Multivariate ageing and dependence properties of generalized order statistics and related stochastic models [Elektronische Ressource] / vorgelegt von Ute Seehafer

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• Mathematics

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2008
• Nombre de lectures: 22
• Langue: English

Extrait

Multivariate Ageing and Dependence Properties
of Generalized Order Statistics
and Related Stochastic Models
Von der Fakult¨at fur¨ Mathematik, Informatik und
Naturwissenschaften der Rheinisch-Westf¨alischen Technischen
Hochschule Aachen zur Erlangung des akademischen Grades
eines Doktors der Naturwissenschaften genehmigte Dissertation
vorgelegt von
Diplom-Mathematikerin
Ute Seehafer
aus Potsdam
Berichter: Universit¨atsprofessor Dr. Erhard Cramer
Universit¨atsprofessor Dr. Udo Kamps
Tag der mundlic¨ hen Prufung¨ : 17. Juli 2008
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar.¨ii
Acknowledgement
First of all, I would like to thank my supervisor Professor Erhard Cramer for
giving me the opportunity to work in this interesting field, for constructive
criticism and for constant encouragement. Furthermore, I am grateful to
Professor Udo Kamps for accepting to be referee of this thesis.
I also thank all my former collegues for their kindness, their support and for
an enjoyable time together. Special thanks go to my parents who promoted
me and stood by me all my life.
The gratefulness I owe to Bj¨orn Lenz and our daughter Olivia can not be
expressed in words. Thank you for your support, your confidence and your
love.Contents
1 Introduction 1
2 Ageing and dependence concepts 9
2.1 Stochastic orders . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Some univariate stochastic orders . . . . . . . . . . . . . 9
2.1.2 Some multivariate stochastic orders . . . . . . . . . . . 11
2.2 Univariate ageing properties . . . . . . . . . . . . . . . . . . . . 13
2.3 The NBU(2) property for discrete distributions . . . . . . . . . 18
2.4 Multivariate ageing properties . . . . . . . . . . . . . . . . . . . 23
2.4.1 Multivariate IFR . . . . . . . . . . . . . . . . . . . . . . 23
2.4.2 Multivariate IFRA . . . . . . . . . . . . . . . . . . . . . 26
2.4.3 Multivariate NBU . . . . . . . . . . . . . . . . . . . . . 28
2.5 Dependence notions . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Some Stochastic models 33
3.1 Generalized order statistics . . . . . . . . . . . . . . . . . . . . 33
3.2 Mixture models . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Order statistics and record values from exchangeable sequences 39
3.4 Order statistics from INID progressive censoring . . . . . . . . 40
4 Ageing properties 47
4.1 Results for ordinary order statistics . . . . . . . . . . . . . . . . 47
4.2 Closure properties of generalized order statistics with respect
to the IFR, IFRA and NBU classes . . . . . . . . . . . . . . . . 50
4.2.1 Univariate properties . . . . . . . . . . . . . . . . . . . . 50
4.2.2 Multivariate properties. . . . . . . . . . . . . . . . . . . 51
4.3 Closure properties of generalized order statistics with respect
to the NBU(2) and NBUC classes . . . . . . . . . . . . . . . . . 63
iiiiv CONTENTS
4.3.1 NBU(2) and NBUC property of the first generalized or-
der statistic . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.2 NBU(2) property of the second generalized order statistic 65
4.4 Characterizations of ageing classes . . . . . . . . . . . . . . . . 67
4.5 Preservation of ageing properties under mixtures . . . . . . . . 71
4.6 Ageing properties of order statistics and record values from
exchangeable sequences . . . . . . . . . . . . . . . . . . . . . . 73
5 Dependence Properties 77
5.1 Dependence structure of generalized order statistics. . . . . . . 77
5.2 Dependence str of mixture models . . . . . . . . . . . . . 78
5.2.1 PUOD and association for mixture models. . . . . . . . 79
5.2.2 RCSI and MTP for mixture models . . . . . . . . . . . 842
5.2.3 Mixture models with conditional independence . . . . . 95
5.2.4 Anextensionoftheproportionalhazardsregressionmodel 99
5.3 Dependenceoforderstatisticsandrecordvaluesfromexchange-
able sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4 Dependence of progressively censored order statistics . . . . . . 103
A Conditional distributions 115
Bibliography 119¤
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Chapter 1
Introduction
Life times and failure times of individuals or technical units are considered in
many areas, e.g. in the analysis of product or system reliability (cf. Meeker
