Stochastic comparisons of series, parallel and k -out-of-n systems with heterogeneous bathtub failure rate type components

Journal Pre-proof Stochastic comparisons of series, parallel and k-out-of-n systems with heterogeneous bathtub failure rate type components Dhrubasish Bhattacharyya, Ruhul Ali Khan, Murari Mitra

PII: DOI: Reference:

S0378-4371(19)31761-3 https://doi.org/10.1016/j.physa.2019.123124 PHYSA 123124

To appear in:

Physica A

Received date : 24 April 2019 Revised date : 27 September 2019 Please cite this article as: D. Bhattacharyya, R.A. Khan and M. Mitra, Stochastic comparisons of series, parallel and k-out-of-n systems with heterogeneous bathtub failure rate type components, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123124. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

*Highlights (for review)

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Highlights 1. The paper deals with lifetimes of systems with heterogeneous Chen distributed components.

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2. Stochastic comparison results based on vector majorization are established for series and parallel systems. 3. Analogous results are also proved under multivariate majorization setup.

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4. Ageing comparisons between parallel systems with Chen and exponentially distributed components are studied.

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5. A smorgasbord of interesting results are also established for k-out-of-n systems with Chen distributed components.

1

Journal Pre-proof *Manuscript Click here to view linked References

Stochastic comparisons of series, parallel and k-out-of-n systems with heterogeneous bathtub failure rate type components Dhrubasish Bhattacharyyaa , Ruhul Ali Khana , Murari Mitraa Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur P.O. - Botanic Garden, Howrah - 711103, West Bengal, India

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a

Abstract

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In this paper, we explore stochastic comparison of order statistics arising from a two parameter distribution with bathtub shaped failure rate. We obtain detailed results in the contexts of the usual stochastic order, the hazard rate order, the reversed hazard rate order and the likelihood ratio order for series, parallel and k-out-of-n systems. Results involving multivariate chain majorization are also established. Keywords: k-out-of-n system, Stochastic order, Schur-convexity, Chain majorization, Convex transform order. AMS 2010 classification code : 62G30; 60E15 1. Introduction

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Distributions with one or two parameters such as exponential, Weibull, gamma and generalized exponential exhibit only monotone hazard rate and have been studied quite extensively in the literature. However, their inability to model the initial ‘burn-in’ phase typically exhibited by real lifetime data is a serious limitation. On the other hand, more flexible distributions usually comprise four or more parameters which makes the study of estimation from small samples a rather hopeless numerical task. Chen (2000) introduced a two parameter lifetime distribution which exhibits both increasing and bathtub failure rate function depending on the restriction on model parameters, which makes it very useful for modeling of real life data and to describe component lifetime. The cumulative distribution function (cdf) of the distribution is given by F (x) = 1 − e−λ(e

xβ −1)

, x > 0, (λ, β > 0)

(1)

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where λ, β are model parameters. When a random variable X has cdf given by (1), we say that X follows a Chen distribution with parameters β and λ, and write X ∼ Chen(β, λ). Order statistics play a prominent role in various branches of statistics such as reliability theory, survival analysis and actuarial science. (For a comprehensive discussion and relevant applications of order statistics, one can see Balakrishnan and Zhao (2013) and the references therein.) Beyond statistical and probability theory, order statistics have significant applications in various applied areas such as flood-risk methodology, empirical studies of price dispersion on the internet (Warin and Leiter (2012)), utility maximization frameworks for fair and efficient multicasting in multicarrier Email addresses: [email protected] (Dhrubasish Bhattacharyya), [email protected] (Ruhul Ali Khan), [email protected] (Murari Mitra)

Preprint submitted to Physica A

September 27, 2019

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2. Preliminaries

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wireless cellular networks (Liu et al. (2013)). Let X1 , . . . , Xn be n independent random variables representing the component lifetimes of a system with n components. It is well-known that the k-th order statistic Xk:n represents the lifetime of a (n − k + 1)-out-of-n system which fails if and only if at least k components fail. The well studied series and parallel systems are special cases of this system corresponding to k = 1 and k = n respectively. Comparison of ageing characteristics of such systems is of profound interest in the context of reliability engineering as well as in actuarial science since it enables one to obtain bounds on various ageing characteristics (such as survival function and hazard rate function) of a system. Stochastic orderings play a very important role in this context. A substantial body of work is available in the literature dealing with stochastic comparisons of order statistics arising from exponential, gamma, Weibull, generalized exponential etc., see, for example, Dykstra et al. (1997), Misra and Misra (2013), Zhao and Balakrishnan (2011), Torrado and Kochar (2015), Gupta et al. (2015), Kundu and Chowdhury (2016), Kundu et al. (2016). In this paper, we consider two series systems (or parallel systems or k-out-of-n systems) with heterogeneous Chen distributed components. The main focus of our work is to investigate comparisons between corresponding system lifetimes in terms of various notions of stochastic ordering such as usual stochastic order, hazard rate order, reversed hazard rate order, likelihood ratio order, convex transform order and so on. The organization of the rest of the paper is as follows. In Section 2, we briefly review the notions of stochastic orderings and majorization. We also present important results which are frequently used in the sequel. Section 3 consists of our main results. Subsection 3.1 provides various stochastic ordering results for series and parallel systems when component lifetimes are independent and heterogeneous with heterogeneity in one parameter. In Subsection 3.2, we compare survival functions of series and parallel systems for independent and heterogeneous component lifetimes with heterogeneity in both parameters. Finally, in Subsection 3.3, we extend some interesting results in the literature related to stochastic comparisons of k-out-of-n systems with exponentially distributed components to the case when the components are Chen distributed. Throughout the paper, the word increasing (resp. decreasing) and nondecreasing (resp. noninsign creasing) are used interchangeably. We also write A = B to indicate that A and B have the same sign.

