Stochastic Comparisons of Lifetimes of Two Series and Parallel Systems with Location-Scale Family Distributed Components having Archimedean Copulas
aa r X i v : . [ m a t h . S T ] O c t Stochastic Comparisons of Lifetimes of Two Seriesand Parallel Systems with Location-Scale FamilyDistributed Components having ArchimedeanCopulas
Amarjit KunduDepartment of MathematicsSantipur CollegeWest Bengal, IndiaShovan Chowdhury ∗ Quantitative Methods and Operations Management AreaIndian Institute of Management, KozhikodeKerala, India.October 3, 2017
Abstract-
In this paper, we compare the lifetimes of two series and two parallel systemsstochastically where the lifetime of each component follows location-scale ( LS ) family of dis-tributions. The comparison is carried out under two scenarios: one, that the components ofthe systems have a dependent structure sharing Archimedean copula and two, that the compo-nents are independently distributed. It is shown that the systems with components in series orparallel sharing Archimedean copula with more dispersion in the location or scale parametersresults in better performance in the sense of the usual stochastic order. It is also shown that ifthe components are independently distributed, it is possible to obtain more generalized resultsas compared to the dependent set-up. The results in this paper generalizes similar results inboth independent and dependent set up for exponential and Weibull distributed components. Keywords
Location-scale family of distributions; Series system; Parallel system; Archimedeancopula; stochastic order; likelihood ratio order; ageing faster order; multiple outlier model;majorization ∗ Corresponding author e-mail: [email protected]; [email protected] Introduction
Stochastic comparison of system lifetimes has always been a relevant topic in reliabilityoptimization and life testing experiments. These comparisons can be used to choose the bestsystem structure under different criteria or to study where to place the different components ina system structure. If X n ≤ X n ≤ . . . ≤ X n : n denote the order statistics corresponding to therandom variables X , X , . . . , X n , then the lifetime of a series and parallel system correspondto the smallest ( X n ), and the largest ( X n : n ) order statistic respectively. Classical theory ofsystems assumes that the lifetimes of the components are iid (see David and Nagaraja [1]).Considerable amount of work has also been carried out in the past years in comparing thelifetimes of heterogeneous independent components of systems (largely on the smallest and thelargest order statistics) with certain underlying distributions on both finite and infinite rangewith respect to usual stochastic ordering, hazard rate ordering, reversed hazard rate orderingand likelihood ratio ordering. One may refer to Dykstra et al. [2], Zhao and Balakrishnan ([3]),Balakrishnan et al. [4], Torrado and Kochar [5], Torrado [6], Fang and Balakrishnan [7], Kundu et al. [8], Kundu and Chowdhury ([9],[11]), Chowdhury and Kundu [12] and Hazra et al. [13]for more detail. There are few work in the same area where the authors have compared systemsstochastically through relative ageing, also known as ageing faster ordering in terms of hazardrate or reversed hazard rate ordering. One may refer to Sengupta and Deshpande [14], Rezaei et al. [15] and Li and Li [16] for more detail.However, in practical situations, the components of a system may have a structural depen-dence which result in a set of statistically dependent observations. The dependence structure ofthe components are investigated by researchers very recently with the help of copulas. Navarroand Spizzichino [17] studied stochastic orders of series and parallel systems with componentssharing a common copula. Rezapour and Alamatsaz [18] investigated stochastic orders on or-der statistics from samples with different survival Archimedean copulas. Li and Li [19] studiedstochastic ordering of the sample minimums of Weibull samples sharing a common Archimedeansurvival copula. Li and Fang [20] compared the lifetimes of parallel systems with proportionalhazard rate (PHR) components following the Archimedean copula which was further investi-gated by Li et al. [21] and Fang et al. [22].The location-scale family of distributions is commonly used in lifetime studies. The mostwidely used statistical distributions are either members of this class or closely related to thisclass of distributions; such as exponential, normal, Weibull, lognormal, loglogistic, logistic, andextreme value distributions. Methods of inference and statistical theory for the general familycan be applied to this large, important class of models. A random variable X is said to fol-low location-scale family distribution, written as LS ( λ, σ, F ) and will be termed as LS familyhereafter, if the distribution function of X is given by F ( x ; λ, σ, F ) = F (cid:18) x − λσ (cid:19) , x > λ, σ > x, λ, σ ∈ R , (1)2here λ and σ are the location and the scale parameter respectively and F ( · ) is the baselinedistribution function of the rv X . Although significant previous research has compared seriesor parallel systems of heterogenous independent components including scale family of distribu-tions (see Khaledi et al. [23], Li et al. [21], Kochar and Torrado [24] and Li and Li [16]), therehas been few work examining similar comparisons for dependent components; furthermore, allsuch comparisons for dependent components assume either PHR components or Weibull dis-tributed components. As LS family of distributions cover a large pool of lifetime distributions,we are motivated to assume the component lifetimes to follow LS family and to compare thelifetimes of two series or parallel systems stochastically, assuming a dependence structure inthe components. In this sense, the paper distinguishes itself from the other few existing work.It generalizes the results on stochastic comparison of lifetimes of two series or parallel systemswith heterogeneous dependent and independent components. Moreover, the comparisons arecarried out under different baseline distributions of LS family with stochastic ordering, hazardrate orderings and reversed hazard rate orderings between them. The rest of the paper is orga-nized as follows. In Section 2, we have given the required definitions and some useful lemmaswhich are used throughout the paper. Results related to stochastic comparison of series systemswith heterogeneous independent and dependent components are derived in Section 3. Section 4discusses some results on parallel systems with dependent components. For two absolutely continuous random variables X and Y with distribution functions F ( · ) and G ( · ), survival functions F ( · ) and G ( · ), density functions f ( · ) and g ( · ), hazard ratefunctions r ( · ) and s ( · ) and reversed hazard rate functions ˜ r ( · ) and ˜ s ( · ) respectively, X is saidto be smaller than Y in i ) likelihood ratio order (denoted as X ≤ lr Y ), if, for all t , g ( t ) f ( t ) increasesin t , ii ) hazard rate order (denoted as X ≤ hr Y ), if, for all t , G ( t ) F ( t ) increases in t or equivalently r ( t ) ≥ s ( t ), iii ) reversed hazard rate order (denoted as X ≤ rh Y ), if, for all t , G ( t ) F ( t ) increases in t or equivalently ˜ r ( t ) ≤ ˜ s ( t ), iii ) Ageing faster order in terms of the hazard rate order (denotedas X ≤ R − hr Y ), if, for all t , r X ( t ) r Y ( t ) increases in t , and iv ) usual stochastic order (denoted as X ≤ st Y ), if F ( t ) ≥ G ( t ) for all t . In the following diagram we present a chain of implicationsof the stochastic orders. For more on stochastic orders, see Shaked and Shanthikumar [27]. X ≤ hr Y ↑ ց X ≤ lr Y → X ≤ st Y. ↓ ր X ≤ rh Y The notion of majorization (Marshall et al. [25]) is essential for the understanding of the3tochastic inequalities for comparing order statistics. Let R n be an n -dimensional Euclideanspace. Further, for any two real vectors x = ( x , x , . . . , x n ) ∈ R n and y = ( y , y , . . . , y n ) ∈ R n ,write x (1) ≤ x (2) ≤ · · · ≤ x ( n ) and y (1) ≤ y (2) ≤ · · · ≤ y ( n ) as the increasing arrangements ofthe components of the vectors x and y respectively. The following definitions may be found inMarshall et al. [25]. Definition 1 i) The vector x is said to majorize the vector y (written as x m (cid:23) y ) if j X i =1 x ( i ) ≤ j X i =1 y ( i ) , j = 1 , , . . . , n − , and n X i =1 x ( i ) = n X i =1 y ( i ) . ii) The vector x is said to weakly supermajorize the vector y (written as x w (cid:23) y ) if j X i =1 x ( i ) ≤ j X i =1 y ( i ) for j = 1 , , . . . , n. iii) The vector x is said to weakly submajorize the vector y (written as x (cid:23) w y ) if n X i = j x ( i ) ≥ n X i = j y ( i ) for j = 1 , , . . . , n. (iv) The vector x is said to be p -larger than the vector y (written as x p (cid:23) y ) if j Y i =1 x ( i ) ≤ j Y i =1 y ( i ) for j = 1 , , . . . , n. (v) The vector x is said to reciprocally majorize the vector y (written as x rm (cid:23) y ) if j X i =1 x ( i ) ≥ j X i =1 y ( i ) for j = 1 , , . . . , n. It is not so difficult to show that x m (cid:23) y ⇒ x w (cid:23) y ⇒ x p (cid:23) y ⇒ x rm (cid:23) y . Definition 2
A function ψ : R n → R is said to be Schur-convex (resp. Schur-concave) on R n if x m (cid:23) y implies ψ ( x ) ≥ ( resp. ≤ ) ψ ( y ) f or all x , y ∈ R n . Definition 3
For any integer r , a function ψ : R → R is said to be r-convex (resp. r-concave)on R if d r ψ ( x ) dx r ≥ ( ≤ )0 f or all x ∈ R . Notation 1
Let us introduce the following notations.(i) D + = { ( x , x , . . . , x n ) : x ≥ x ≥ . . . ≥ x n > } . ii) E + = { ( x , x , . . . , x n ) : 0 < x ≤ x ≤ . . . ≤ x n } . Let us first introduce the following lemmas which will be used in the next sections to prove theresults.
