A Concept of Generalized Order Statistics by Udo Kamps

By Udo Kamps

Order information and checklist values look in lots of statistical functions and are regularly occurring in statistical modeling and inference. as well as those standard versions, numerous different types of ordered random variables, recognized and new ones, are brought which are successfully utilized, e.g., in reliability thought. the most objective of this booklet is to offer an idea of generalized order information as a unified method of quite a few versions of ordered random variables. various similar effects on distributional and second houses of order information and list values are present in the literature that are deduced individually. the concept that of generalized order data, even if, enablesa universal method of structural similarities and analogies. renowned effects should be subsumed, generalized, and built-in inside of a common framework. consequently, the idea that of generalized order facts offers a wide classification of versions with many fascinating, vital and invaluable homes for either the outline and the research of useful difficulties. Contents: types of ordered random variables (with functions in reliability theory): order data, order facts with nonintegral pattern measurement, sequential order records, list values, krecords, Pfeifer's list version, knrecords from nonidentical distributions, ordering through truncation of distributions, censoring schemes / generalized order records / distribution thought of generalized order records / moments of generalized order records / life of moments / characterization of distributions by way of sequences of moments / recurrence family for moments and characterizations of distributions / inequalities for moments and characterizations of distributions / reliability houses: transmission of getting older houses, partial ordering of generalized order statistics

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4. ), the stronger assumption turns out to be useful. An important special case in the concept of g OS' s is choosing m according to m1 = ... = m0 _ 1 = m E IR . Here, the main regularity conditions imposed on k, n and r k which imply ~ 1 and k + (n-1)(m+1) k + (n-r)(m+1) ~ ~ ~ n are 1 1 for r E {1, ... ,n}. Generalized order statistics based on some distribution function F are now defined by means of the quantile transformation X(r,n,m,k) DEFINITION = F-\ U(r,n,iil,k) ) , r = 1, ... ,n . 23. 1.

37 I Generalized Order Statistics Ordinary record values or k-th records (cf. 's. The assumption of identical distributions is weakened in Pfeifer's model. In relation to other modifications and approaches, we refer to the survey article by Nevzorov (1987, p 214-220). 1. A ,P) with P x(n) . J =P x(n) 1 . ,n,JEIN. l = 1 ' lJ. n+l = min { j E IN · x(n+l) > X ( n) } ' ll. 0 J n E IN ' ' and record values by This model is extensively examined in Pfeifer (1979, 1982a,b). , it is shown that, under suitable conditions, the sequence of jump times of an elementary pure birth process and the sequence of record values are identically distributed.

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