In this section, we review the definitions of some well-known stochastic orderings which are used to compare the magnitude and ageing characteristics of two random variables. We also recall the concepts of majorization and related orders. 2.1. Stochastic Orders

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Let X and Y be non-negative random variables with cdf’s FX and FY , survival functions F¯X (= 1−FX ) and F¯Y (= 1−FY ), pdf’s fX and fY , hazard rate functions hX (= fX /F¯X ) and hY (= fY /F¯Y ) and reversed hazard rate functions rX (= fX /FX ) and rY (= fY /FY ) respectively. Also let the supports of X and Y be [lX , uX ] and [lY , uY ] respectively. Definition 2.1. We say that X is smaller than Y in the ¯ 1. usual stochastic order, denoted by X ≤st Y , if F¯ (t) ≤ G(t) or F (t) ≥ G(t) ∀ t.

2. hazard rate order, denoted by X ≤hr Y , if hX (t) ≥ hY (t) or this ratio is well defined.

2

¯ G(t) F¯ (t)

increases in t ∀ t for which

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3. reversed hazard rate order, denoted by X ≤rh Y , if rX (t) ≤ rY (t) or G(t) F (t) increases in t ∀ t for which this ratio is well defined. 4. likelihood ratio order, denoted by X ≤lr Y , if fg(t) (t) increases in t ∀ t for which this ratio is well defined.

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The following implications are well-known. X ≤lr Y ⇒ X ≤hr Y ⇓ ⇓ X ≤rh Y ⇒ X ≤st Y. In the following definition we present some important notions of stochastic order which compare relative ageing characteristics or skewness as well as dispersion of two random variables. Definition 2.2. We say that X is larger than Y in the

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(i) convex transform order, denoted by X ≥c Y , if F −1 G(x) is convex in x ≥ 0. Equivalently X ≥c Y if and only if G−1 F (x) is concave in x ≥ 0. (ii) star order, denoted by X ≥∗ Y , if F −1 G(x)/x is increasing in x ≥ 0. (iii) superadditive order, denoted by X ≥su Y , if F −1 G(x + y) ≥ F −1 G(x) + F −1 G(y) for all x ≥ 0 and y ≥ 0. (iv) New Better than Used in Expectation (NBUE) order, denoted by X ≥N BU E Y , if Z ∞ Z ∞ 1 1 ¯ ¯ F (x)dx ≥ G(x)dx for t ∈ [0, 1]. E(X) F −1 (t) E(Y ) G−1 (t) (v) Lorenz order, denoted by X ≥Lorenz Y , if Z

F −1 (t)

0

F¯ (x)dx ≤

1 E(Y )

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1 E(X)

Z

G−1 (t)

0

¯ G(x)dx

for t ∈ [0, 1].

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(vi) Decreasing Mean Residual Life (DMRL) order, denoted by X ≥DM RL Y , if R∞ 1 ¯ E(X) F −1 (t) F (x)dx R is increasing in t ∈ [0, 1]. ∞ 1 ¯ G(x)dx −1 E(Y )

G

(t)

(vii) right-spread order, denoted by X ≥RS Y , if Z ∞ Z ∞ F¯ (x)dx ≥ F −1 (t)

¯ G(x)dx

G−1 (t)

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(viii) increasing convex order, denoted by X ≥icx Y , if Z ∞ Z ∞ ¯ ¯ F (x)dx ≥ G(x)dx t

t

for t ∈ (0, 1).

for all t ≥ 0.

2.2. Majorization and Related Orders In this section we briefly review the concepts of majorization and related orders. Let {x(1) , x(2) , . . . , x(n) } and {x[1] , x[2] , . . . , x[n] } denote respectively the increasing and decreasing arrangements of the components of the vector x = (x1 , x2 , . . . , xn ). Definition 2.3. Let x = (x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ) be two vectors. Then 3

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Pk (i) the vector x is said to be majorized by the vector y, denoted by x m y, if i=1 x(i) ≥ Pn Pn Pk y . x = y for k = 1, . . . , n − 1 and i=1 (i) i=1 (i) i=1 (i) (ii) P the vector xPis said to be weakly supermajorized by the vector y, denoted by x w y, if k k i=1 x(i) ≥ i=1 y(i) , for k = 1, . . . n. P (iii) the vector x is said to be weakly submajorized by the vector y, denoted by x w y, if ki=1 x[i] ≤ Pk i=1 y[i] , for k = 1, . . . n.

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Definition 2.4. A real-valued function φ defined on a set A ∈ Rn is said to be Schur-convex (Schur-concave) on A if x ≤m y on A implies φ(x) ≤ (≥)φ(y). Lemma 2.1. If I ⊆ R is an interval and h(x) : I → R is convex, then g(x) = Schur-convex on In , where x = (x1 , . . . , xn ).

Pn

i=1 h(xi )

is

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Lemma 2.2. A permutation symmetric differentiable function φ defined on a set A ∈ Rn is Schurconvex (Schur-concave) iff   ∂φ(x) ∂φ(x) (xi − xj ) − ≥ (≤)0 ∂xi ∂xj ∀ i 6= j, where x = (x1 , . . . xn ).

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For extensive and comprehensive details on the theory of majorization orders and their applications, see Marshall et al. (1979). Before presenting the definition of chain majorization, we introduce the following notation. A square matrix Π is called a permutation matrix if each of its rows and columns contains exactly a single unity and all the remaining entries are zero. A square matrix Tw of order n is said to be a T-transform matrix if it can be written as Tw = wIn + (1 − w)Π, where 0 ≤ w ≤ 1, and Π is a permutation matrix which interchanges exactly two coordinates. Two T-transform matrices Tw1 = w1 In +(1−w1 )Π1 and Tw2 = w2 In +(1−w2 )Π2 are said to be of same structure if Π1 = Π2 . It is also well known that the product of two T-transform matrices of same structure is a T-transform matrix of the same structure. In the following definition, the idea of multivariate chain majorization is presented. Definition 2.5. Let A = (aij )m×n and B = (bij )m×n be two m × n matrices. Then B is said to be chain mejorized by A, denoted by B  A, if there exists a finite number of n × n T-transform matrices Tw1 , . . . Twr such that B = ATw1 Tw2 . . . Twr .

