Read e-book online A Concept of Generalized Order Statistics PDF

By Udo Kamps

ISBN-10: 3519027364

ISBN-13: 9783519027362

ISBN-10: 3663091961

ISBN-13: 9783663091967

Order facts and list values look in lots of statistical purposes and are familiar in statistical modeling and inference. as well as those standard versions, a number of different types of ordered random variables, identified and new ones, are brought which are successfully utilized, e.g., in reliability idea. the most goal of this e-book is to give an idea of generalized order information as a unified method of numerous types of ordered random variables. a number of comparable effects on distributional and second homes of order statistics and list values are present in the literature that are deduced individually. the idea that of generalized order facts, in spite of the fact that, enablesa universal method of structural similarities and analogies. popular effects should be subsumed, generalized, and built-in inside a common framework. hence, the concept that of generalized order information presents a wide classification of versions with many fascinating, very important and invaluable houses for either the outline and the research of functional difficulties. Contents: versions of ordered random variables (with functions in reliability theory): order facts, order information with nonintegral pattern measurement, sequential order data, checklist values, krecords, Pfeifer's checklist version, knrecords from nonidentical distributions, ordering through truncation of distributions, censoring schemes / generalized order data / distribution concept of generalized order facts / moments of generalized order facts / life of moments / characterization of distributions by means of 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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N : n-1 n) n (x 1, ... ,x ) = k ( II j=1 n ~f. : The g OS' s X(r,n,rn,k) , r (F(s) < 1): = 2, ... ). In its general version, Pfeifer's model (as well as the model of sequential order statistics) is not suitable for our purposes. It does not provide a common approach to distributional properties and to results on moments of g OS's (see Ch. II, III, IV; cf. 3. 6. ). 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.

T ( r-1) ,k r-1 r-1 I XI:!. 2' 2 c 10rm r 1(1). h transltlon .. a Markov cha'm Wit - ( 1 - F /t) ) kr , 1 _ F (s) r s) - r ~ 2. In the distribution theoretical sense, the k -record modd and Pfeifer' s model are identical, n since the following transformation can be used: G (t) r =1- (1 - F (t)) r k r. However, the interpretations of the models are obviously different. Putting k r =n - r + 1 , we obtain the Markovian structure of sequential order statistics. 6. , we restriet ourselves to a particular choice of the distribution functions F 1, ...

X. e. XR . = X. for all j E IN. arge numbers, central Iimit theorems, functional laws of the iterated logarithm and strong invariance principles for the number N(t) of records in the time interval (O,t) as weil as for the record times. Moreover, they consider martingales based on L(n) and moment properties of the record times. In Section 1. 7. we will introduce another model of kn -records in connection with a line scheme of r. v. 1 s with possibly different underlying distributions. 6. Pfeifer's Record Model In his dissertation, Pfeifer establishes a model of record values based on a double sequence of non-identically distributed random variables (cf.

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