Bivariate Limit Theorems For Random Generalized Order Statistics And Its Dual In A Stationary Gaussian Sequences

Most of the traditional statistical theory assume that the sample size is fixed or known beforehand when making statistical inference.However,in many practical problems we often come across situations where the sample size n is a positive integer-valued random variable(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[47].In this thesis,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[48]poiuted 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 are not random.Extreme value theory is mainly a model building tool,but it can also be utilized in statistical evaluations.It concerns the largest or the smallest in a set of rv's,where the rv's in question are either actual observations or just hypothetical quantities for describing a model.Hence,extreme value theory is more than the study of the largest or smallest order statistics(os)since values other than the extremes may become mean-ingless in certain situations,for example,a spacecraft may be destroyed by the first failure of essential components[6][51].Kalps[58]introduced the concept of generalized order statistics(gos)as a unified approach t.o a variety of models of ordered rv'^with different interpret.ations.Ordi-nary order statistics(oos),k-records(ordinary record values when k = 1),sequential order statistics(sos),ordering via truncated distributions and censoring schemes can be discussecd as they a.re special cases of the gos.Since Kalls[59]had introduced the unifying model of gos,the use of such amodel has been steadily growing along the years because it is more flexible in reliability theory,statistical modeling and inference.Burkschat et al.[34]introduced the dual model of gos,which is called dual gener-alized order statistics(dgos).The dgos model enables us to study descending ordered rv slike reversed os,lower k-records and lower Pfeifer records,through a common approach.Burkschat et al.[34]showed the relationship between gos and dgos and they illustrated this relationship by some examples.Vasudeva and Moridani[76]studied the limit distribution of upper extreme of stationary Gaussian sequence(sGs)the sample size is itself a rv vn,which is independent of the basic variables.However,in the work of Vasudeva and Moridani[76]there is a restrictive condition that the random sequence of the correlation coefficient Pvn converges in probability to a positive constant or infinity.More recently,Barakat et al.[27]gos reduced this restrictive condition and obtained the parallel results for limit distributions of oos with random indices in a sGs.Moreover,Barakat et al.[28]studied the limit distributions of extreme,intermediate and central m-gos,when the random sample size is assumed to converge weakly.Recently,Barakat[11]studied the limit joint probability function(jpf)of any two extreme,as well as central,m-gos,when the sample size is non-random.The main purpose of this thesis,we will extend the recent,work of Barakat[11]to the case when the sample size is assumed to be a positive integer-valued rv vn independent of the basic variables.As an application of this result,the sufficient conditions for the weak convergence of random generalized quasi-range,generalized quasi-midrange,generalized extremal product,generalized extremal quotient and dual generalized quasi-range,dual generalized quasi-midrange,dual generalized extremal product,dual generalized extremal quotient.It is worth mentioning that,the results of this thesis contribute not only to a critical assessment of existing statistical method-ology,but also help to address their limitations within different contexts.This thesis consists of six chapters,the first of them is an introductory chapter,besides the list of references.It is worth mentioning that the material from the chapters second and four of this thesis is prepared as the one paper[44].Chapter one:In this chapter,deals with general introduction for distributional the-ory of os,gos and dgos with a review for basic probability limit theorems which will be used in the sequel description on the topic that would facilitate the reader to follow all the other chapters of this thesis.Also,we review of the limit theory os with variable rank,the sGs,the extremes with random sample size and a review of gos and dgos,of the limit theory of some important functions,quasi-range,quasi-mid-range,extremal quotient and extremal product and its dual are presented.Chapter two:In this chapter,we study the limit df's bivariate random sample size of the extreme,central and intermediate m-gos of a sGs under equi-correlated set up,when the random sample size is assumed to converge weakly and vn is independent of the basic variables.Moreover,the sufficient conditions for the weak convergence with random indices is obtained.Chapter three:In this chapter,we study the limit df's bivariate random sample size of the extreme,central and intermediate m-dgos of a sGs under equi-correlated set up,when the random sample size is assumed to converge weakly and vn is independent of the basic variables.Moreover,the sufficient conditions for the weak convergence with random indices is obtained.Chapter four:In this chapter,we study the limit df's bivariate random sample size of the generalized quasi-range,generalized quasi-midrange,generalized extremal prod-uct,generalized extremal quotient in a sGs of the sufficient conditions for the weak convergence are obtained.Moreover,the classes of the non-degenerate limit df's of these statistics are derived.Chapter five:In this chapter,we study the limit df's bivariate random sample size of the dual generalized quasi-range,dual generalized quasi-midrange,dual generalized ex-tremal product,dual generalized extremal quotient in a sGs of the sufficient conditions for the weak convergence are obtained.Moreover,the classes of the non-degenerate limit df's of these statistics are derived.Chapter six:Summary and Conclusion.