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The following two theorems are useful to prove the main results of this paper. Let     x1 . . . xn Sn = (x, y) = y . . . y : xi > 0, yj > 0 and (xi − xj )(yi − yj ) ≤ 0, i, j = 1, . . . n , 1

    x1 . . . xn Sn = (x, y) = y . . . y : xi > 0, yj > 0 and (xi − xj )(yi − yj ) ≥ 0, i, j = 1, . . . n , 1 n     x ...x Tn = (x, y) = y1 . . . y n : xi ≥ 1, yj > 0 and (xi − xj )(yi − yj ) ≤ 0, i, j = 1, . . . n , 0

1

and

n

0

Tn =



(x, y) =



n

x1 . . . xn y1 . . . yn



 : xi ≥ 1, yj > 0 and (xi − xj )(yi − yj ) ≥ 0, i, j = 1, . . . n . 4

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Theorem 2.3. A differentiable function φ : R+ → R+ satisfies φ(A) ≥ φ(B) for all A, B such that A ∈ S2 , and B  A

(2)

iff

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1. φ(A) = φ(AΠ) for all permutation matrices Π and for all A ∈ S2 ; and P2 2. i=1 (aik − aij ) [φik (A) − φij (A)] ≥ 0 for all j, k = 1, 2, and for all A ∈ S2 , where φij (A) = ∂φ(A) ∂aij . 2

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Theorem 2.4. Let theQfunction ψ : R+ → R+ be differentiable and the function ψn : R+ → R+ be defined as ψn (A) = ni=1 ψ(a1i , a2i ). Further assume that ψ satisfies (2). Then for A ∈ Sn (Tn ) and B = ATw , we have ψn (A) ≥ ψn (B).

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Remark 2.5. The proofs of the Theorems 2.3 and 2.4 are quite similar to those of Theorems 2 and 3 of Balakrishnan et al. (2015) and therefore omitted for the sake of conciseness. The result 0 0 of the Theorem 2.3 remains valid even if S2 is replaced by any one of S2 , T2 and T2 . Similarly, 0 0 Theorem 2.4 also remains valid when Sn is replaced by any one of Sn , Tn and Tn . 3. Main Results

3.1. Comparison Results Based on Vector Majorization In this subsection, we compare series and parallel systems where the component lifetimes are independent and follow Chen distribution with one parameter fixed and the comparison results have been established with respect to the other varying parameter. In order to establish the subsequent results, we first prove the following lemma.

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Lemma 3.1. Let the function φ1 : (0, ∞) × (0, ∞) × (0, ∞) → (0, ∞) be defined as φ1 (x, λ, β) =

β

eλ(ex

−1)

−1

.

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Then,

β

λex xβ

1. φ1 (x, λ, β) is strictly decreasing in λ for fixed x > 0, β > 0. 2. φ1 (x, λ, β) is increasing in β for fixed x ∈ (0, 1] and decreasing in β for fixed x ∈ [1, ∞), when λ ≥ 1. Proof:.

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1. For fixed x > 0 and β > 0, we have

and hence

  β β xβ xβ (1 + λ)eλ(ex −1) − λexβ eλ(ex −1) − 1 e ∂φ1 (x, λ, β) =  2 β ∂λ eλ(ex −1) − 1 ∂φ1 (x, λ, β) sign λ(exβ −1) β =e (1 + λ − λex ) − 1. ∂λ xβ

β

Now, since e−x > 1−x ∀ x > 0, we have e−λ(e −1) > 1+λ−λex which implies eλ(e β λ − λex ) < 1. Thus ∂φ1 (x,λ,β) < 0 ∀ x > 0, λ > 0 and hence the assertion follows. ∂λ 5

xβ −1)

(1+

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2. For fixed x > 0 and λ ≥ 1, it is easy to note that   β β ∂φ1 (x, λ, β) sign λ(ex −1) β β β xβ +λ(ex −1) = ln x e (1 + x ) − (1 + x ) − λx e . ∂β Substituting xβ = y, we get,

0

y −1)

f (y) = eλ(e

and

00

y −1)

f (y) = (1 − λey )λey+λ(e 00

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  ∂φ1 (x, λ, β) sign 1 y y = ln y eλ(e −1) (1 + y) − (1 + y) − λyey+λ(e −1) , y > 0. ∂β β
y y Now, setting, f (y) = eλ(e −1) (1 + y) − (1 + y) − λyey+λ(e −1) , we obtain, − λ2 ye2y+e

λ(ey −1)

λ(ey −1)

− 2λ2 ye2y+e

− 1;

λ(ey −1)

− λ3 ye3y+e

.

0

∂φ1 (x,λ,β) ∂β

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It is easy to note that f (y) ≤ 0 ∀ y > 0 when λ ≥ 1. Also, we have, f (0) = 0. Hence 0 0 f (y) is a decreasing function of y and consequently f (y) ≤ 0 for all y > 0. Next noting ≥ 0 ∀ x ∈ (0, 1] and that f (0) = 0, we get, f (y) ≤ 0 for all y > 0. Thus, we have, ∂φ1 (x,λ,β) ∂β ≤ 0 ∀ x ∈ [1, ∞). This completes the proof of the lemma.



Lemma 3.2. Let the function φ2 : (0, ∞) × (0, ∞) × (0, ∞) → (0, ∞) be defined as β

ex − 1

φ2 (x, λ, β) =

−1)

−1

.

