题目：Mixed (Bayesian) models for the statistical regularities in data, their mathematical foundation and application to the analysis of extremal precipitation
报告人：Victor Korolev（Moscow State University, Russia）
时间：4月25日14:00
地点：闵行校区统计楼103室
摘要：Mixed probability models are proposed for statistical regularities in the behavior of such characteristics of precipitation as the duration of a wet period, maximum daily precipitation within a wet period and total precipitation volume per a wet period. The base for the models is the generalized negative binomial (GNB) distribution. The results of fitting the GNB distribution to real data are presented and demonstrate excellent concordance of the GNB model with the empirical distribution of the duration of wet periods measured in days. Based on this GNB model, asymptotic approximations are proposed for the distributions of the maximum daily precipitation volume within a wet period and of the total precipitation volume for a wet period. The asymptotic distribution of the maximum daily precipitation volume within a wet period turns out to be a tempered scale mixture of the gamma distribution in which the scale factor has the Weibull distribution, whereas the asymptotic approximation for the total precipitation volume for a wet period turns out to be the generalized gamma (GG) distribution. Both approximations appear to be very accurate. These asymptotic approximations are deduced using limit theorems for statistics constructed from samples with random sizes having the generalized negative binomial distribution. Based on these models, two approaches are proposed to the definition of abnormally extremal precipitation. These approaches improve the existing ones (Zolina et al., 2013), (Korolev et al., 2017). The first approach to the definition (and determination) of abnormally extreme precipitation is based on the distribution of the maximum daily precipitation of the form of a tempered scale mixture of the gamma distribution in which the scale factor has the Weibull distribution. The analytic and asymptotic properties of this distribution are discussed. According to the first approach, a daily precipitation volume is considered to be abnormally extremal, if it exceeds a certain (pre-defined) quantile of this distribution. The second approach is based on that the total precipitation volume for a wet period has the GG distribution. This model is deduced as a version of the law of large numbers for random sums in which the number of summands has the GNB distribution. Hence, the hypothesis that the total precipitation volume during a certain wet period is abnormally large at a given time horizon can be formulated as the homogeneity hypothesis of a sample from the GG distribution. Two equivalent tests are proposed for testing this hypothesis. One of them is based on the beta distribution whereas the second is based on the Snedecor—Fisher distribution. Both of these tests deal with the relative contribution of the total precipitation volume for a wet period to the considered set (sample) of successive wet periods. Within the second approach it is possible to introduce the notions of relatively abnormal and absolutely abnormal precipitation volumes. The results of the application of this test to real data are presented yielding the conclusion that the intensity of wet periods with abnormally large precipitation volume increases.
Victor Yu. Korolev简介
Korolev教授是大数学家柯尔莫果洛夫的学生，现任莫斯科国立大学计算和控制系副主任，数理统计教研室主任。1954年生，1981年获俄罗斯科学院数理学副博士学位，论文题目为《多发过程和极限定理》，1994年获俄罗斯科学院博士学位，论文题目为《具有独立随机指数的随机序列极限分布及其若干应用》。
曾荣获洛莫诺索夫大奖（2005年），被授予莫斯科国立大学功勋教授（2006年），荣获莫斯科建城850周年金质奖（1997年）。已发表学术论文230余篇，专著及教材二十余本。