Lemma 1 ( Lemma 3.1 of Kundu et al. [8] ) Let ϕ : D + → R be a function, continuously dif-ferentiable on the interior of D + . Then, for x , y ∈ D + , x m (cid:23) y implies ϕ ( x ) ≥ ( resp. ≤ ) ϕ ( y ) if, and only if, ϕ ( k ) ( z ) is decreasing (resp. increasing) in k = 1 , , . . . , n, where ϕ ( k ) ( z ) = ∂ϕ ( z ) /∂z k denotes the partial derivative of ϕ with respect to its k th argument. Lemma 2 ( Lemma 3.3 of Kundu et al. [8] ) Let ϕ : E + → R be a function, continuously dif-ferentiable on the interior of E + . Then, for x , y ∈ E + , x m (cid:23) y implies ϕ ( x ) ≥ ( resp. ≤ ) ϕ ( y ) if, and only if, ϕ ( k ) ( z ) is increasing (resp. decreasing) in k = 1 , , . . . , n, where ϕ ( k ) ( z ) = ∂ϕ ( z ) /∂z k denotes the partial derivative of ϕ with respect to its k th argument. Lemma 3 ( Lemma 3.1 of Khaledi and Kochar [ ]) Let S ⊆ R n + . Further, let ψ : S → R be afunction. Then, for x , y ∈ S , x p (cid:23) y implies ψ ( x ) ≥ ( resp. ≤ ) ψ ( y ) if, and only if,(i) ψ ( e a , . . . , e a n ) is Schur-convex (resp. Schur-concave) in ( a , . . . , a n ) ∈ S ,(ii) ψ ( e a , . . . , e a n ) is decreasing (resp. increasing) in a i , for i = 1 , . . . , n, where a i = ln x i , for i = 1 , . . . , n. Lemma 4 ( Lemma 4.1 of Hazra et al. [13] ) Let S ⊆ R n + . Further, let ψ : S → R be afunction. Then, for x , y ∈ S , x rm (cid:23) y implies ψ ( x ) ≥ ( resp. ≤ ) ψ ( y ) if, and only if, i) ψ ( a , . . . , a n ) is Schur-convex (resp. Schur-concave) in ( a , . . . , a n ) ∈ S ,(ii) ψ ( a , . . . , a n ) is increasing (resp. decreasing) in a i , for i = 1 , . . . , n, where a i = x i , for i = 1 , . . . , n. Lemma 5 ( Theorem A.8 of Marshall et al. [25] p.p.
Let S ⊆ R n . Further, let ϕ : S → R be a function. Then for x , y ∈ S , x (cid:23) w y = ⇒ ϕ ( x ) ≥ ( resp. ≤ ) ϕ ( y ) if, and if, ϕ is both increasing (resp. decreasing) and Schur-convex (resp. Schur-concave) on S . Similarly, x w (cid:23) y = ⇒ ϕ ( x ) ≥ ( resp. ≤ ) ϕ ( y ) if, and if, ϕ is both decreasing (resp. increasing) and Schur-convex (resp. Schur-concave) on S . Now, let us recall that a copula associated with a multivariate distribution function F is a function C : [0 , n [0 ,
1] satisfying: F ( x ) = C ( F ( X ) , ..., F n ( X n )) , where the F i ’s,1 ≤ i ≤ n are the univariate marginal distribution functions of X i s. Similarly, a survival copulaassociated with a multivariate survival function F is a function C : [0 , n [0 ,
1] satisfying: F ( x ) = P ( X > x , ..., X n > x n ) = C (cid:0) F ( x ) , ..., F n ( x n ) (cid:1) , where, for 1 ≤ i ≤ n , F i ( · ) = 1 − F i ( · ) are the univariate survival functions. In particular, acopula C is Archimedean if there exists a generator ψ : [0 , ∞ ] [0 ,
1] such that C ( u ) = ψ (cid:0) ψ − ( u ) , ..., ψ − ( u d ) (cid:1) . For C to be Archimedean copula, it is sufficient and necessary that ψ satisfies i ) ψ (0) = 1and ψ ( ∞ ) = 0 and ii ) ψ is d − monotone, i.e. ( − k d k ψ ( s ) ds k ≥ k ∈ { , , ..., d − } and ( − d − d d − ψ ( s ) ds d − is decreasing and convex. Archimedean copulas cover a wide range of depen-dence structures including the independence copula and the Clayton copula. For more de-tail on Archimedean copula, see, Nelsen [28] and McNeil and Nˇ e slehov´ a [29]. In this paper,Archimedean copula is specifically employed to model on the dependence structure among ran-dom variables in a sample. The following important lemma is used in the next sections to provesome of the important theorems. Lemma 6 ( Li and Fang [20] ) For two n-dimensional Archimedean copulas C ψ ( u ) and C ψ ( u ) ,with φ = ψ − = sup { x ∈ R : ψ ( x ) > u } , the right continuous inverse, if φ ◦ ψ is super-additive, then C ψ ( u ) ≤ C ψ ( u ) for all u ∈ [0 , n . Recall that a function f is said to besuper-additive if f ( x + y ) ≥ f ( x ) + f ( y ) , for all x and y in the domain of f . Comparison of Series Systems with LS Distributed Compo-nents
This section is devoted to the comparison of two series systems with heterogenous LS family distributed components. The comparison is carried out under two scenarios: one, thatthe components have a dependent structure sharing Archimedean copulas and the other is thatthe components are independently distributed. Some Results on Heterogenous Dependent Components
Let, X and Y be two random variables having distribution functions F ( · ) and G ( · ) respec-tively. Also suppose that X i ∼ LS ( λ i , σ i , F ) and Y i ∼ LS ( µ i , ξ i , G ) ( i = 1 , , . . . , n ) be two setsof n dependent random variables with Archimedean copulas having generators ψ (cid:0) φ = ψ − (cid:1) and ψ (cid:0) φ = ψ − (cid:1) respectively. Also suppose that G n ( · ) and H n ( · ) be the survival func-tions of X n and Y n respectively. Then, G n ( t ) = ψ " n X k =1 φ (cid:26) − F (cid:18) t − λ k σ k (cid:19)(cid:27) , t > max ( λ k , ∀ k ) , and H n ( t ) = ψ " n X k =1 φ (cid:26) − G (cid:18) t − µ k ξ k (cid:19)(cid:27) , t > max ( µ k , ∀ k ) . Let r X ( u ) and r Y ( u ) are the hazard rate functions of the random variables X and Y respectively. The first two theorems show that usual stochastic ordering exists between X n and Y n under weak majorization order of the scale parameters and stochastic ordering between X and Y . Theorem 1
Let X , X , ..., X n be a set of dependent random variables sharing Archimedeancopula having generator ψ such that X i ∼ LS ( λ i , σ i , F ) , i = 1 , , ..., n . Let Y , Y , ..., Y n beanother set of dependent random variables sharing Archimedean copula having generator ψ such that Y i ∼ LS ( λ i , ξ i , G ) , i = 1 , , ..., n . Assume that σ , ξ and λ ∈ D + (or E + ). Furthersuppose that φ ◦ ψ is super-additive, ψ or ψ is log-concave and X ≤ st Y . If either r X ( u ) or r Y ( u ) is decreasing in u , then σ w (cid:22) ξ implies X n ≤ st Y n . Proof:
By Lemma 6, super-additivity of φ ◦ ψ implies that ψ " n X k =1 φ (cid:26) − F (cid:18) t − λ k σ k (cid:19)(cid:27) ≤ ψ " n X k =1 φ (cid:26) − F (cid:18) t − λ k σ k (cid:19)(cid:27) . (1)As X ≤ st Y and φ and ψ are decreasing in x , it can be easily shown that ψ " n X k =1 φ (cid:26) − F (cid:18) t − λ k σ k (cid:19)(cid:27) ≤ ψ " n X k =1 φ (cid:26) − G (cid:18) t − λ k σ k (cid:19)(cid:27) . (2)71) and (2) together implies that ψ " n X k =1 φ (cid:26) − F (cid:18) t − λ k σ k (cid:19)(cid:27) ≤ ψ " n X k =1 φ (cid:26) − G (cid:18) t − λ k σ k (cid:19)(cid:27) . Therefore, to prove the result it suffices to prove that ψ " n X k =1 φ (cid:26) − G (cid:18) t − λ k σ k (cid:19)(cid:27) ≤ ψ " n X k =1 φ (cid:26) − G (cid:18) t − λ k ξ k (cid:19)(cid:27) . Let us assume that ψ " n X k =1 φ (cid:26) − G (cid:18) t − λ k σ k (cid:19)(cid:27) = ψ " n X k =1 φ { − G ( p k ( t − λ k )) } = Ψ( p ) , where p = ( p , p , ..., p n ) = (cid:16) σ , σ , ..., σ n (cid:17) . Hence, by Lemma 5, to prove the result, it sufficesto prove that Ψ( p ) is decreasing and s-convex in p . Now, if σ , λ ∈ D + ( or E + ) and r Y ( u ) isdecreasing in u , then, for all i ≤ j , r Y ( p i ( t − λ i )) ≥ ( ≤ ) r Y ( p j ( t − λ j )) . Again, as ψ is log-concave, giving that ψ ( u ) ψ ′ ( u ) is increasing in u, then, for all i ≤ j , it can bewritten that − r Y ( p i ( t − λ i )) ψ ( u i ) ψ ′ ( u i ) ≥ ( ≤ ) − r Y ( p j ( t − λ j )) ψ ( u j ) ψ ′ ( u j ) , (3)where u i = φ [1 − G ( p i ( t − λ i ))] . Now, differentiating Ψ( p ) with respect to p i , we get ∂ Ψ ∂p i = − ψ ′ " n X k =1 φ { − G ( p k ( t − λ k )) } ( t − λ i ) r Y ( p i ( t − λ i )) ψ [ φ { − G ( p i ( t − λ i )) } ] ψ ′ [ φ { − G ( p i ( t − λ i )) } ] ≤ , proving that Ψ( p ) is decreasing in each p i . Moreover, using (3), it can be easily shown that ∂ Ψ ∂p i − ∂ Ψ ∂p j ≤ ( ≥ )0. Thus, by Lemma 2 (Lemma 1) it can be written that Ψ( p ) is s-convex in p .This proves the result. ✷ The next theorem discusses about the stochastic ordering between X n and Y n under weakmajorization order of the scale parameters when ψ or ψ is log-convex. The theorem can beproved in the similar line as of the previous one and hence the proof is omitted. Theorem 2
Let X , X , ..., X n be a set of dependent random variables sharing Archimedeancopula having generator ψ such that X i ∼ LS ( λ i , σ i , F ) , i = 1 , , ..., n . Let Y , Y , ..., Y n beanother set of dependent random variables sharing Archimedean copula having generator ψ such that Y i ∼ LS ( λ i , ξ i , G ) , i = 1 , , ..., n . Assume that σ , ξ and λ ∈ D + (or E + ). Furthersuppose that φ ◦ ψ is super-additive, ψ or ψ is log-convex and X ≤ st Y . If either r X ( u ) or r Y ( u ) is increasing in u , then σ (cid:23) w ξ implies X n ≤ st Y n .