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Then,

β

eλ(ex

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1. φ2 (x, λ, β) is decreasing in λ for fixed x > 0, β > 0. 2. φ2 (x, λ, β) is increasing in β for fixed x ∈ (0, 1] and decreasing in β for fixed x ∈ [1, ∞), when λ > 0.

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Proof:. The proof follows from the arguments used in the proof of Lemma 3.1 and therefore omitted for the sake of conciseness.  In the following theorem, we establish stochastic ordering result for series system with independent heterogeneous Chen distributed components when the parameter β is varying and the parameter λ is held constant. Theorem 3.3. Let X1 , . . . , Xn be a set of independent random variables with Xi ∼ Chen(βi , λ), i = 1, . . . , n. Further, let Y1 , . . . , Yn be another set of independent random variables with Yi ∼ Chen(βi? , λ), i = 1, . . . n. Then, β = (β1 , . . . βn ) m β ? = (β1? , . . . , βn? ) =⇒ X1:n ≥st Y1:n . Proof:. The survival function of X1:n is given by F¯X1:n (x) =

n Y i=1

6

  β λ 1−ex i

e

.

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Clearly, F¯X1:n (x) is permutation symmetric with respect to βi and n ∂ F¯X1:n (x) Y λ = e ∂βi



1−ex

βj

j=1 j6=i





−λe

βi

xβi +λ(1−ex

x ln x



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β = −λex i xβi F¯X1:n (x) ln x.

) βi

Thus,

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 ¯    β ∂ FX1:n (x) ∂ F¯X1:n (x) β j (βi − βj ) − = λ(βi − βj )F¯X1:n (x) ex xβj − ex i xβi ln x. ∂βi ∂βj β

Since, ex xβ is increasing in β if x ≥ 1 and decreasing in β if 0 ≤ x < 1, we have   ¯ ∂ FX1:n (x) ∂ F¯X1:n (x) − ≤ 0, ∀ i 6= j. (βi − βj ) ∂βi ∂βj

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Now, from Lemma 2.2, it follows that F¯X1:n (x) is Schur-concave in β and hence the theorem follows.  The following example illustrates that the condition β m β ? in Theorem 3.3 is not sufficient for the hazard rate ordering between X1:n and Y1:n . Example 3.1. Let Xi and Yi be two sets of independent random variables with Xi ∼ Chen(βi , 1) and Yi ∼ Chen(βi? , 1), i = 1, 2, 3. If (β1 , β2 , β3 ) = (1, 3, 4) and (β1? , β2? , β3? ) = (1, 2, 5), then from Fig. 1, it is clear that

F¯X1:n (x) F¯Y1:n (x)

is not monotone. Hence, X1:n hr Y1:n , although β m β ? .

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1.2

1:3

F¯X1:3 (x) F¯Y (x)

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1.15

1.1

1.05

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1

0

0.2

0.4

0.6

0.8

1

x

Figure 1: Plot of

¯X F (x) 1:n ¯Y F (x) 1:n

Although the condition of Theorem 3.3 is not sufficient for hazard rate ordering between X1:n and Y1:n , it is still possible to compare their hazard rate functions in a certain range. The following result is an attempt in this direction. Theorem 3.4. Let X1 , . . . Xn be a set of independent random variables with Xi ∼ Chen(βi , λ) and Y1 , . . . Yn be another set of independent random variables with Yi ∼ Chen(βi? , λ). Then, β m β ? =⇒ hX1:n (x) ≤ hY1:n (x) for all x ≥ 1. 7

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Proof:. The hazard rate function of X1:n is given by n

n

X X fX (x) β hX1:n (x) = ¯ 1:n = λβi ex i xβi −1 = g(x, λ, βi ). FX1:n (x) i=1 i=1 Now,

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∂g β = λxβi −1 ex i [1 + (1 + xβi )βi ln x] ∂βi

and

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∂ 2 g sign = 2(1 + xβi ) ln x + βi (ln x)2 [1 + xβi (xβi + 3)]. ∂βi2 2

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∂ g Clearly, ∂β 2 ≥ 0, if x ≥ 1. Thus, g(x, λ, βi ) is convex in βi and hence from Lemma 2.1, it follows Pn i that i=1 g(x, λ, βi ) is Schur-convex in β if x ≥ 1. Thus, β m β ? =⇒ hX1:n (x) ≤ hY1:n (x) for x ≥ 1 and the result follows.  The following theorem deals with the stochastic comparison of series systems for independent Chen distributed components with varying parameter λ.

Theorem 3.5. Let Xi , Yi be two setsP of independent P random variables with Xi ∼ Chen(β, λi ) and Yi ∼ Chen(β, λ?i ), i = 1, . . . , n. Then ni=1 λi ≥ ni=1 λ?i =⇒ X1:n ≤lr Y1:n . Proof:. The density function of X1:n is given by   fX1:n (x) =

n X

j=1 j6=i

i=1

n Pn X β xβ =( λi )βxβ−1 ex +(1−e ) i=1 λi .

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i=1

n n X Y  ¯ ¯   fXi (x)  FXi (x) = FX1:n (x) hXi (x)

Thus,

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Pn λ?i (1−exβ )(Pn λ? −Pn λi ) fY1:n i=1 i i=1 e = Pi=1 , n fX1:n i=1 λi P P which is increasing in x if ni=1 λi ≥ ni=1 λ?i and hence the result follows.  Pn Pn Remark 3.6. The condition i=1 λi ≥ i=1 λ?i is even weaker than λ w λ? and consequently the conclusion of the Theorem 3.5 holds if λ w λ? .

