Семинар Теорија вероватноћа и математичка статистика, 9, 10. и 11. јул 2018.

Наредни састанаци Семинара биће одржани у 9, 10. и 11. јула 2018. Предавања на Семинару ће одржати професор Gennady Samorodnitsky ca Cornell University, Ithaca по следећем распореду:

понедељак 9. јул 2018. у 16 часова, сала 830

Наслов предавања: FROM INFINITE URN SCHEMES TO SELF-SIMILAR STABLE PROCESSES

Апстракт: We investigate the randomized Karlin model with parameter $\beta\in(0,1)$, which is based on an infinite urn scheme. It has been shown before that when the randomization is bounded, the so-called odd-occupancy process scales to a fractional Brownian motion with Hurst index $\beta/2\in(0,1/2)$. We show that when the randomization is heavy-tailed with index $\alpha\in(0,2)$, then the odd-occupancy process scales to a  new $(\beta/\alpha)$-self-similar symmetric $\alpha$-stable process with stationary increments.

уторак 10. јул 2018. у 16 часова, сала 830

Наслов предавања: ЕXTREMAL THEORY FOR LONG RANGE DEPENDENT INFINITELY DIVISIBLE PROCESSES

Апстракт: We prove limit theorems of an entirely new type for certain long memory regularly varying stationary  infinitely divisible random processes. These theorems involve multiple phase transitions governed by how long the memory is. Apart from one regime, our results exhibit limits that are not among the classical extreme value distributions. Restricted to the one-dimensional case, the distributions we obtain interpolate, in the appropriate parameter range, the alpha-Frechet distribution and the skewed \alpha-stable  distribution. In general, the limit is  a new family of stationary and self-similar random sup measures with parameters  alpha in (0,infty)  and beta in (0,1), with  representations based on intersections of independent beta-stable regenerative sets. The tail of the limit random sup-measure on each interval with finite positive length is regularly varying with index -alpha. The intriguing structure of these random sup-measures is due to intersections of independent beta-stable regenerative sets and the fact  that the number of such sets intersecting simultaneously increases to infinity as  beta increases to one.

среда 11. јул 2018. у 16 часова, сала 830

Наслов предавања: EXTREME VALUE ANALYSIS WITHOUT THE LARGEST VALUES: WHAT CAN BE DONE?

Апстракт: Motivated by an analysis of the degree distributions in a large social network, we are concerned with the analysis of heavy-tailed data when a portion of the extreme values are unavailable.  We focus on the Hill estimator, which plays a starring role in heavy tailed modeling. The Hill estimator for this data exhibited a smooth and increasing “sample path” as a function of the number of upper order statistics used in constructing the estimator. This behavior became more apparent as we artificially removed more of the upper order statistics. Building on this observation, we introduce a new parameterization into the Hill estimator that is a function of ? and ?, that correspond, respectively, to the proportion of extreme values that are unavailable and the proportion of upper order statistics used normalized Hill estimator to a Gaussian random field. An estimation procedure is developed based on the limit theory to estimate the number of missing extremes and extreme value parameters including the tail index and the bias of Hill’s estimate. We illustrate how this approach works in both simulations and real data examples.

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