8n the next theorem usual stochastic ordering between X n and Y n has been establishedunder p and rm orderings of the scale parameters. Theorem 3
Let X , X , ..., X n be a set of dependent random variables sharing Archimedeancopula having generator ψ such that X i ∼ LS ( λ i , σ i , F ) , i = 1 , , ..., n . Let Y , Y , ..., Y n beanother set of dependent random variables sharing Archimedean copula having generator ψ such that Y i ∼ LS ( λ i , ξ i , G ) , i = 1 , , ..., n . Assume that σ , ξ , λ ∈ D + (or E + ). Further supposethat φ ◦ ψ is super-additive, ψ or ψ is log-concave and X ≤ st Y , then,i) σ p (cid:22) ξ ⇒ X n ≤ st Y n , if ur X ( u ) or ur Y ( u ) is decreasing in u ;ii) σ rm (cid:22) ξ ⇒ X n ≤ st Y n , if u r X ( u ) or u r Y ( u ) is decreasing in u . Proof: i) As φ ◦ ψ is super-additive, by Lemma 6 it can be written that ψ " n X k =1 φ (cid:26) − F (cid:18) t − λ k σ k (cid:19)(cid:27) ≤ ψ " n X k =1 φ (cid:26) − F (cid:18) t − λ k σ k (cid:19)(cid:27) . (4)Again noticing the fact that X ≤ st Y and φ and ψ are decreasing function of x , it can beeasily shown that ψ " n X k =1 φ (cid:26) − F (cid:18) t − λ k σ k (cid:19)(cid:27) ≤ ψ " n X k =1 φ (cid:26) − G (cid:18) t − λ k σ k (cid:19)(cid:27) . (5)So, (4) and (5) together gives, ψ " n X k =1 φ (cid:26) − F (cid:18) t − λ k σ k (cid:19)(cid:27) ≤ ψ " n X k =1 φ (cid:26) − G (cid:18) t − λ k σ k (cid:19)(cid:27) . Therefore, to prove the result, it suffices to prove that ψ " n X k =1 φ (cid:26) − G (cid:18) t − λ k σ k (cid:19)(cid:27) ≤ ψ " n X k =1 φ (cid:26) − G (cid:18) t − λ k ξ k (cid:19)(cid:27) . Let us assume that ψ hP nk =1 φ n − G (cid:16) t − λ k σ k (cid:17)oi = ψ [ P nk =1 φ { − G ( e p k ( t − λ k )) } ] = Ψ( p ), where p = ( p , p , ..., p n ) = (cid:16) log σ , log σ , ..., log σ n (cid:17) . Hence, by Lemma 3, it suffices to prove that Ψ( p ) is decreasing ands-convex in p . Now, if λ , σ ∈ D + ( or E + ) and ur Y ( u ) is decreasing in u , then for all i ≤ j , e p i ( t − λ i ) r Y ( e p i ( t − λ i )) ≥ ( ≤ ) e p j ( t − λ j ) r Y ( e p j ( t − λ j )) . Again, if ψ is log-concave, then ψ ( u ) ψ ′ ( u ) is increasing in u, which implies that − e p i ( t − λ i ) r Y ( e p i ( t − λ i )) ψ ( u i ) ψ ′ ( u i ) ≥ ( ≤ ) − e p j ( t − λ j ) r Y ( e p j ( t − λ j )) ψ ( u j ) ψ ′ ( u j ) ; i ≤ j, (6)9ith u i = φ [1 − G ( e p i ( t − λ i ))] . Now, differentiating Ψ( p ) with respect to p i , we get, ∂ Ψ ∂p i = − ψ ′ " n X k =1 φ { − G ( e p k ( t − λ k )) } e p i ( t − λ i ) r Y ( e p i ( t − λ i )) ψ [ φ { − G ( e p i ( t − λ i )) } ] ψ ′ [ φ { − G ( e p i ( t − λ i )) } ] ≤ , proving that Ψ( p ) is decreasing in each p i . Moreover, using (6), it can be easily shown that ∂ Ψ ∂p i − ∂ Ψ ∂p j ≤ ( ≥ )0 . Thus, by Lemma 2 (Lemma 1), Ψ( p ) is s-convex in p . This proves the result.To prove ii), using Lemma 4, it is to prove that ψ " n X k =1 φ (cid:26) − G (cid:18) t − λ k σ k (cid:19)(cid:27) = Ψ( σ ) , is increasing in each σ i and s-convex in σ . Now, if λ , σ ∈ D + ( E + ) and u r Y ( u ) is decreasingin u , following similar argument as in the previous result, it can be shown that, for all i ≤ j , − (cid:18) t − λ i σ i (cid:19) r Y (cid:18) t − λ i σ i (cid:19) t − λ i ψ ( u i ) ψ ′ ( u i ) ≥ ( ≤ ) − (cid:18) t − λ j σ j (cid:19) r Y (cid:18) t − λ j σ j (cid:19) t − λ j ψ ( u j ) ψ ′ ( u j ) , (7)with u i = φ h − G (cid:16) t − λ i σ i (cid:17)i . Now, differentiating Ψ( σ ) with respect to σ i , we get, ∂ Ψ ∂σ i = ψ ′ " n X k =1 φ (cid:26) − G (cid:18) t − λ k σ k (cid:19)(cid:27) t − λ i σ i (cid:19) r Y (cid:18) t − λ i σ i (cid:19) t − λ i ψ h φ (cid:16) − G (cid:16) t − λ i σ i (cid:17)(cid:17)i ψ ′ h φ (cid:16) − G (cid:16) t − λ i σ i (cid:17)(cid:17)i ≥ , proving that Ψ( σ ) is increasing in each σ i . Moreover, using (7), it can be easily shown that ∂ Ψ ∂σ i − ∂ Ψ ∂σ j ≥ ( ≤ )0 . Thus, by Lemma 1 (Lemma 2), Ψ( σ ) is s-convex in σ . ✷ The next two theorems show that usual stochastic ordering exists between X n and Y n under majorization order of the location parameters. Proof of the second theorem follows fromthe first one, and hence is omitted. Theorem 4
Let X , X , ..., X n be a set of dependent random variables sharing Archimedeancopula having generator ψ such that X i ∼ LS ( λ i , σ i , F ) , i = 1 , , ..., n . Let Y , Y , ..., Y n beanother set of dependent random variables sharing Archimedean copula having generator ψ such that Y i ∼ LS ( µ i , σ i , G ) , i = 1 , , ..., n . Assume that λ , µ , σ ∈ D + (or E + ). Further supposethat φ ◦ ψ is super-additive, ψ or ψ is log-concave and either ur X ( u ) or ur Y ( u ) is decreasingin u, then, X ≤ st Y and λ (cid:22) w µ ⇒ X n ≤ st Y n . roof: Following the same argument as in (4) and (5) of Theorem 3, it can be shown that ψ " n X k =1 φ (cid:26) − F (cid:18) t − λ k σ k (cid:19)(cid:27) ≤ ψ " n X k =1 φ (cid:26) − G (cid:18) t − λ k σ k (cid:19)(cid:27) . (8)As before, to prove the result, it suffices to prove that ψ " n X k =1 φ (cid:26) − G (cid:18) t − λ k σ k (cid:19)(cid:27) ≤ ψ " n X k =1 φ (cid:26) − G (cid:18) t − µ k σ k (cid:19)(cid:27) . Using Lemma 5, it can be said that the above relation will hold ifΨ( λ ) = ψ " n X k =1 φ (cid:26) − G (cid:18) t − λ k σ k (cid:19)(cid:27) is increasing in each λ i and s-convex in λ . Now, as λ , σ ∈ D + ( or E + ), ur Y ( u ) is decreasing in u and ψ is log-concave, following the same argument as of Theorem 3, we can show that − t − λ i (cid:18) t − λ i σ i (cid:19) r Y (cid:18) t − λ i σ i (cid:19) ψ ( u i ) ψ ′ ( u i ) ≥ ( ≤ ) − t − λ j (cid:18) t − λ j σ j (cid:19) r Y (cid:18) t − λ j σ j (cid:19) ψ ( u j ) ψ ′ ( u j ) ; i ≤ j, (9)with u i = φ h − G (cid:16) t − λ i σ i (cid:17)i . Differentiating Ψ( λ ) with respect to λ i , we get ∂ Ψ ∂λ i = ψ ′ " n X k =1 φ (cid:26) − G (cid:18) t − λ i σ i (cid:19)(cid:27) t − λ i t − λ i σ i r Y (cid:18) t − λ i σ i (cid:19) ψ h φ n − G (cid:16) t − λ i σ i (cid:17)oi ψ ′ h φ n − G (cid:16) t − λ i σ i (cid:17)oi ≥ , proving that Ψ( λ ) is increasing in each λ i . Moreover, using (9), it can be easily shown that ∂ Ψ ∂λ i − ∂ Ψ ∂λ j ≥ ( ≤ )0 . Thus, by Lemma 1 (Lemma 2), Ψ( λ ) is s-convex in λ . This proves the result. ✷ Theorem 5