and Escobar [1998]). In the stochastic modeling, such life or failure times are
represented by random variables resp. their distributions. Ageing properties
are an important component in the analysis of life time distributions.
Positiveageingdenotestheadverseeffectofageonlifetimes. Itmaybecaused
by wearout or fatigue. A positive ageing class contains life time distributions
that show a characteristic behavior of positive ageing. Ageing properties are
oftenexpressedintermsoftheresiduallifetime(cf. p.14)ofunitsofdifferent
ages. In particular, a probability distribution may belong to a certain posi-
tive ageing class if the (random) residual life time of the corresponding unit
decreases with increasing age. Thus, it is necessary to compare probability
distributions. For this reason, stochastic orders are an important tool for the
analysis of ageing concepts. Standard monographs dealing with this subject
are Mul¨ ler and Stoyan [2002] and Shaked and Shanthikumar [2007].
The best studied (univariate) ageing class is the IFR (increasing failure rate)
¯class. A life time distribution with survival functionF belongs to this class if
the residual life time of a unit of less age dominates the residual life time of a
unit of greater age with respect to the stochastic order or equivalently ifst
¯ ¯ ¯ ¯F x t F s F x s F t for 0 s t, x 0
(cf. p. 9 and p. 14). Distributions with a Lebesgue density belong to the IFR
class if and only if their hazard rates (cf. p. 14) are increasing (cf. Barlow
and Proschan [1978]). Given that a unit with an IFR life time has survived
until time t 0, the probability that it fails in the next h 0 units of time
12 CHAPTER 1. INTRODUCTION
increasesint. LifetimeswithIFRdistributionsoccurinmanysituations. Ex-
amples are failure times of diverse mechanical units and life times of humans
after some initial period (cf. Barlow and Proschan [1978] and Crowder et al.
[1991]).
Besides the IFR class, there is a variety of different (positive) ageing classes
introduced in the literature. In particular, closure properties of these classes
with respect to the formation of coherent systems, convolutions and mixtures
are considered. For details, we refer, e.g., to Barlow and Proschan [1978].
It is worth mentioning that the IFR class is not closed under the formation
of coherent systems with independent components. The smallest (univariate)
class that contains the exponential distributions and that is closed under the
formation of coherent systems with independent components and under limits
in distribution is the IFRA (increasing failure rate average) class (cf. Barlow
and Proschan [1978]). Notice that exponential distributions are the only dis-
tributions with a constant hazard rate. Therefore, they play an exceptional
role. For example, exponential distributions belong to the IFR class and to
the DFR (decreasing failure rate) class. The latter is a negative ageing class.
Negative ageing is the dual concept to positive ageing and denotes the bene-
ficial effect of age on random life times.
The treatment of ageing concepts is predominantly restricted to univariate
properties. Nevertheless, there were different approaches to find suitable mul-
tivariateversionsofexistingunivariateclasses(cf.,e.g.,Marshall[1975],Block
and Savits [1980], Savits [1985] or Marshall and Shaked [1986]). We summa-
rize some of these approaches in Section 2.4.
Besidesthebehaviorofageing,thestructureofdependenceisofinterestinthe
analysis of vectors of random life times. In many situations where several life
timesareconsideredsimultaneously, itisnotrealistictoassumeindependence
of the life times. Dependence arises since the considered units are influenced
by similar environmental conditions and loads. We may think of clinical tri-
als where life times of family members are analyzed or of technical systems
consisting of several components (cf. Barlow and Proschan [1978] and Meeker
and Escobar [1998]).
Positivedependencedescribesthetendencyofcomponentsofarandomvector
to assume concordant values. There is a variety of positive dependence no-
tions that measure this tendency, e.g. association or the MTP (multivariate2
totally positive of order two) property. For detailed information on MTP2
functions, we refer to Karlin and Rinott [1980].
As for ageing properties, dual concepts of negative dependence exist for most
of the positive dependence notions. Throughout this thesis, we concentrate
on positive ageing and positive dependence. In particular, several stochasticP
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