Download PDF by Udo Kamps: A Concept of Generalized Order Statistics

By Udo Kamps

ISBN-10: 3519027364

ISBN-13: 9783519027362

ISBN-10: 3663091961

ISBN-13: 9783663091967

Order information and checklist values seem in lots of statistical purposes and are normal in statistical modeling and inference. as well as those standard versions, numerous different versions of ordered random variables, identified and new ones, are brought which might be successfully utilized, e.g., in reliability idea. the most function of this publication is to give an idea of generalized order facts as a unified method of numerous versions of ordered random variables. quite a few similar effects on distributional and second homes of order statistics and list values are present in the literature that are deduced individually. the concept that of generalized order data, notwithstanding, enablesa universal method of structural similarities and analogies. popular effects should be subsumed, generalized, and built-in inside of a normal framework. as a result, the concept that of generalized order records presents a wide type of versions with many fascinating, very important and priceless homes for either the outline and the research of useful difficulties. Contents: versions of ordered random variables (with purposes in reliability theory): order information, order information with nonintegral pattern dimension, sequential order information, checklist values, krecords, Pfeifer's list version, knrecords from nonidentical distributions, ordering through truncation of distributions, censoring schemes / generalized order facts / distribution idea of generalized order information / moments of generalized order facts / lifestyles of moments / characterization of distributions by means of sequences of moments / recurrence kin for moments and characterizations of distributions / inequalities for moments and characterizations of distributions / reliability homes: transmission of getting older houses, partial ordering of generalized order statistics

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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. It should be noted that the numbering of records is modified here. In the special case of ordinary record values it coincides with the notation of Nevzorov (1987) (cf.

Pfeifer' s model is too extensive to explain the analogies in the behaviour of order statistics and record values (see Introduction) as well as to generalize known results and to integrate them within a general framework. Therefore, we restriet ourselves to a particular 57 I Generalized Order Statistics choice of the underlying distributions (cf. 8. ). However, the Markovian structure of Pfeifer' s records does provide a helpful toolleading to a unified approach to a variety of results on independence or identical distributions of certain statistics.

S with distribution function F : fX(1,n,-l,k), ... ,X(n,n,-1,k) (x1, ... g. 4. for k E IN). 5. for k The term 'k-th record values' is used in the literature. 1. instead of m + 1 . n 53 I Generalized Order Statistics v) Given positive real numbers ßl' ... ,ßn, we choose mi = ßi- ßi+ 1 - 1, i = 1, ... e. 'Yr = ßr , 1 ~ r ~ n-1 ) to obtain the joint density X(1,n,m,ß ), ... ,X(n,n,ffi,ß ) n (x1, ... ,xn) n f m. ) j= 1 J i =1 } ß -1 (1-F(x )) n f(x) I n n I of Pfeifer' s record values from non-identically distributed r.

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A Concept of Generalized Order Statistics by Udo Kamps

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