Let X , X , ..., X n be a set of dependent random variables sharing Archimedeancopula having generator ψ such that X i ∼ LS ( λ i , σ i , F ) , i = 1 , , ..., n . Let Y , Y , ..., Y n beanother set of dependent random variables sharing Archimedean copula having generator ψ such that Y i ∼ LS ( µ i , σ i , G ) , i = 1 , , ..., n . Assume that λ , µ , σ ∈ D + (or E + ). Further supposethat φ ◦ ψ is super-additive, ψ or ψ is log-convex and either r X ( u ) or r Y ( u ) is increasing in u, then, X ≤ st Y and λ w (cid:23) µ ⇒ X n ≤ st Y n . .2 Heterogenous independent Components
In this subsection, we will compare two series systems with heterogenous independent LS family distributed components. For i = 1 , , . . . , n , let X i and Y i be two sets of n independentrandom variables following LS distribution with parameters ( λ , σ ) and ( µ , ξ ) respectively, asgiven in (1). If F n ( · ) and G n ( · ) be the survival functions of X n and Y n respectively, thenclearly F n ( x ) = n Y k =1 (cid:20) − F (cid:18) t − λ k σ k (cid:19)(cid:21) and G n ( x ) = n Y k =1 (cid:20) − G (cid:18) t − µ k ξ k (cid:19)(cid:21) . Again, if r n ( · ) and s n ( · ) are the hazard rate functions of X n and Y n respectively then, r n ( t ) = n X k =1 σ k f (cid:16) t − λ k σ k (cid:17) − F (cid:16) t − λ k σ k (cid:17) = n X k =1 σ k r X (cid:18) t − λ k σ k (cid:19) (10)and s n ( t ) = n X k =1 ξ k g (cid:16) t − µ k ξ k (cid:17) − G (cid:16) t − µ k ξ k (cid:17) = n X k =1 ξ k r Y (cid:18) t − µ k ξ k (cid:19) , (11)In the next two theorems, stochastic comparison between minimum order statistics fromthe LS family with respect to hazard rate ordering has been discussed. These results strengthenTheorems 1 - 4 for the independent case, as hazard rate ordering implies usual stochasticordering. Theorem 6
Let X , X , ..., X n be a set of independent random variables such that X i ∼ LS ( λ i , σ i , F ) , i = 1 , , ..., n . Let Y , Y , ..., Y n be another set of independent random variables such that Y i ∼ LS ( λ i , ξ i , G ) , i = 1 , , ..., n . Suppose that σ , ξ , λ ∈ D + (or E + ). Then, if X ≤ hr Y andi) either ur X ( u ) or ur Y ( u ) is concave in u, then, σ m (cid:22) ξ ⇒ X n ≤ hr Y n ,ii) either ur X ( u ) or ur Y ( u ) is increasing (decreasing) and concave in u, then, σ w (cid:22) ( (cid:22) w ) ξ ⇒ X n ≤ hr Y n ,iii) either ur X ( u ) or ur Y ( u ) is increasing in u, and u ddu ( ur X ( u )) or u ddu ( ur Y ( u )) is decreasingin u, then, σ p (cid:22) ξ ⇒ X n ≤ hr Y n ,iv) either ur X ( u ) or ur Y ( u ) is increasing in u, and u ddu ( r X ( u )) or u ddu ( r Y ( u )) is decreasingin u, then, σ rm (cid:22) ξ ⇒ X n ≤ hr Y n . roof: If X ≤ hr Y , then r X (cid:16) t − λ k σ k (cid:17) ≥ r Y (cid:16) t − λ k σ k (cid:17) , which results in n X k =1 r X (cid:16) t − λ k σ k (cid:17) σ k ≥ n X k =1 r Y (cid:16) t − λ k σ k (cid:17) σ k . So, to prove the result it suffices to prove that P nk =1 r Y (cid:16) t − λkσk (cid:17) σ k ≥ P nk =1 r Y (cid:16) t − λkξk (cid:17) ξ k .P roof of i ): Here we have to prove that P nk =1 r Y (cid:16) t − λkσk (cid:17) σ k = P nk =1 p k r Y ( p k ( t − λ k )) = Ψ( p ),is s-concave in p . Now, if λ , σ , ξ ∈ D + ( or E + ) then p i ( t − λ i ) ≤ ( ≥ ) p j ( t − λ j ) . Thus, theconcavity of the function ur Y ( u ) gives ddu [ ur Y ( u )] u = p i ( t − λ i ) − ddu [ ur Y ( u )] u = p j ( t − λ j ) ≥ ( ≤ )0 . Now, differentiating Ψ( p ) with respect to p i , we get ∂ Ψ ∂p i = ddu [ ur Y ( u )] u = p i ( t − λ i ) , (12)proving that ∂ Ψ ∂p i − ∂ Ψ ∂p j ≥ ( ≤ )0 . Thus, by Lemma 2 (Lemma 1) Ψ( p ) is s-concave in p . Thisproves the result. P roof of ii ) : If ur Y ( u ) is increasing (decreasing) in u, then from (12) it can be written thatthat Ψ( p ) is increasing (decreasing) in p i . Again, as ur Y ( u ) is concave, by the previous resultΨ( p ) is s-concave in p . Thus, by Lemma 5, the result is proved. P roof of iii ) : Let us assume that n X k =1 r Y (cid:16) t − λ k σ k (cid:17) σ k = n X k =1 e p k r Y ( e p k ( t − λ k )) = Ψ( p ) , where p = (cid:16) log σ , log σ , ..., log σ n (cid:17) . So, by Lemma 3, it is suffices to prove that Ψ( p ) isincreasing in each p i and s-concave in p . Now, differentiating Ψ( p ) with respect to p i , we get, ∂ Ψ ∂p i = e p i [ r Y ( e p i ( t − λ i ))] + e p i ( t − λ i ) r Y ( e p i ( t − λ i ))= e p i ddu [ ur Y ( u )] u = e pi ( t − λ i ) = 1 t − λ i (cid:20) u ddu ( ur Y ( u )) (cid:21) u = e pi ( t − λ i ) > . The last inequality follows from the fact that ur Y ( u ) is increasing in u . Again, as u ddu ( ur Y ( u ))isdecreasing in u, and λ , σ ∈ D + ( or E + ) , then for all i ≤ j , ∂ Ψ ∂p i − ∂ Ψ ∂p j = 1 t − λ i (cid:20) u ddu ( ur Y ( u )) (cid:21) u = e pi ( t − λ i ) − t − λ j (cid:20) u ddu ( ur Y ( u )) (cid:21) u = e pj ( t − λ j ) ≥ ( ≤ ) 0 . P roof of iv ) Let us assume that Ψ( σ ) = P nk =1 r Y (cid:16) t − λkσk (cid:17) σ k . Now, differentiating Ψ( σ ) withrespect to σ i and considering the fact that ur Y ( u ) is increasing in u , it can be written that, ∂ Ψ ∂σ i = − σ i r Y (cid:18) t − λ i σ i (cid:19) − (cid:18) t − λ i σ i (cid:19) r ′ Y (cid:18) t − λ i σ i (cid:19) = − t − λ i ) (cid:18) t − λ i σ i (cid:19) ddu [ ur Y ( u )] u = t − λiσi ≤ . Again, as λ , σ ∈ D + ( or E + ) , and u ddu [ ur Y ( u )] is decreasing in u, it can be easily shown that ∂ Ψ ∂σ i − ∂ Ψ ∂σ j ≤ ( ≥ )0 . Thus, by Lemma 1 (Lemma 2), Ψ( σ ) is s-concave and hence the result isproved by Lemma 4. ✷ Theorem 7
