Asymptotic Behavior Of Generalized Order Statistics With Random Indices

Kamps[52]introduced the concept of generalized order statistics(gos)as a unified approach to a variety of models of ascendingly ordered random variables(rv's)with different interpretations.Since Kamps[52]had introduced the unifying model of gos,the use of such a model has been steadily growing along the years.This is due to the fact that this model includes several important practical models that had been separately treated in statistical literature.Examples of such models are the ordinary order statistics(oos),order statistics with non-integral sample size,sequential order statistics(sos),progressive type II censored order statistics(pos),record values,kth record values and Pfeifer's records.These models can be applied in reliability theory.For instance,the rth extreme os represents the life-length of some r-out-of-n system,whereas the sos model is an extension of the oos model and serves as a model describing certain dependencies or interactions among the system components caused by failures of components and the pos model is an important method of obtaining data in lifetime tests.Moreover,the distributional and inferential properties of oos and record values turn out to remain valid for gos(c.f.[37,52]).Burkschat et al.[33]introduced the dual model of gos,which is called dual generalized order statistics(dgos).The dgos model enables us to study descending ordered rv's,like reversed order statistics,lower k-records and lower Pfeifer records,through a common approach.Burkschat et al.[33]showed the relationship between gos and dgos and they illustrated this relationship by some examples.Record values and record times have been of interest to humans throughout history.Meteorologists frequently deal with upper and lower record temperatures and precipitation levels.A seismologist may be interested in earthquakes of record magnitude.Record values appear often in sporting events.For example,an analyst may be concerned with record performances in the Olympic one hundred meter dash.There is something about record breaking performances that makes them fascinating to humans.Actually,no one can't be interested in record values.The upper and lower record model can be obtained as a special case of gos and dgos model,when m =-1 and k=1.In many practical problems we often come across situations where the sample size n is a positive integer-valued rv vn,following a given distribution function(df).Perhaps,one of the major reasons for this phenomenon is that in many biological,agricultural and some quality control problems,it is almost impossible to have a fixed sample size,because some observations always get lost for various reasons.However,random sample sizes naturally arise in such topics as sequential analysis,branching processes,damage models or rarefaction of point processes and records as maxima.In this dissertation,we consider the random sample size as an extension of a model(mainly for statistical inference),one can usually assume that it is independent of the underlying variables.Galamabos[44]pointed out that if we allow linear normalization with the(same)random indices,then the normalizing constants may dominate both the conditions for convergence and the actual form of the limiting distribution.Therefore,the only interesting weak convergence results are those when the normalizing constants ate not random.The main aim of this dissertation is to study the asymptotic behavior of general sequences of univariate extreme,central and intermediate gos and dgos,in general frame work,namely in a subclasses of gos and dgos models,known as m-gos and m-dgos,m>-1 and univariate upper and lower record values,which are connect-ed asymptotically with some regularly varying functions.Moreover,the limit df's of univariate m-gos and its dual,m>-1 and record values,with random indices under general conditions are obtained.The classes of limit df's of bivariate extreme,central and intermediate m-gos and its dual,m>-1,from independent and iden-tically distributed(i.i.d.)rv's with random sample size are fully characterized.Two cases are considered:the first case is when the random sample size is assumed to be independent of all basic rv's(i.e.,the original random sample).The second case is when the interrelation of the random size and the basic rv's is not restricted.Finally,the asymptotic behavior of the joint upper and joint lower record values for fixed and random sample sizes are studied.This problem for fixed sample size is recently tackled by Barakat et al.[23-25]for m-gos and its dual,m>-1,i.e.,the record values case was excluded from this study.In this dissertation,we will fill this gap.Moreover,as an application of this study,we could study the asymptotic behavior of some of simple functions of joint record values for fixed and random sample sizes,which have important applications.Namely,the record quasi-ranges,record quasi-midranges,record extremal quasi-quotients and record extremal quasi-products are obtained.This dissertation consists of five chapters,the first of them is an introductory chapter,besides the list of references.It is worth to mention that the material from the second to the fifth chapters of the dissertation were prepared as the six separated papers[2,3,15,21,22,30].Chapter one:In this chapter,we simply give an elementary introduction to os,gos and its dual and record values,which should be regarded as a bare essential description on the topic that would facilitate the reader to follow all the other chap-ters of this dissertation.The limit theory of os,gos and its dual and record values are discussed.Moreover,the limit theory of some important record functions,e.g.,record range,record mid-range,record extremal quotient and record extremal prod-uct are presented.Chapter two:In this chapter,we study the asymptotic behavior of a general sequences of univariate os,m-gos and its dual,m>-1 and record values,which are connected asymptotically with some regularly varying functions.Moreover,the limit df's of univariate os,m-gos and its dual,m>-1 and record values,with random indices,are studied under general conditions.All the results in the last two sections of this chapter are new and are the extension of the results of Barakat and El-shandidy[171.Chapter three:In this chapter,the class of limit df 's of bivariate extreme,cen-tral and intermediate m-gos,m>-1,from i.i.d.rv's with random indices are fully characterized.Two cases are considered of dependence of the sample size on the basic observations(either independence or general dependence without any con-straints).An illustrative examples are provided,which lend further support to our theoretical results.Chapter four:In this chapter,the class of limit df's of bivariate extreme,cen-tral and intermediate m-dgos,m>-1,from i.i.d.rv's with random indices are fully characterized.Two cases are considered of dependence of the sample size on the basic observations(either independence or general dependence without any con-straints).An illustrative examples are provided,which lend further support to our theoretical results.Chapter five:In this chapter,the class of limit df's of the joint upper record values and the joint of lower record values for fixed and random sample sizes are fully characterized.Sufficient conditions for the weak convergence are obtained.As an application of this result,the sufficient conditions for the weak convergence of the record quasi-ranges,record quasi-midranges,record extremal quasi-quotients and record extremal quasi-products,for fixed and random sample sizes are obtained.Moreover,the classes of the non-degenerate limit df's of these statistics are derived.