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The following two theorems deal with stochastic comparison results for parallel systems when the component lifetimes are independent and heterogeneous with heterogeneity in one parameter. Theorem 3.7. Let X1 , . . . Xn be a set of independent random variables with Xi ∼ Chen(βi , λ) and Y1 , . . . Yn be another set of independent random variables with Yi ∼ Chen(βi? , λ), where λ ≥ 1. Then, β m β ? =⇒ Xn:n ≤st Yn:n .

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Proof:. It is easy to note that the c.d.f of Xn:n is given by  n  Y β λ(1−ex i ) FXn:n (x) = 1−e i=1

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and hence,   n  β β ∂FXn:n (x) Y xβi +λ(1−ex i ) βi λ(1−ex i ) λe x ln x = 1−e ∂βi

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j=1 j6=i

β

ex i xβi

= λFXn:n (x) ln x

βi

eλ(ex −1) − 1 = FXn:n (x)φ1 (x, λ, βi ) ln x.

For any fixed x ∈ (0, 1] and λ ≥ 1, we have from Lemma 3.1,

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φ1 (x, λ, βi ) − φ1 (x, λ, βj ) ≥ (≤)0 according as βi ≥ (≤)βj . Thus, we have, (βi − βj )



∂FXn:n (x) ∂FXn:n (x) − ∂βi ∂βj



≤ 0 for all x ∈ (0, 1] and λ ≥ 1.

Again from Lemma 3.1, for any fixed x ∈ [1, ∞) and λ ≥ 1, we have,

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φ1 (x, λ, βi ) − φ1 (x, λ, βj ) ≤ (≥)0 according as βi ≥ (≤)βj   ∂FXn:n (x) ∂FXn:n (x) ≤ 0. Hence FXn:n is Schur-concave in β and and consequently (βi − βj ) − ∂βi ∂βj therefore the conclusion of the theorem follows. 

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Theorem 3.8. Let X1 , . . . Xn be a set of independent random variables with Xi ∼ Chen(β, λi ), i = 1, . . . , n. Further, let Y1 , . . . , Yn be another set of independent random variables with Yi ∼ Chen(β, λ?i ), i = 1, . . . , n. Then, λ m λ =⇒ Xn:n ≤rh Yn:n . Proof:. The density function of Xn:n is given by " n  # n X b d Y λi (1−ex ) fXn:n = 1−e = Fn:n (x) rXi (x), dx i=1

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i=1

where rXi is the reversed hazard rate function of Xi . Therefore, the reversed hazard rate function of Xn:n is given by n X λi fn:n (x) β = βex xβ−1 . rXn:n (x) = xβ −1) λ (e Fn:n (x) −1 e i Now, let ψ(x, λ, β) =

eλ(e

i=1

λ

xβ −1)

−1

. After simplification and algebraic manipulation we get,

  β β ∂ 2 ψ(x, λ, β) sign xβ λ(ex −1 ) λ(ex −1 ) xβ = {1 − λ + λe − e } + {1 − e 1 + λ − λe } ∂2λ = f1 (x, λ, β) + f2 (x, λ, β). 9

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Since ex ≥ 1 + x ∀x ≥ 0, we have f1 (x, λ, β) ≤ 0. In the proof of Lemma 3.1, we have established that eλ(e

xβ −1 )

β

(1 + λ − λex ) ≤ 1 which implies f2 (x, λ, β) is non-negative. Again, β

f2 (x, λ, β) − f1 (x, λ, β) = λ − λex − eλ(e

xβ −1)

β

(λ − λex ).

xβ

β

of

Now, since λ − λex ≤ 0 and eλ(e −1) ≥ 1, we have f2 (x, λ, β) ≥ f1 (x, λ, β) which implies P β ∂ 2 ψ(x,λ,β) ≥ 0, i.e., ψ(x, λ, β) is convex and hence rXn:n (x) = βex xβ−1 ni=1 ψ(x, λi , β) is Schur∂2λ convex in λ. Thus the assertion of the Theorem follows. 

p ro

3.2. Comparison Results Based on Multivariate Majorization In this subsection, we establish some interesting results comparing survival functions in the context of both series and parallel systems when the component lifetimes are independent and follow Chen distributions with heterogeneity in both the parameters.

Pr e-

Theorem 3.9. Let X1 , X2 be a set of independent random variables with Xi ∼ Chen(βi , λi ), i = 1, 2. Also, let Y1 , Y2 be another set of independent random variables with Yi ∼ Chen(βi? , λ?i ), i = 1, 2. Then,   λ1 λ2 0 1. for ∈ S2 (S2 ), we have β1 β2    ?  λ1 λ2 λ1 λ?2  =⇒ F¯Y1:2 (x) ≥ F¯X1:2 (x), for all x ∈ (0, 1]([1, ∞)); β1? β2? β1 β2

al

and  λ λ2 0 2. for 1 ∈ T2 (T2 ), we have β1 β2    ?  λ1 λ2 λ1 λ?2  =⇒ F¯Y2:2 (x) ≤ F¯X2:2 (x), for all x ∈ (0, 1]([1, ∞)). β1? β2? β1 β2

urn

Proof:.