Let X , X , ..., X n be a set of independent random variables such that X i ∼ LS ( λ i , σ i , F ) , i = 1 , , ..., n . Let Y , Y , ..., Y n be another set of independent random variables such that Y i ∼ LS ( λ i , ξ i , G ) , i = 1 , , ..., n . Suppose that σ , ξ , λ ∈ D + (or E + ). Then, if X ≤ hr Y andi) either ur X ( u ) or ur Y ( u ) is convex in u, then, σ m (cid:23) ξ ⇒ X n ≤ hr Y n ,ii) either ur X ( u ) or ur Y ( u ) is increasing (decreasing) and convex in u, then, σ (cid:23) w ( w (cid:23) ) ξ ⇒ X n ≤ hr Y n ,iii) either ur X ( u ) or ur Y ( u ) is decreasing in u, and u ddu ( ur X ( u )) or u ddu ( ur Y ( u )) is increasingin u, then, σ p (cid:23) ξ ⇒ X n ≤ hr Y n ,iv) either ur X ( u ) or ur Y ( u ) is decreasing in u, and u ddu ( r X ( u )) or u ddu ( r Y ( u )) is increasingin u, then, σ rm (cid:23) ξ ⇒ X n ≤ hr Y n . Proof:
Proof of the theorem follows from the previous theorem and hence is omitted. ✷ In the next theorem, it is shown that under certain restrictions, there exists R-hr orderingbetween two minimum order statistics obtained from two different LS families, when one setof scale parameters majorizes the other. It is to be mentioned here that, due to mathematicalcomplexities we cannot proceed in full generality, where two LS families could be generated fromtwo different baseline distributions and hence we consider that both LS families are generatedfrom the same baseline distribution. Theorem 8
Let X , X , ..., X n be a set of independent random variables such that X i ∼ LS ( λ i , σ i , F ) , i = 1 , , ..., n . Let Y , Y , ..., Y n be another set of independent random variables such that Y i ∼ LS ( λ i , ξ i , F ) , i = 1 , , ..., n . Suppose that λ , σ , ξ ∈ D + (or E + ). If r X ( u ) is r -concavefor r = 1 , and then, σ m (cid:22) ξ ⇒ X n ≤ R − hr Y n . roof: To prove the result we have only to prove that g ( t ) = s n ( t ) r n ( t ) is decreasing in t . Now, g ′ ( t ) = ddt P nk =1 1 ξ k r X (cid:16) t − λ k ξ k (cid:17)P nk =1 1 σ k r X (cid:16) t − λ k σ k (cid:17) sign = P nk =1 1 ξ k r ′ X (cid:16) t − λ k ξ k (cid:17)P nk =1 1 ξ k r ′ X (cid:16) t − λ k ξ k (cid:17) − P nk =1 1 σ k r ′ X (cid:16) t − λ k σ k (cid:17)P nk =1 1 σ k r ′ X (cid:16) t − λ k σ k (cid:17) . So, it is to be proved that P nk =1 r ′ X (cid:16) t − λkσk (cid:17) σ k P nk =1 r ′ X (cid:16) t − λkσk (cid:17) σ k = P nk =1 p k r ′ X ( p k ( t − λ k )) P nk =1 p k r X ( p k ( t − λ k )) = P nk =1 u ( p k , λ k , t ) v ( p k , λ k , t ) P nk =1 u ( p k , λ k , t ) = P nk =1 u k v k P nk =1 u k = Ψ( p ) ( say ) , with p = (cid:16) σ , σ , ..., σ n (cid:17) , is s-concave in p , where u ( p k , λ k , t ) = p k r X ( p k ( t − λ k )) = u k (say)and v ( p k , λ k , t ) = p k r ′ X ( p k ( t − λ k )) r X ( p k ( t − λ k )) = v k (say). Now, differentiating Ψ( p ) with respect to p i weget, ∂ Ψ ∂p i sign = n X k =1 u k (cid:18) v i ∂u i ∂p i + u i ∂v i ∂p i (cid:19) − n X k =1 u k v k (cid:18) ∂u i ∂p i (cid:19) . (13)Now, differentiating u k and v k with respect to p k , it can be written that ∂u k ∂p k = u k w k ; w k = v k ( t − λ k ) p k , and ∂v k ∂p k = v k (cid:20) p k + ( t − λ k ) r ′′ X ( p k ( t − λ k )) r ′ X ( p k ( t − λ k )) − ( t − λ k ) r ′ X ( p k ( t − λ k )) r Y ( p k ( t − λ k )) (cid:21) = v k s k (say),where s k = p k + ( t − λ k ) r ′′ X ( p k ( t − λ k )) r ′ X ( p k ( t − λ k )) − ( t − λ k ) r ′ X ( p k ( t − λ k )) r Y ( p k ( t − λ k )) . Using (13) and expressions for ∂u k ∂p k and ∂v k ∂p k , it can be shown that, ∂ Ψ ∂p i sign = n X k =1 u k ! ( u i v i w i + u i v i s i ) − n X k =1 u k ! v k u i w i , which gives, for all i ≤ j∂ Ψ ∂p i − ∂ Ψ ∂p j sign = n X k =1 u k ! [ u i v i ( w i + s i ) − u j v j ( w j + s j )] − n X k =1 u k v k ! ( u i w i − u j w j ) , (14)with u i v i ( w i + s i ) − u j v j ( w j + s j ) = 2 h p i r ′ X ( p i ( t − λ i )) − p j r ′ X ( p j ( t − λ j )) i + h p i ( t − λ i ) r ′′ X ( p i ( t − λ i )) − p j r ′′ X ( p j ( t − λ j )) i . (15)Now as λ , σ ∈ D + , ( or E + ), then for all i ≤ j , p i , < p j , which gives p i ( t − λ i ) ≤ ( ≥ ) p j ( t − λ j )and p i ( t − λ i ) ≤ ( ≥ ) p j ( t − λ j ) . Moreover, as r X ( u ) is s-concave for s = 1 ,
2, it is decreasingand concave and hence it can be concluded that p i r ′ X ( p i ( t − λ i )) ≥ ( ≤ ) p j r ′ X ( p j ( t − λ j )). So,the first term of (15) is positive (negative).Again, as r X ( u ) is also 3-concave and thus r ′′ X ( u ) decreasing in u , then for all i ≤ j it can be15ritten that, r ′′ X ( p i ( t − λ i )) ≥ ( ≤ ) r ′′ X ( p j ( t − λ j )), which gives p i ( t − λ i ) r ′′ X ( p i ( t − λ i )) ≥ ( ≤ ) p j ( t − λ j ) r ′′ X ( p j ( t − λ j )). So the second term of (15) is also positive (negative). Thus the firstterm of (14) is positive (negative).Now, following the arguments as before it can be easily shown that, u i w i − u j w j = [ r X ( p i ( t − λ i )) − r X ( p j ( t − λ j ))]+ h p i ( t − λ i ) r ′ X ( p i ( t − λ i )) − p j ( t − λ j ) r ′ X ( p j ( t − λ j )) i ≥ ( ≤ )0 . Now as r X ( u ) is decreasing in u , v k = p k r ′ X ( p k ( t − λ k )) r X ( p k ( t − λ k )) is negative. So, the second term of (14)is also positive (negative). Hence, ∂ Ψ ∂p i − ∂ Ψ ∂p j ≥ ( ≤ )0. Thus by Lemma 2 (Lemma 1) it can beconcluded that Ψ( p ) is s-concave in p . This proves the result. ✷ Systems are also compared stochastically in situations where the components are frommultiple-outlier models. This is to be mentioned here that, a multiple-outlier model is a setof independent random variables X , ..., X n of which X i st = X, i = 1 , ..., n and X i st = Y, i = n + 1 , ..., n where 1 ≤ n < n and X i st = X means that cdf of X i is same as that of X. In otherwords, the set of independent random variables X , ..., X n is said to constitute a multiple-outliermodel if two sets of random variables ( X , X , . . . , X n ) and ( X n +1 , X n +2 , . . . , X n + n ) (where n + n = n ), are homogenous among themselves and heterogenous between themselves. Formore details on multiple-outlier models, readers may refer to Kochar and Xu ([30]), Zhao andBalakrishnan ([31]), Balakrishnan and Torrado ([32]), Zhao and Zhang [33], Kundu et al. [8],Kundu and Chowdhury [9] and the references there in. Two theorems given below show that,under certain conditions, majorized scale parameter vectors of LS distributed components ofseries system for multiple outlier model, leads to smaller system life in terms of R-hr ordering. Theorem 9
