# A Concept of Generalized Order Statistics by Udo Kamps

By Udo Kamps

Order data and checklist values look in lots of statistical purposes and are well-known in statistical modeling and inference. as well as those general types, a number of different types of ordered random variables, identified and new ones, are brought which might be successfully utilized, e.g., in reliability idea. the most objective of this ebook is to provide an idea of generalized order data as a unified method of a number of versions of ordered random variables. a number of similar effects on distributional and second homes of order data and list values are present in the literature that are deduced individually. the idea that of generalized order records, even though, enablesa universal method of structural similarities and analogies. renowned effects could be subsumed, generalized, and built-in inside a basic framework. as a result, the concept that of generalized order information offers a wide classification of versions with many attention-grabbing, vital and helpful houses for either the outline and the research of useful difficulties. Contents: versions of ordered random variables (with functions in reliability theory): order information, order facts with nonintegral pattern measurement, sequential order information, checklist values, krecords, Pfeifer's checklist version, knrecords from nonidentical distributions, ordering through truncation of distributions, censoring schemes / generalized order facts / distribution idea of generalized order facts / moments of generalized order information / lifestyles of moments / characterization of distributions by means of sequences of moments / recurrence relatives for moments and characterizations of distributions / inequalities for moments and characterizations of distributions / reliability homes: transmission of getting older homes, partial ordering of generalized order statistics

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The random variables X(r,n,m,k) = F-1( U(r,n,m,k) ) , r = l, ... ,n, are called generalized order statistics ( based on the distribution function F ). In the sequel we write' g OS's' for brevity. In the case m 1 = ... = m0 _ 1 = m , say, they are denoted by X(r,n,m,k) , r = 1, ... ,n . 51 I Generalized Order Statistics REMARK 24. 3. be absolutely continuous with density function f. P( X(1,n,Iil,k) ~ x1 , Noticing that P( r ••• , X(n,n,Iil,k) ~ xn ) 1(U(1,n,Iil,k)) ~ x1 , ... , F-1(U(n,n,Iil,k)) ~ xn) P( U(1,n,Iil,k)) ~ F(x 1) , ...

I) The Markovian structure of xF~ ' ... ). Hence, in the distribution theoretical sense, these models are identical and the records may be interpreted as certain line minima. 47 I Generalized Order Statistics ü) ki =n-i + 1, 1~i~n, Putting r we get the structure of sequential order statistics ( li k. ) 1 ). j=l üi) J n-r . I = 1 forall i Choosing yields Pfeifer 1 s record model (d. 6. ). That is, record values from non-identically distributed r. v. V. 1 s Yii) , 1 ~ i ~ n , which are distributed according to Fi, respectively: x(Il = y(Il *,1 iv) 1 ' In the case of identical distribution functions F 1 = ...

ExAMPLES i) 25. In the case m 1 = ... e. 's X1' ... ,Xn with distribution function F: fX(1,n,O, 1), ... ,X(n,n,O, 1) (x 1, •.. ,x ) = n! g. 1. ). ii) Choosing m 1 = ... e. Ir= a -r + 1, 1 ~ r ~ n-1 ), we describe OS's with 52 I Generalized Order Statistics non-integral sample size : fX(l,n,O,a-n+l), ... ,X(n,n,O,G'-n+l) ( xl, ... ). iii) Given positive real numbers a 1, ... 1 - (n-i) a. 1 - 1 , i = 1, ... e. 1r = (n-r+ 1) ar , 1 -< r -< n-1 ) to obtain the joint density X(1,n,ffi,a ), ... ,X(n,n,m,a ) f n n (xl, ...