1. The survival function of X1:2 is given by F¯X1:2 (x) =

2 Y

xβi )

eλi (1−e

.

i=1

Jo

It is easy to see that F¯X1:2 (x) is permutation invariant in λi and βi , hence the condition 1 of Theorem 2.3 is satisfied. The partial derivative of F¯X1:2 (x) with respect to βi and λi ¯ ∂ F¯ 1:2 (x) ¯X (x)λi exβi xβi ln x and ∂ FX1:2 (x) = F¯X (x)(1 − exβi ). Now, λ1 λ2 ∈ = − F are X∂β 1:2 1:2 ∂λi i β1 β2 S2 =⇒ (λ1 − λ2 )(β1 − β2 ) ≤ 0 which implies λ2 ≥ λ1 & β2 ≤ β1 or λ2 ≤ λ1 & β2 ≥ β1 . Here we assume  λ¯2 ≥ λ1 & ¯β2 ≤ β1 , since the proof for the  ¯other case ¯is quite similar. Setting ∂ FX1:2 (x) ∂ FX1:2 (x) ∂ FX1:2 (x) ∂ F 1:2 (x) (β1 − β2 ) − ∂β2 = ζ1 (x) and (λ1 − λ2 ) − X∂λ = ζ2 (x), we get, ∂β1 ∂λ1 2 ζ1 (x) = F¯X1:2 (x)(β1 , β2 )(λ2 xβ2 ex

and

β2

β1

− λ1 xβ1 ex ) ln x;

β2 β1 ζ2 (x) = F¯X1:2 (x)(λ1 − λ2 )(ex − ex ).

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β

β

β

i=1

1−e

λi (1−ex

βi

)



,

p ro

FX2:2 (x) =

n  Y

of

Now, xβ ex is decreasing in β for any fixed x ∈ (0, 1], which implies λ2 xβ2 ex 2 ≥ λ1 xβ1 ex 1 β and consequently, we have ζ1 (x) ≤ 0 for all x ∈ (0, 1]. Also, since ex is decreasing in β for fixed x ∈ (0, 1], we have ζ2 (x) ≤ 0 for all x ∈ (0, 1]. Thus, FX1:2 (x) satisfiescondition  2 of λ1 λ2 0 Theorem 2.3 for x ∈ (0, 1] and hence the result follows. When x ∈ [1, ∞) and ∈ S2 , β1 β2 the proof of the theorem follows similarly and therefore omitted for the sake of conciseness. 2. The distribution function of X2:2 is given by

which is permutation invariant in (λi , βi ). Hence condition (1) of Theorem 2.3 is satisfied. The partial derivatives of FX2:2 (x) with respect to λi and βi are given by β

and

Pr e-

∂FX2:2 (x) ex i − 1 = FX2:2 (x) = FX2:2 (x)φ2 (x, λi , βi ); βi ∂λi eλi (ex −1) − 1 β

∂FX2:2 (x) λ i e x i x βi = FX2:2 (x) ln x = FX2:2 (x)φ1 (x, λi , βi ) ln x. β ∂βi eλi (x i − 1) − 1

Thus, We have,



+ FX2:2 (x)(β1 − β2 )(φ1 (x, λ1 , β1 ) − φ1 (x, λ2 , β2 )) ln x.

 λ1 λ2 Now, ∈ T2 =⇒ λ2 ≥ λ1 & β2 ≤ β1 or λ2 ≤ λ1 & β2 ≥ β1 and λ1 , λ2 ≥ 1. Here we β1 β2 assume, λ2 ≤ λ1 & β2 ≥ β1 , since the proof for the case of other alternative follows similarly. For fixed x ∈ (0, 1], we have from Lemma 3.2,

urn



al

 ∂FX2:2 (x) ∂FX2:2 (x) φ(λ, β) = (λ1 − λ2 ) − ∂λ1 ∂λ2   ∂FX2:2 (x) ∂FX2:2 (x) + (β1 − β2 ) − ∂β1 ∂β2 = FX2:2 (x)(λ1 − λ2 )(φ2 (x, λ1 , β1 ) − φ2 (x, λ2 , β2 ))

φ2 (x, λ2 , β2 ) ≥ φ2 (x, λ2 , β1 ) ≥ φ2 (x, λ1 , β1 ),

Jo

which implies (λ1 −λ2 )(φ2 (x, λ1 , β1 )−φ2 (x, λ2 , β2 )) ≤ 0. Again, from Lemma 3.1, we conclude that (β1 − β2 )(φ1 (x, λ1 , β1 ) − φ1 (x, λ2 , β2 )) ln x ≤ 0 for all x ∈ (0, 1] and consequently, we get ¯ ¯ φ(λ, β) ≤ 0 for all x ∈ (0, 1]. Hence from Theorem 2.3,  we get  FY2:2 (x) ≥ FX2:2 (x) for all λ λ2 0 x ∈ (0, 1]. The proof of the theorem for x ∈ [1, ∞) and 1 ∈ T2 follows similarly. This β1 β2 completes the proof of the theorem. 

We now extend the result of Theorem 3.9 to the case when n > 2. Theorem 3.10. Let X1 , . . . , Xn and Y1 , . . . , Yn be two sets of independent random variables with Xi ∼ Chen(βi , λi ) and Yi ∼ Chen(βi? , λ?i ), i = 1, . . . , n. Then, 11

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  ?    λ1 . . . λn λ1 . . . λ?n λ1 . . . λn 0 1. for ∈ Sn (Sn ) and we = T , we have, β1 . . . β n β1? . . . βn? β1 . . . β n w F¯Y1:n (x) ≥ F¯X1:n (x), for all x ∈ (0, 1]([1, ∞));

of

and   ?    λ1 . . . λn λ1 . . . λ?n λ1 . . . λn 0 2. for ∈ Tn (Tn ) and = T , we have, β1 . . . β n β1? . . . βn? β1 . . . βn w

p ro

F¯Yn:n (x) ≤ F¯Xn:n (x), for all x ∈ (0, 1]([1, ∞)).

β

Pr e-

x Proof:. For Q any fixed x > 0, set ψn (λ, β) = F¯X1:n (x) and ψ(λ, β) = eλ(1−e ) . Then it follows that ψn (λ, β) = ni=1 ψ(λi , βi ). According to Theorem 3.9, we have ψ(λi , βi ) satisfies (2). Hence, the result follows from Theorem 2.4. The proof of the other part of the theorem is similar, and therefore omitted for conciseness.  Recall that the product of finite number of T-transform matrices with same structure is again a T-transform matrix. Hence, the following corollary become evident.