Let { X , X , . . . , X n } and { Y , Y , . . . , Y n } be two sets of independent random vari-ables each following the multiple outlier LS model such that X i ∼ LS ( λ , σ , F ) and Y i ∼ LS ( λ , ξ , F ) for i = 1 , , . . . , n and X i ∼ LS ( λ , σ , F ) and Y i ∼ LS ( λ , ξ , F ) for i = n + 1 , n + 2 , . . . , n + n = n . Suppose that ( λ , λ ) , ( σ , σ ) , ( ξ , ξ ) ∈ D + ( or E + ) . Now if ur X ( u ) is decreasing in u and r X ( u ) is log-concave and -log-convex in u , then ( 1 σ , σ , . . . , σ , | {z } n σ , σ , . . . , σ | {z } n ) m (cid:22) ( 1 ξ , ξ , . . . , ξ , | {z } n ξ , ξ , . . . , ξ | {z } n ) ⇒ X n ≥ R − hr Y n . Proof:
To prove the result we have only to prove that g ( t ) = s n ( t ) r n ( t ) is increasing in t . Now, g ′ ( t ) = ddt P nk =1 1 ξ k r X (cid:16) t − λ k ξ k (cid:17)P nk =1 1 σ k r X (cid:16) t − λ k σ k (cid:17) sign = P nk =1 1 ξ k r ′ X (cid:16) t − λ k ξ k (cid:17)P nk =1 1 ξ k r ′ X (cid:16) t − λ k ξ k (cid:17) − P nk =1 1 σ k r ′ X (cid:16) t − λ k σ k (cid:17)P nk =1 1 σ k r ′ X (cid:16) t − λ k σ k (cid:17) . So, to prove the result it is to be shown that P nk =1 r ′ X (cid:16) t − λkσk (cid:17) σ k P nk =1 r ′ X (cid:16) t − λkσk (cid:17) σ k = P nk =1 p k r ′ X ( p k ( t − λ k )) P nk =1 p k r X ( p k ( t − λ k )) = P nk =1 u ( p k , λ k , t ) v ( p k , λ k , t ) P nk =1 u ( p k , λ k , t ) = P nk =1 u k v k P nk =1 u k = Ψ( p ) ( say ) , p = p , . . . , p | {z } n , p , . . . , p | {z } n = σ , σ , . . . , σ | {z } n , σ , σ . . . σ | {z } n , is s-convex in p , where u ( p k , λ k , t ) = p k r X ( p k ( t − λ k )) = u k (say) and v ( p k , λ k , t ) = p k r ′ X ( p k ( t − λ k )) r X ( p k ( t − λ k )) = v k (say).Let i ≤ j . Now three cases may arise: Case ( i ) : If 1 ≤ i, j ≤ n , i.e. , if p i = p j = p and λ i = λ j = λ , then ∂ Ψ ∂p i − ∂ Ψ ∂p j = ∂ Ψ ∂p − ∂ Ψ ∂p = 0 .Case ( ii ) : If n + 1 ≤ i, j ≤ n , i.e. , if p i = p j = p and λ i = λ j = λ , then ∂ Ψ ∂p i − ∂ Ψ ∂p j = ∂ Ψ ∂p − ∂ Ψ ∂p = 0 .Case ( iii ) : If 1 ≤ i ≤ n and n + 1 ≤ j ≤ n , then p i = p , p j = p , λ i = λ and λ j = λ . Then, ∂ Ψ ∂p i ( n u + n u ) = n u ∂u ∂p ( v − v ) + w ( n u + n u ) , and ∂ Ψ ∂p j ( n u + n u ) = n u ∂u ∂p ( v − v ) + w ( n u + n u ) , where, for i = 1 , w i = u i ∂v i ∂p i . Thus ∂ Ψ ∂p i − ∂ Ψ ∂p j sign = ( v − v ) (cid:18) n u ∂u ∂p + n u ∂u ∂p (cid:19) + ( w − w ) ( n u + n u ) . (16)Now, as ur X ( u ) is decreasing in u , then for i = 1 , ∂u i ∂p i = ddu ( ur X ( u )) | u =( t − λ i ) p i <
0. Now,if ( λ , λ ) , ( σ , σ ) ∈ D + ( or E + ), then p ≤ ( ≥ ) p and p ( t − λ ) ≤ ( ≥ ) p ( t − λ ). Thus, as r X ( u ) is log-concave in u , by considering the fact that r X ( u ) is decreasing in u , it can be writtenthat − p r ′ X ( p ( t − λ )) r X ( p ( t − λ )) ≤ ( ≥ ) − p r ′ X ( p ( t − λ )) r X ( p ( t − λ )) , giving that v ≥ ( ≤ ) v . So, the first term of (16) is negative (positive). Again, for i = 1 , w i = p i r X ( t − λ i ) ∂∂p i r ′ X ( p i ( t − λ i )) r X ( p i ( t − λ i )) ! = p i ( t − λ i ) r X ( t − λ i ) ∂ ∂u (log r X ( u )) | u = p i ( t − λ i ) . So, considering the fact that r X ( u ) is log-concave giving ∂ ∂u log r X ( u ) <
0, and r X ( u ) is also2-log-convex, it can be written that, − p ( t − λ ) r X ( t − λ ) ∂ ∂u (log r X ( u )) | u = p ( t − λ ) ≥ ( ≤ ) − p ( t − λ ) r X ( t − λ ) ∂ ∂u (log r X ( u )) | u = p ( t − λ ) , w ≤ ( ≥ ) w . So, the second term of (16) is also negative (positive). Thus, as ∂ Ψ ∂p i − ∂ Ψ ∂p j ≤ ( ≥ )0, by Lemma 2 (Lemma 1) it can be concluded that Ψ( p ) is s-convex in p . Thisproves the result. ✷ The next theorem can be proved in the similar line as of previous theorem.
Theorem 10
Let { X , X , . . . , X n } and { Y , Y , . . . , Y n } be two sets of independent randomvariables each following the multiple outlier LS model such that X i ∼ LS ( λ , σ , F ) and Y i ∼ LS ( λ , ξ , F ) for i = 1 , , . . . , n and X i ∼ LS ( λ , σ , F ) and Y i ∼ LS ( λ , ξ , F ) for i = n + 1 , n + 2 , . . . , n + n = n . Suppose that ( λ , λ ) , ( σ , σ ) , ( ξ , ξ ) ∈ D + ( or E + ) . Now if ur X ( u ) is increasing in u and r X ( u ) is decreasing, log-concave and -log-concave in u , then ( 1 σ , σ , . . . , σ , | {z } n σ , σ , . . . , σ | {z } n ) m (cid:23) ( 1 ξ , ξ , . . . , ξ , | {z } n ξ , ξ , . . . , ξ | {z } n ) ⇒ X n ≥ R − hr Y n . Now, combining Theorem 6 i) and Theorem 9 it can be shown that, for multiple outlier LS model, majorized scale parameter vector of minimum order statistics leads to smaller systemlife in terms of lr ordering. The statement of the theorem is given below. Theorem 11
Let { X , X , . . . , X n } and { Y , Y , . . . , Y n } be two sets of independent randomvariables each following the multiple outlier LS model such that X i ∼ LS ( λ , σ , F ) and Y i ∼ LS ( λ , ξ , F ) for i = 1 , , . . . , n and X i ∼ LS ( λ , σ , F ) and Y i ∼ LS ( λ , ξ , F ) for i = n + 1 , n + 2 , . . . , n + n = n . Suppose that ( λ , λ ) , ( σ , σ ) , ( ξ , ξ ) ∈ D + ( or E + ) . Now if ur X ( u ) is decreasing andconcave in u and r X ( u ) is log-concave and -log-convex in u , then ( 1 σ , σ , . . . , σ , | {z } n σ , σ , . . . , σ | {z } n ) m (cid:22) ( 1 ξ , ξ , . . . , ξ , | {z } n ξ , ξ , . . . , ξ | {z } n ) ⇒ X n ≤ lr Y n . Again combining Theorem 7 i) and Theorem 10 the following theorem can be obtained.
