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A variety of aging concepts have been defined, studied, and applied inreliability, in areas of statistics and operations research such as sequentialtesting, replacement and maintenance models, and probability inequalitytheory. (See, e. g., Bryson and Siddiqui (1969), Block et al. (1976), Whitt(1980), Muth (1980), Barlow and Proschan (1981), Klefsjo (1982), Shaked(1983), Loh(1984), Deshpand et al. (1986), Hollander et al. (1986),Abouammoh( 1988), Abouammoh and Ahmed (1988), Ahmed (1990),Deshpande et al. (1990), Abouammoh and Ahmed (1991,1992), Cao andwang (1991), Abouammoh et al. (1993), Shaked and Shantihkumar (1994),Ahmed (2000), Li et al. (2000), Ahmed (2001), Belzunce et al. (2001),Bassan et al. (2002), Finkelstein (2002), Ahmed et al. (2003), Bradley andGupta (2003), and Mugdai and Ahmed (2004).In reliability theory, various concepts of aging and wear have beenproposed to study lifetime of systems, or components, in terms of theconditional distribution of lifetimes, failure rate, and renewal failure rates.Examples of these classes are:(i) Increasing failure rate (IFR),(ii) increasing failure rate average (IFRA),(iii) new better than used failure rate (NBUFR),(iv) new better than average failure rate (NBAFR),2or through the conditional mean remaining life as :(i) decreasing mean remaining life (DMRL),(ii) decreasing mean remainig life average (DMRLA),(iii) decreasing harmonic mean remainig life average (DHMRLA),(iv) new better than used in expectation (NBUE),(v) new better than average mean remaining life (NBAMRL), and(vi) harmonic new better than used in expectation (HNBUE).The corresponding dual classes are :DFR, DFRA, NWUFR, NWAFR, NWU, IMRL, IMRLA, IHMRLA,NWUE, NWAMRL, and HNWUE, respectively. Note that D, I, W standfor the decreasing, increasing and worse, respectively.Recently, several authors have introduced new classes of life distributions.Such as the classes NBU(2), IFR(2), DMRLHA, NBUC and DMRLclasses.Let F be the cumulative distribution function of X.−F(x) = 1 – F(x) iscalled its survival function.[ ]( )( ) ( )F tF x P X x t X t F x t t+= > + > =is the survival function at age t, i.e., the conditional probability that a unitof age t (i.e., a unit having already survived up to time t ) will survive for3an additional x units of time. Obviously, any study of the phenomenon ofaging has to be based on Ft (x) and functions related to it.In the context of reliability, “ no aging ” is equivalent to the phenomenonthat age has no effect on the residual survival function of a unit; i.e.,F (x | t) = F (x) , for all t, x > 0.The last equation is satisfied only by exponential survivalfunction, F (x) = e−λx , x > 0, λ > 0, among continuous survival functions. Noaging has thus been equivalently described as:(i) Constant failure rate,(ii) constant mean residual life, or(iii) exponential renewal distribution.Stochastic orders and inequalities are being used at an accelerated rate inmany diverse areas of probability and statistics. In the literature severalconcepts of stochastic orderings between random variables have beenconsidered. They are useful in modeling for reliability and economicsapplications and as a mathematical tool to prove important results inapplied probability (see, e.g., Barlow and Proschan (1981), Alzaid (1988),Ahmed (1988), Alzaid et al. (1991), Fagiuoli and Pellerey (1993), Shakedand Shanthikumar (1994), Kebir (1994), Shaked and Wong (1997),Belzunce et al. (1997), Belzunce et al. (1999), and Ahmed et al. (2004),among others)4If A denotes some aging property, a general procedure to give definitionand characterization of A (as has been pointed out in Pellerey and Shaked(1997)) is by means of stochastic orders of residual lifetimes of the form,t st ord t X A X X ′−∈ ⇔ ≥ whenever t < t′, t,t′∈(0,∞)andst ord tX A X X−∈ ⇔ ≥ whenever t∈(0,∞) ,wherest−ord≥ denotes some stochastic order and X ∈ A denotes that X has theaging property A.Other results on characterizations of this kind can be found in Deshpand etal. (1986), Belzunce et al. (1997), and Pellerey and Shaked (1997).Other characterizations and definitions are of the formX A Y Xst−ord∈ ⇔ ≥where Y is an exponential random variable with mean E(X).The comparison of the exponential random variable Y with a randomvariable X, is an indication of an aging property associated with X.In Chapter 2 of this dissertation we present definitions and basic factsregarding failure rate, the conditional survival function and the remaininglife. We present additional theoretical consequences of theses notions toprobability and statistical theory by considering particular types ofordering.5In Section 2.2, stochastic orders including the usual ordering, dispersiveordering, super-additive ordering, convex ordering, hazard ordering,likelihood ordering, mean residual life ordering are presented. Thevariability ordering and peakedness ordering are considered in Section 2.3.Monotone convex (concave) orderings are introduced in Section 2.4 andinter-relationships among these orderings are displayed.Most of the well-known families of life distribution defined andcharacterized in terms of different concepts of stochastic dominance aregiven in Section 2.5.Chapter 3 contains stationary renewal processes based on i.i.d randomvariables. Several aging notions arising from observing renewal processesare studied.We derive useful closure properties for these processes under commonlyoccurring operations in reliability (such as convolution, mixture andformation of coherent structures).In Section 3.2, we develop the new renewal better than used (NRBU) andthe renewal new better than used (RNBU) classes of life distributions.Inter-connections between these classes and the RIFRA and DMRLHAclasses are established.Section 3.3 presents preservation properties of these classes. In particular,we show that both the RIFRA and RNBU classes are closed underformation of coherent structures (see Theorem 3.2). We also prove that theRIFR class is closed under convolution (see Theorem 3.3 and Theorem3.4).6In Chapter 4, new life distributions based on tail properties and concepts ofasymptotic decay are introduced and their properties exploited.In Section 4.2, we define the newly introduced UBA, UBAE, HUBAE,GHUBAE, the HUBAET and the UBALT classes of life distributions. Ineach one of the introduced families the life of a used unit is compared withthat of a unit of a specific age.The preservation under convolutions and mixtures of the HUBAE and theGHUBAE classes are studied in Section 4.3. It should be emphasized thatthe development of this section are merely a sample of potentialdevelopments possible. Separate papers are in process more fullyexploiting properties of other classes introduced in Chapter 4.Chapter 5 contains new classes of life distribution at a specific age t0. Inparticular the NBU ∗ t0 is introduced.In Section 5.2 it is shown that the new class is preserved under convolution,coherent systems and shoch models. We emphasize that we have been ableto give a new and much shorter proof to the interesting result of Li and Li(1998). Stochastic comparisons of excess lifetime of renewal processes arestudied in Section 5.3. In particular, we show the γ (t), the time of the nextevent after time t is a renewal process with an 0 NBU ∗ t underlyingdistribution is decreasing.In Section 5.4 0 IFR ∗ t is studied and its relationships with 0 NBU ∗ t and0 DMRL ∗ t , are established.7Finally, in Chapter 6, we point out several open points for future research.These open points are explained, and suggestions for directions of futureresearch are introduced.
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