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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: STRONG CONVERGENCE BOUNDS OF THE HILL-TYPE ESTIMATOR UNDER SECOND-ORDER REGULARLY VARYING CONDITIONS | STRONG CONVERGENCE BOUNDS OF THE HILL-TYPE ESTIMATOR UNDER SECOND-ORDER REGULARLY VARYING CONDITIONS ZUOXIANG PENG AND SARALEES NADARAJAH Received 22 April 2005 Revised 7 July 2005 Accepted 10 July 2005 Bounds on strong convergences of the Hill-type estimator are established under second-order regularly varying conditions Copyright 2006 Z. Peng and S. Nadarajah. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Suppose X1 X2 . are independent and identically distributed iid random variables with common distribution function df F. Let Mn max X1 . Xn denote the maximum of the first n random variables and let w F sup x F x 1 denote the upper end point of F. The extreme value theory seeks norming constants an 0 bn e and a nondegenerate df G such that the df of a normalized version of Mn converges to G that is x Pr iMn - an bn Fn anX bn G x 1.1 as n - 00. If this holds for suitable choices of an and bn then it is said that G is an extreme value df and F is in the domain of attraction of G written as F e D G . For suitable constants a 0 and b e R one can write G ax b Gy x exp ị - 1 yx 1 q 1.2 for all 1 yx 0 and y e . For y 0 1.1 is equivalent to m U x 1.3 where U t 1 1 - F t inf t e R 1 1 - F x t that is U t is a regularly varying function at infinity with index y. Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 95124 Pages 1-7 DOI 10.1155 JIA 2006 95124 2 Strong convergence bounds of the Hill-type estimator The distribution given by 1.2 is known as the extreme value distribution. Its practical applications have been wide-ranging fire protection and insurance problems model for the extremely high temperatures prediction of the high return levels of wind speeds relevant for the design of civil engineering structures model for the extreme occurrences in