Theorem 12
Let { X , X , . . . , X n } and { Y , Y , . . . , Y n } be two sets of independent randomvariables each following the multiple outlier LS model such that X i ∼ LS ( λ , σ , F ) and Y i ∼ LS ( λ , ξ , F ) for i = 1 , , . . . , n and X i ∼ LS ( λ , σ , F ) and Y i ∼ LS ( λ , ξ , F ) for i = n + 1 , n + 2 , . . . , n + n = n . Suppose that ( λ , λ ) , ( σ , σ ) , ( ξ , ξ ) ∈ D + ( or E + ) . Now if ur X ( u ) is increasing andconvex in u and r X ( u ) is decreasing, log-concave and -log-convex in u , then ( 1 σ , σ , . . . , σ , | {z } n σ , σ , . . . , σ | {z } n ) m (cid:23) ( 1 ξ , ξ , . . . , ξ , | {z } n ξ , ξ , . . . , ξ | {z } n ) ⇒ X n ≤ lr Y n . Now the question arises − what will happen if location parameter vector of one LS familymajorizes the other when the scale parameter vector remains same? The next few theoremsdeal with such cases.The next theorem shows that under some restrictions, majorization ordering between locationparameter vectors of minimum order statistics from two different LS family of distributionsimplies hr ordering between them. 18 heorem 13 Let { X , X , . . . , X n } and { Y , Y , . . . , Y n } be two sets of independent randomvariables each following LS model such that X i ∼ LS ( λ i , σ i , F ) and Y i ∼ LS ( µ i , σ i , G ) for i =1 , , . . . , n . Suppose further that λ , µ , σ ∈ D + (or E + ). Then, if X ≤ hr Y , either r X ( u ) or r Y ( u ) is increasing (decreasing) in u and either u r ′ X ( u ) or u r ′ Y ( u ) is decreasing (increasing)in u then, λ (cid:22) w ( (cid:23) w ) µ ⇒ X n ≤ hr Y n . Proof: If X ≤ hr Y , then r X (cid:16) t − λ k σ k (cid:17) ≥ r Y (cid:16) t − λ k σ k (cid:17) , which results in n X k =1 r X (cid:16) t − λ k σ k (cid:17) σ k ≥ n X k =1 r Y (cid:16) t − λ k σ k (cid:17) σ k . So, to prove the result it suffices to prove that P nk =1 r Y (cid:16) t − λkσk (cid:17) σ k ≥ P nk =1 r Y (cid:16) t − µkσk (cid:17) σ k , or equivalentlyit is required to show that P nk =1 r Y (cid:16) t − µkσk (cid:17) σ k = Ψ( µ ) is decreasing (increasing) in each µ i ands-concave (s-convex) in µ . Now, differentiating Ψ with respect to µ i it can be written that ∂ Ψ ∂µ i = − σ i r ′ Y (cid:16) t − µ i σ i (cid:17) , which is clearly decreasing (increasing) in µ i if r Y ( u ) is increasing (decreasing)in u . Again, if µ , σ ∈ D + , then for all i ≤ j , µ i ≥ µ j , σ i ≥ σ j giving t − µ i σ i ≤ t − µ j σ j . So, if r Y ( u )is increasing (decreasing) and u r ′ Y ( u ) is decreasing (increasing) in u , then1( t − µ i ) (cid:18) t − µ i σ i (cid:19) r ′ Y (cid:18) t − µ i σ i (cid:19) ≥ ( ≤ ) 1( t − µ j ) (cid:18) t − µ j σ j (cid:19) r ′ Y (cid:18) t − µ j σ j (cid:19) giving that ∂ Ψ ∂µ i − ∂ Ψ ∂µ j = 1 σ j r ′ Y (cid:18) t − µ j σ j (cid:19) − σ i r ′ Y (cid:18) t − µ i σ i (cid:19) ≤ ( ≥ )0 . So, by Lemma 1 it can be concluded that Ψ( µ ) is s-concave (s-convex) in µ . This proves theresult.When µ , σ ∈ E + , then the theorem can be proved in similar line. ✷ The theorem given below shows that for two LS families of distribution functions generatedfrom the same baseline distribution, there exists R-hr ordering between their correspondingminimum order statistics. Theorem 14
Let { X , X , . . . , X n } and { Y , Y , . . . , Y n } be two sets of independent randomvariables each following LS model such that X i ∼ LS ( λ i , σ i , F ) and Y i ∼ LS ( µ i , σ i , F ) for i =1 , , . . . , n . Suppose further that λ , µ , σ ∈ D + (or E + ). Then, if r X ( u ) is increasing and concave(decreasing and convex) in u ; u r ′ X ( u ) is decreasing (increasing) in u and u r ′′ X ( u ) is increasing(decreasing) in u then, λ (cid:22) w ( (cid:23) w ) µ ⇒ X n ≥ R − hr Y n . Proof:
To prove the result we have only to prove that g ( t ) = s n ( t ) r n ( t ) is increasing in t . Now, g ′ ( t ) = ddt P nk =1 1 σ k r X (cid:16) t − µ k σ k (cid:17)P nk =1 1 σ k r X (cid:16) t − λ k σ k (cid:17) sign = P nk =1 1 σ k r ′ X (cid:16) t − µ k σ k (cid:17)P nk =1 1 ξ k r ′ X (cid:16) t − µ k σ k (cid:17) − P nk =1 1 σ k r ′ X (cid:16) t − λ k σ k (cid:17)P nk =1 1 σ k r ′ X (cid:16) t − λ k σ k (cid:17) . P nk =1 r ′ X (cid:16) t − µkσk (cid:17) σ k P nk =1 r ′ X (cid:16) t − µkσk (cid:17) σ k = Ψ( µ )is increasing (decreasing) in each µ i and s-convex (s-concave) in µ . Now differentiating Ψ withrespect to µ i , ∂ Ψ ∂µ i " n X k =1 σ k r X (cid:18) t − µ k σ k (cid:19) = − " n X k =1 σ k r X (cid:18) t − µ k σ k (cid:19) σ i r ′′ X (cid:18) t − µ i σ i (cid:19) + " n X k =1 σ k r ′ X (cid:18) t − µ k σ k (cid:19) σ i r ′ X (cid:18) t − µ i σ i (cid:19) which is increasing (decreasing) in each µ i if r X ( u ) is increasing and concave (decreasing andconvex) in u . Again, ∂ Ψ ∂µ i − ∂ Ψ ∂µ j sign = " n X k =1 σ k r X (cid:18) t − µ k σ k (cid:19) σ j r ′′ X (cid:18) t − µ j σ j (cid:19) − σ i r ′′ X (cid:18) t − µ i σ i (cid:19) + " n X k =1 σ k r ′ X (cid:18) t − µ k σ k (cid:19) σ i r ′ X (cid:18) t − µ i σ i (cid:19) − σ j r ′ X (cid:18) t − µ j σ j (cid:19) . (17)Now if µ , σ ∈ D + , then for all i ≤ j , t − µ i ≤ t − µ j , σ i ≥ σ j and t − µ i σ i ≤ t − µ j σ j . So, as u r ′ X ( u )is decreasing (increasing) in u then for all i ≤ j , (cid:18) t − µ i σ i (cid:19) r ′ X (cid:18) t − µ i σ i (cid:19) ≥ ( ≤ ) (cid:18) t − µ j σ j (cid:19) r ′ X (cid:18) t − µ j σ j (cid:19) . As, r X ( u ) is increasing (decreasing) in u , then from the above relation it can be written that1( t − µ i ) (cid:18) t − µ i σ i (cid:19) r ′ X (cid:18) t − µ i σ i (cid:19) ≥ ( ≤ ) 1( t − µ j ) (cid:18) t − µ j σ j (cid:19) r ′ X (cid:18) t − µ j σ j (cid:19) . So, the second term of (17) is positive (negative). Again, u r ′′ X ( u ) is increasing (decreasing) in u gives (cid:18) t − µ i σ i (cid:19) r ′′ X (cid:18) t − µ i σ i (cid:19) ≤ ( ≥ ) (cid:18) t − µ j σ j (cid:19) r ′′ X (cid:18) t − µ j σ j (cid:19) . Considering the fact that r X ( u ) is concave (convex) in u , from the above relation it can bewritten that, 1( t − µ i ) (cid:18) t − µ i σ i (cid:19) r ′′ X (cid:18) t − µ i σ i (cid:19) ≥ ( ≤ ) 1( t − µ j ) (cid:18) t − µ j σ j (cid:19) r ′′ X (cid:18) t − µ j σ j (cid:19) . So, the first term of (17) is also positive (negative). Thus ∂ Ψ ∂µ i − ∂ Ψ ∂µ j ≥ ( ≤ )0. So by Lemma 1 itcan be concluded that Ψ is s-convex (s-concave) in µ . Thus, using Lemma 5 the required resultcan be obtained.When µ , σ ∈ E + , then the theorem can be proved in similar line. ✷ Combining Theorem 13 and Theorem 14 the following theorem can easily be obtained.20 heorem 15
Let { X , X , . . . , X n } and { Y , Y , . . . , Y n } be two sets of independent randomvariables each following LS model such that X i ∼ LS ( λ i , σ i , F ) and Y i ∼ LS ( µ i , σ i , F ) for i =1 , , . . . , n . Suppose further that λ , µ , σ ∈ D + (or E + ). Then, if r X ( u ) is increasing and concave(decreasing and convex) in u ; u r ′ X ( u ) is decreasing (increasing) in u and u r ′′ X ( u ) is increasing(decreasing) in u then, λ (cid:22) w ( (cid:23) w ) µ ⇒ X n ≤ lr Y n . Similar results for the parallel system are furnished in this section for dependent setup.These results are the extension of the Theorems 3.1-3.5 of Hazra et al. [13], where the parallelsystem consists of independent LS family distributed components. The following theorems16-18 can be proved in the same line as of Theorem 1, Theorem 3 and Theorem 4 respectivelyand hence the proofs are omitted. For results of independent random variables, one may referto Hazra et al. [13].Let X and Y be two random variables having distribution functions F ( · ) and G ( · ) respec-tively. Suppose that X i ∼ LS ( λ i , σ i , F ) and Y i ∼ LS ( µ i , ξ i , G ) ( i = 1 , , . . . , n ) be two sets of n dependent random variables with Archimedean copulas having generators ψ (with φ = ψ − )and ψ ( φ = ψ − ) respectively. Also suppose that G n : n ( · ) and H n : n ( · ) be the distributionfunctions of X n : n and Y n : n respectively. Then, G n : n ( t ) = ψ " n X k =1 φ (cid:26) F (cid:18) t − λ k σ k (cid:19)(cid:27) , t > max ( λ k , ∀ k ) , and H n : n ( t ) = ψ " n X k =1 φ (cid:26) G (cid:18) t − µ k ξ k (cid:19)(cid:27) , t > max ( µ k , ∀ k ) . Let ˜ r X ( u ) and ˜ r Y ( u ) are the reversed hazard rate functions of the random variables X and Y respectively. Theorem 16
