| Kamps(1995a)suggested a new theoretical approach,which is called general-ized order statistics(GOS).This new model includes ordinary order statistics(OOS),sequential order statistics(SOS),progressive type II censored order statistics(POS),record values,kth record values and Pfeifers records.The concept of generalized order statistics enables a common approach to structural similarities and analogies.Known results in submodels can be subsumed,generalized,and integrated within a general framework.Moreover the distributional and inferential properties of or-dinary order statistics turn out to remain valid for generalized order statistics(cf.Kamps(1995a,b),Cramer and Kamps(2001)).Thus,the concept of GOS provides a large class of models with many interesting and useful properties for both the description and the analysis of practical problems.Due to this reason,the question arises whether the general distribution theory of GOS as well as their properties can be obtained by analogy with that for OOS.The latter has been extensively investi-gated in the literature,e.g.,see,David(1981),Arnold et al.(1992),Balakrishnan and Rao(1998a,b).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 random variables(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 since values other than the extremes may become meaningless in certain situations,for example,a spacecraft may be destroyed by the first failure of essential components.Both for mathematical purposes and for widening the field of prospective ap-plications,a number of extreme value models have been developed in which the assumptions of classical models are violated.The primary objective of the current study is to focus on one of the most developed important extreme value models in which the extremes are based on a random sample with random size.Actually,in many applications of extremes,the sample size itself is frequently a rv.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 process-es,damage models or rarefaction of point processes and records as maxima,while their introduction in an applied model permits the user to select samples of varying sizes on different occasions(Galambos(1978,1987)).In the first group of examples,the random sample size is generated by the problem itself,hence the statistician has no control over the dependence between the sample size and the underlying rv's.On the other hand,if one introduces 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.Most statistical methods assume that you have a sample of observations,all of which come from the same distribution,and that you are interested in modeling that one distribution.If you actually have data from more than one distribution with no information to identify which observation goes with which distribution,standard models wont help you.However,finite mixture models(FMMs)might come to the rescue.They use a mixture of parametric distributions to model data,estimating both the parameters for the separate distributions and the probabilities of component membership for each observation.The main aim of this thesis divides into two parts:the first one is to investi-gate the asymptotic behavior of some important functions of GOS when the sample size is assumed to be a positive integer-valued rv.The second part is to study the asymptotic distribution of the linear normalized maximum under multivariate finite mixture models from independent but non-identical distributed random vectors.This thesis consists of five chapters,the first of them is an introductory chapter,containing historical study.It is worth to mention that the material of the second,third,fourth and the fifth chapters of the thesis were prepared as the four separated papers see Alawady et al.(2016a,b,c)and Hu et al.(2016).Chapter one:In this Chapter,we simply give an elementary introduction to ordered random variables and GOS,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 thesis.The asymptotic theory of extreme OOS,GOS and its dual is discussed.The asymptotic behavior of functions of OOS,e.g.,quasi-range,quasi-midrange,extremal quotient and extremal product is presented.An introduction to extremes with random indices and the limit theory of extreme of independent and non-identical distributed random variable are given.Finally,we study the rela-tionship between the normalizing constants and the sample size for order statistics,GOS and its dual.Chapter two:In this chapter,the limit distribution functions are obtained for the extremal ratio and product with random indices under nonrandom normaliza-tion based on GOS and DGOS.Moreover,this chapter considers the conditions under which the cases of random and nonrandom indices give the same asymptotic results.Some illustrative examples are obtained,which lend further support to our theoretical results.Chapter three:In this chapter,the limit distribution functions are obtained for the quasi-range and quasi-midrange based on generalized order statistics and its dual with random sample size.Some illustrative examples for the most important distribution functions are obtained,which lend further support to our theoretical results.Chapter four:In this chapter,we study the asymptotic distribution of the linear normalized maximum and minimum under finite mixture models from independent and non-identical random variables.Two cases in this study are considered:the first case,we obtain sufficient conditions for this weak convergence,as well as the limit forms.The second case gives sufficient conditions for this convergence when the components of the mixture have different linear normalization.Illustrative ex-amples are given.Chapter five:In this chapter,we study the asymptotic distribution of the linear normalized maximum under multivariate finite mixture models from independent but non-identical distributed random vectors.We obtain sufficient conditions for this weak convergence,as well as the limit forms.Sufficient conditions for this con-vergence when the components of the mixture have different linear normalization have also been derived.Illustrative examples are provided,which lend further sup-port to our theoretical results. |
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