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Corollary 3.11. Let X1 , . . . , Xn be independent random variables with Xi ∼ Chen(βi , λi ), i = 1, . . . , n. Also, let Y1 , . . . , Yn be another set of independent random variables with Yi ∼ Chen(βi? , λ?i ), i = 1, . . . , n.      ?  λ1 . . . λn λ1 . . . λn λ1 . . . λ?n = T . . . Twk , where Twi , i = 1. If ∈ Sn (Sn∗ ) and β1? . . . βn? β1 . . . βn w1 β1 . . . βn 1, . . . , k have the same structure. Then, we have F¯Y1:n (x) ≥ F¯X1:n (x) for all x ∈ (0, 1]([1, ∞)).      ?  λ1 . . . λn λ1 . . . λn λ1 . . . λ?n ∗ 2. If = ∈ Tn (Tn ) and T . . . Twk , where Twi , i = β1 . . . β n β1? . . . βn? β1 . . . βn w1 1, . . . , k have the same structure. Then, we have F¯Yn:n (x) ≤ F¯Xn:n (x) for all x ∈ (0, 1]([1, ∞)).

urn

Since the product of two T-transform matrices of different structures is not necessarily a Ttransform matrix, it is of interest to know if the the result of Corollary 3.11 can be extended to the case when the matrices Twi , i = 1, . . . , k are of different structures. The following result provides an answer.

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Corollary 3.12. Let X1 , . . . , Xn and Y1 , . . . , Yn be two sets of independent random variables with Xi ∼ Chen(βi , λi ) and Yi ∼ Chen(βi? , λ?i ), i = 1, . . . , n.     λ1 . . . λn λ1 . . . λn ∗ 1. If ∈ Sn (Sn ), T . . . Twi ∈ Sn (Sn∗ ) for i = 1, . . . , k − 1, where β1 . . . β n β1 . . . βn w1  ?    λ1 . . . λ?n λ1 . . . λn k ≥ 2; and = T . . . Twk . Then, we have F¯Y1:n (x) ≥ F¯X1:n (x) β1? . . . βn? β1 . . . βn w1 for all x ∈ (0, 1]([1, ∞)).     λ1 . . . λ n λ1 . . . λ n ∗ 2. If ∈ Tn (Tn ), T . . . Twi ∈ Tn (Tn∗ ) for i = 1, . . . , k − 1, where β1 . . . βn β1 . . . βn w1  ?    λ1 . . . λ?n λ1 . . . λn k ≥ 2; and = T . . . Twk . Then, we have F¯Yn:n (x) ≤ F¯Xn:n (x) β1? . . . βn? β1 . . . βn w1 for all x ∈ (0, 1]([1, ∞)).

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Proof:. Here we assume x ∈ (0, 1]. Now, Set (j)

"

(j)

(j)

λ1 . . . λn (j) (j) β1 . . . βn

(j)

#

=



 λ1 . . . λn T . . . Twj , for β1 . . . βn w1

of

j = 1, . . . , k − 1. Let Y1 , . . . , Yn , j = 1, . . . , k − 1 be a set of independent random variables (j) (j) (j) with Yi ∼ Chen(λi , βi ), i = 1, . . . , n and j = 1, . . . k − 1. From the result of Theorem 3.10, (k−1) (1) it follows that F¯X1:n (x) ≤ F¯Y1:n (x) ≤ . . . ≤ F¯Y1:n (x) ≤ F¯Y1:n (x). Hence, the conclusion follows. When x ∈ [1, ∞), the result follows similarly. The other part of the corollary follows similarly and therefore omitted for conciseness. 

Pr e-

p ro

3.3. Some Further Comparison Results Stochastic comparisons of series and parallel systems have been studied extensively in this article when component lifetimes follow heterogeneous Chen distributions. We now wish to compare such systems under some stochastic orders such as convex transform order, star order, and so on. These types of orders were not studied until Kochar and Xu (2009) posed and answered the following question. “Suppose there are two parallel systems, one consisting of independent homogeneous exponential components and the other comprising independent heterogeneous exponential components. Which parallel system ages faster?” Theorem 3.13 (Kochar and Xu (2009)). Let X1 , . . . , Xn be independent exponential RVs with respective hazard rates λ1 , . . . , λn . Let Y1 , . . . , Yn be a random sample of size n from the exponential distribution with common hazard rate λ. Then Xn:n ≥c Yn:n .

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Now we want to investigate whether the result of Theorem 3.13 holds when the component lifetimes follow Chen distribution. Suppose Y1 , . . . , Yn be independent exponential random variables with respective hazard rates λ1 , . . . , λn and Y1∗ , . . . , Yn∗ be a random sample of size n from the exponential distribution with common hazard rate λ. Also let X1 , . . . , Xn be independent r.v.’s with Xi ∼ Chen(β1 , λi ) , i = 1, . . . , n and X1∗ , . . . , Xn∗ be a random sample of size n from Chen distribution with parameters β2 , λ. 1

1

st

st

Jo

urn

∗ ) = Setting h1 (x) = [ln (1 + x)] β1 and h2 (x) = [ln (1 + x)] β2 , we get h1 (Yn:n ) = Xn:n and h2 (Yn:n st ∗ , where = ∗ ). Xn:n denotes equality in the usual stochastic order. Let, if possible, h1 (Yn:n ) ≥c h2 (Yn:n ∗ and hence from Lemma 3.1 of Balakrishnan et al. (2018) From Theorem 3.13 we have Yn:n ≥c Yn:n we can conclude that h1 (x) is a strictly increasing convex function whereas h2 (x) is a strictly increasing concave function. But, h1 (x) and h2 (x) are both strictly increasing concave functions for β1 ≥ 1 and β2 ≥ 1 which leads to a contradiction. Hence convex transform order does not hold in the situation when β1 ≥ 1 and β2 ≥ 1. when β1 , β2 ∈ (0, 1), we are unable to make any conclusion. Now we want to compare ageing characteristics of two parallel systems, one consisting of independent and homogeneous Chen components and the other comprising independent heterogeneous exponential components.

