value Distribution; Quantile Estimation, Risk Measures; Maximum Likelihood Estimation; Profile Likelihood Confidence Intervals.

Extreme events and event size fluctuations in biased random walks on networks

Random walk on discrete lattice models is important to understand various types of transport processes. The extreme events, defined as exceedences of the flux of walkers above a prescribed threshold, have been studied recently in the context of complex networks. This was motivated by the occurrence of rare events such as traffic jams, floods, and power black-outs which take place on networks. In this work, we study extreme events in a generalized random walk model in which the walk is preferentially biased by the network topology. The walkers preferentially choose to hop toward the hubs or small degree nodes. In this setting, we show that extremely large fluctuations in event-sizes are possible on small degree nodes when the walkers are biased toward the hubs. In particular, we obtain the distribution of event-sizes on the network. Further, the probability for the occurrence of extreme events on any node in the network depends on its 'generalized strength', a measure of the ability of a node to attract walkers. The 'generalized strength' is a function of the degree of the node and that of its nearest neighbors. We obtain analytical and simulation results for the probability of occurrence of extreme events on the nodes of a network using a generalized random walk model. The result reveals that the nodes with a larger value of 'generalized strength', on average, display lower probability for the occurrence of extreme events compared to the nodes with lower values of 'generalized strength'

Extreme Value Theory, Generalized Pareto Distribution, Maximum Likelihood Estimation, Peaks Over Threshold, Perubahan Iklim

Jawa Timur memiliki potensi besar dalam sektor pertanian,khususnya produksi padi. Lima kabupatenpenghasil padi terbesar di Jawa Timur adalah Kabupaten Banyuwangi, Bojonegoro,Jember, Lamongan, dan Ngawi. Produksi padi rentanterhadap keberagaman iklim terutama iklim ekstrim dan terjadinya Perubahan iklim. Dalam upaya meminimalkan kerugian akibat iklim, makaperlu mempelajari karakteristik iklim ekstrim dan identifikasi adanya Perubahaniklim. Penelitian inidilakukan untuk mengetahui cara mengestimasi parameter bentuk dan skalamenggunakan Maximum Likelihood Estimation (MLE), mengeksplorasi karakteristik curah hujan di wilayah penelitian, dan mengidentifikasiperubahan iklim. Metode yang digunakan menganalisis curahhujan ekstrim adalah Extreme Value Theory.Salah satu pendekatan untuk mengidentifikasi nilai ekstrim adalah Peaks Over Threshold yang mengikuti distribusi GeneralizedPareto Distribution (GPD). Data Curah hujan dibagi menjadi dua periode yaitu periode 1 (1981-1990)dan periode 2 (1991-2010). Estimasi parameter bentuk dan skala didapatkanmelalui MLE yang selanjutnya diselesaikan dengan Newton Raphson, karena menghasilkan persamaan yang tidakclosed form. Penelitian inimenghasilkan estimasi parameter bentuk dan skala distribusi GPD, serta confidence interval (1-α)100%dengan α sebesar 5%. Di samping itu disimpulkan bahwa terjadinya Perubahan iklim dikelima kabupaten, khususnya pada musim kemarau dan transis