Let X , X , ..., X n be a set of dependent random variables sharing Archimedeancopula having generator ψ such that X i ∼ LS ( λ, σ i , F ) , i = 1 , , ..., n . Let Y , Y , ..., Y n beanother set of dependent random variables sharing Archimedean copula having generator ψ such that Y i ∼ LS ( λ, ξ i , G ) , i = 1 , , ..., n . Assume that λ , σ , ξ ∈ D + (or E + ). Further supposethat φ ◦ ψ is super-additive, ψ or ψ is log-convex and X ≥ st Y . If either ˜ r X ( u ) or ˜ r Y ( u ) isdecreasing in u then, σ w (cid:23) ξ ⇒ X n : n ≥ st Y n : n . Theorem 17
Let X , X , ..., X n be a set of dependent random variables sharing Archimedeancopula having generator ψ such that X i ∼ LS ( λ i , σ i , F ) , i = 1 , , ..., n . Let Y , Y , ..., Y n beanother set of dependent random variables sharing Archimedean copula having generator ψ uch that Y i ∼ LS ( λ i , ξ i , G ) , i = 1 , , ..., n . Assume that λ , σ , ξ ∈ D + (or E + ). Further supposethat φ ◦ ψ is super-additive, ψ or ψ is log-convex and X ≥ st Y , then,i) σ p (cid:23) ξ ⇒ X n : n ≥ st Y n : n , if either ur X ( u ) or ur Y ( u ) is decreasing in u ;ii) σ rm (cid:23) ξ ⇒ X n : n ≥ st Y n : n , if either u r X ( u ) or u r Y ( u ) is decreasing in u . Theorem 18
Let X , X , ..., X n be a set of dependent random variables sharing Archimedeancopula having generator ψ such that X i ∼ LS ( λ i , σ i , F ) , i = 1 , , ..., n . Let Y , Y , ..., Y n beanother set of dependent random variables sharing Archimedean copula having generator ψ such that Y i ∼ LS ( µ i , σ i , G ) , i = 1 , , ..., n . Assume that λ , σ , µ ∈ D + (or E + ). Further supposethat φ ◦ ψ is super-additive, ψ or ψ is log-convex and either ur X ( u ) or ur Y ( u ) is decreasingin u, then, X ≥ st Y and λ (cid:23) w µ ⇒ X n : n ≥ st Y n : n . References [1] H.A. David and H.N. Nagaraja, Order Statistics, 3rd edition, Wiley, New Jersey, 2003.[2] R. Dykstra, S.C. Kochar and J. Rojo, “Stochastic comparisons of parallel systems of het-erogeneous exponential components,”
Journal of Statistical Planning and Inference , vol.65, pp. 203-211, 1997.[3] P. Zhao and N. Balakrishnan, “New results on comparison of parallel systems with hetero-geneous gamma components,”
Statistics and Probability Letters , vol. 81, pp. 36-44, 2011.[4] N. Balakrishnan, G. Barmalzan and A. Haidari, “On usual multivariate stochastic orderingof order statistics from heterogeneous beta variables,”
Journal of Multivariate Analysis ,vol. 127, pp. 147-150, 2014.[5] N. Torrado and S.C. Kochar, “Stochastic order relations among parallel systems fromWeibull distributions,”
Journal of Applied Probability , vol. 52, pp. 102-116, 2015.[6] N. Torrado, “On magnitude orderings between smallest order statistics from heterogeneousbeta distributions,”
Journal of Mathematical Analysis and Applications , , pp. 824-838,2015.[7] L. Fang and N. Balakrishnan, “Ordering results for the smallest and largest order statisticsfrom independent heterogeneous exponential − Weibull random variables,”
Statistics , vol.50 (6), pp. 1195-1205, 2016.[8] A. Kundu, S. Chowdhury, A.K. Nanda and N. Hazra, “Some Results on Majorization andTheir Applications,”
Journal of Computational and Applied Mathematics , vol. 301, pp.161-177, 2016. 229] A. Kundu and S. Chowdhury, “Ordering properties of order statistics from heterogeneousexponentiated Weibull models,”
Statistics and Probability Letters , vol. 114, pp. 119-127,2016.[10] A. Kundu and S. Chowdhury, “Ordering properties of sample minimum fromKumaraswamy-G random variables,”
Statistics , http://dx.doi.org/10.1080/02331888.2017.1353516, 2017.[11] A. Kundu and S. Chowdhury, “Ordering properties of sample minimum fromKumaraswamy-G random variables, arXiv:1608.08535.[12] S. Chowdhury and A. Kundu, “Stochastic Comparison of Parallel Systems with Log-Lindley Distributed Components,”
Operations Research Letters , vol. 45 (3), pp. 199-205.[13] N.K. Hazra, M.R. Kuiti, M. Finkelstein and A.K. Nanda, “On stochastic comparisonsof maximum order statistics from the location-scale family of distributions,”
Journal ofMultivariate Analysis , vol. 160, pp. 31-41, 2017.[14] D. Sengupta and J. V. Deshpande, “Some results on the relative ageing of two life distri-butions,”
Journal of Applied Probability , vol. 31, pp. 991-1003, 1994.[15] M. Rezaei, B. Gholizadeh, and S. Izadkhah,“On relative reversed hazard rate order,”
Com-munications in Statistics-Theory and Methods , vol. 44, pp. 300-308, 2015.[16] C. Li, and X. Li, “Relative ageing of series and parallel systems with statistically indepen-dent and heterogeneous component lifetimes,”
IEEE Transactions on Reliability , vol. 65,pp. 1014-1021, 2016.[17] J. Navarro and F. Spizzichino, “Comparisons of series and parallel systems with compo-nents sharing the same copula,”
Applied Stochastic Models in Business and Industry , vol.26 (6), pp. 775-791, 2010.[18] M. Rezapour and M.H. Alamatsaz, “Stochastic comparison of lifetimes of two (n-k+1)-out-of-n systems with heterogeneous dependent components,”
Journal of Multivariate Analysis ,vol. 130, pp. 240-251, 2014.[19] C. Li, and X. Li, “Likelihood ratio order of sample minimum from heterogeneous Weibullrandom variables,”
Statistics and Probability Letters , vol. 97, pp. 46-53, 2015.[20] X. Li, and R. Fang, “Ordering properties of order statistics from random variables ofArchimedean copulas with applications,”
Journal of Multivariate Analysis , vol. 133, pp.304-320, 2015.[21] C. Li, R. Fang, and X. Li, “Stochastic comparisons of order statistics from scaled andinterdependent random variables,”
Metrika , vol. 79 (5), pp. 553-578, 2015.2322] R. Fang, C. Li, and X. Li, “Stochastic comparisons on sample extremes of dependent andheterogenous observations,”
Statistics , vol. 50, pp. 930-955, 2016.[23] B.E. Khaledi, S. Farsinezhad, and S.C. Kochar, “Stochastic comparisons of order statisticsin the scale model,”
Journal of Statistical Planning and Inference , vol. 141, pp. 276-286,2011.[24] S.C. Kochar, and N. Torrado, On Stochastic comparisons of largest orderstatistics in the scale model,
Communications in Statistics-Theory and Methods ,DOI:10.1080/03610926.2014.985839, 2015.[25] A.W. Marshall, I. Olkin and B.C. Arnold, Inequalities: Theory of Majorization and ItsApplications, Springer series in Statistics, New York, 2011.[26] B.E. Khaledi, S.C. Kochar, “Dispersive ordering among linear combinations of uniformrandom variables,” Journal of Statistical Planning and Inference, vol. 100, pp. 1321, 2002.[27] M. Shaked, M. and J.G. Shanthikumar, Stochastic Orders, Springer, New York, 2007.[28] R.B. Nelsen, An Introduction to Copulas, Springer: New York, 2006.[29] A.J. McNeil and J. Nˇ e slehov´ a , “Multivariate Archimedean copulas, D-monotone functionsand l1-norm symmetric distributions,” Annals of Statistics , vol. 37, pp. 3059-3097, 2009.[30] S. Kochar, and M. Xu, On the skewness of order statistics in multiple-outlier models.
Journal of Applied Probability , vol. 48, pp. 271-284, 2011.[31] P. Zhao, and N. Balakrishnan, Stochastic comparison of largest order statistics frommultiple-outlier exponential models.
Probability in the Engineering and Informational Sci-ences , vol. 26, pp. 159-182, 2012.[32] N. Balakrishnan, and N. Torrado, Comparisons between largest order statistics frommultiple-outlier models.
Statistics , vol. 50.1, pp. 176-189, 2016.[33] P. Zhao, and Y. Zhang, On Sample Ranges in Multiple-Outlier Models.