By Udo Kamps
Order information and list values seem in lots of statistical functions and are usual in statistical modeling and inference. as well as those trendy versions, a number of different types of ordered random variables, identified and new ones, are brought which are successfully utilized, e.g., in reliability thought. the most goal of this e-book is to give an idea of generalized order records as a unified method of a number of versions of ordered random variables. various comparable effects on distributional and second homes of order information and checklist values are present in the literature that are deduced individually. the concept that of generalized order information, even if, enablesa universal method of structural similarities and analogies. renowned effects might be subsumed, generalized, and built-in inside of a basic framework. consequently, the concept that of generalized order records offers a wide type of versions with many fascinating, very important and helpful homes for either the outline and the research of functional difficulties. Contents: types of ordered random variables (with purposes in reliability theory): order facts, order records with nonintegral pattern measurement, sequential order records, checklist values, krecords, Pfeifer's checklist version, knrecords from nonidentical distributions, ordering through truncation of distributions, censoring schemes / generalized order facts / distribution thought of generalized order data / moments of generalized order records / lifestyles of moments / characterization of distributions via sequences of moments / recurrence family members 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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Additional resources for A Concept of Generalized Order Statistics
N Pfeifer (1979, p 81) shows that the distribution of the n-th record XA n), with \ = i + v n for all i E IN and v E IN 0 fixed, coincides with the distribution of the n-th order statistic based on n + v iid random variables with distribution function F, letting F be the standard exponential distribution function (without loss of generality). (x) I =1- (1- F(x))n--i+l , i = 1, ... ,n, the Markovian structures of records and ordinary order statistics coincide. ( n) f 1 n (xl, ... ,xn) = n Y 1 , ...
S via truncation of distributions Ieads to alternative interpretations of the record models which are shown in the previous sections. 3. 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.
F-1(U(n,n,Iil,k)) ~ xn) P( U(1,n,Iil,k)) ~ F(x 1) , ... , U(n,n,Iil,k)) ~ F(xn) ) , the joint density function of the generalized order statistics X(1,n,Iil,k) , ... , X(n,n,Iil,k) is given by f X(1,n,iit,k), ... 's are obviously included in the distribution theoretical sense. Not only these models, but all the structures introduced in Section 1 are contained in the model of g OS's. 3. appropriately, we obtain the corresponding joint density functions shown in Section 1. ExAMPLES i) 25. In the case m 1 = ...