Lemma 3.14. Let X be a nonnegative r.v’s with continuous distribution function F . Then X ≥c h(X) for all strictly increasing concave functions h(x) : R+ → R+ . Proof:. The distribution function of Z = h(X) is FZ (x) = F (h−1 (x)). Now F −1 (FZ (x)) = h−1 (x) is a concave function as h(x) is a strictly increasing concave function. Hence X ≥c h(X).  Theorem 3.15. Let Y1 , . . . , Yn be independent exponential r.v.’s with hazard rates λ1 , . . . , λn respectively and X1∗ , . . . , Xn∗ be a random sample of size n from Chen(β, λ). Then Yn:n ≥c Xn:n for β > 1. 13

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Proof:. Let Y1∗ , . . . , Yn∗ be a random sample of size n from the exponential distribution with common 1

hazard rate λ. Setting h(x) = ln (1 + x) β , we get from Lemma 3.14, ∗ ∗ Yn:n ≥c h(Yn:n ),

of

∗ . since h(x) is a strictly increasing concave function for β > 1. Now Theorem 3.13 implies Yn:n ≥c Yn:n st ∗ )=X Hence Yn:n ≥c Xn:n , since h(Yn:n  n:n . The next corollary is a direct consequence of Theorem 3.15, which enables us to compare parallel systems with Chen components in the sense of other ageing ordering.

Yn:n ≥order Xn:n where ≥order is any one of the orders

and

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Corollary 3.16. Under the assumption of Theorem 3.15, we have

cv(Yn:n ) ≥ cv(Xn:n )

≥DMRL , ≥* , ≥su , ≥NBUE , ≥Lorenz and cv(X) =

√

V ar(X) E(X) .

Pr e-

In the following theorem we obtain a necessary and sufficient condition for the right-spread ordering as well as the increasing convex ordering of two parallel systems. Theorem 3.17. Under the assumption of Theorem 3.15 the following statements are equivalent: (i) Yn:n ≥RS Xn:n (ii) Yn:n ≥icx Xn:n (iii) E(Yn:n ) ≥ E(Xn:n )

urn

al

Proof:. Let (i) holds. Then from Corollary 4.A.32 of Shaked and Shanthikumar (2007), we get for nonnegative r.v.’s, the right-spread order implies the increasing convex order. Hence, (i) implies (ii). Now assuming (ii) holds, we get (iii) immediately. Finally, suppose (iii) holds. Then from Corollary 3.16, we have Yn:n ≥NBUE Xn:n . Now using Theorem 4.3 of Fernandez-Ponce et al. (1998) we get (iii) implies (i).  Suppose Y1 , . . . , Yn be independent exponential r.v.’s with respective hazard rates λ1 , . . . , λn and Y1∗ , . . . , Yn∗ be a random sample of size n from the exponential distribution with common hazard rate λ. Set  1 Pn−k+1 k   X λ(i) n 0   mk (n, λ) = λi1 . . . λik , mk (n, λ) = i=1 k n−k+1 1≤i1 ···≤ik ≤n

.

Jo

m (n, λ) =

1 2n − 1

  Λ3 − Λ1 Λ2 2Λ1 + Λ21 − Λ2

where

Λr =

n X

(λi )r

i=1

Theorem 3.18. let X1 , . . . , Xn be independent r.v.’s with Xi ∼ Chen(β, λi ) , i = 1, . . . , n and X1∗ , . . . , Xn∗ be a random sample of size n from Chen(β, λ). Then for all k = 1, 2, . . . , n the following statements are equivalent: (i) λ ≥ mk (n, λ) ∗ (ii) Xk:n ≥st Xk:n ∗ (iii)Xk:n ≥hr Xk:n

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Proof:. Suppose Y1 , . . . , Yn are independent exponential r.v.’s with respective hazard rates λ1 , . . . , λn and Y1∗ , . . . , Yn∗ be a random sample of size n from the exponential distribution with common hazard rate λ. Now from Theorem 2 of Bon and P˘ alt˘ anea (2006) we have for all k = 1, 2, . . . , n ∗ Yk:n ≥st Yk:n

if and only if λ ≥ mk (n, λ).

(3)

∗ Yk:n ≥hr Yk:n

of

Now, Theorem 1 of Yu (2016)Yu (2016) implies that for all k = 1, 2, . . . , n if and only if λ ≥ mk (n, λ).

(4)

Pr e-

p ro

Suppose (i) holds. Using (3) and Theorem 1.A.3 of Shaked and Shanthikumar (2007) we get 1 1 1 ∗ ) β or, equivalently X ∗ β is an increasing ln (1 + Yk:n ) β ≥st ln (1 + Yk:n k:n ≥st Xk:n as [ln (1 + x)] function for all β > 0. Hence (i) implies (ii). β ∗ β ∗ Let (ii) hold. Then we observe that eXk:n − 1 ≥st eXk:n − 1 or, equivalently Yk:n ≥st Yk:n β using Theorem 1.A.3 of Shaked and Shanthikumar (2007) and the fact that ex − 1 is an increasing function for β > 0. Hence (ii) implies (i) from (3). From Theorem 1.B.1 of Shaked and Shanthikumar (2007) (iii) implies (ii) immediately. Now using Theorem 1.B.2 of Shaked and Shanthikumar (2007) and (4) we prove that (iii) and (i) is equivalent similarly as we have done to show the equivalence of (i) and (ii).  Theorem 3.19. Let X1 , . . . , Xn be independent r.v.’s with Xi ∼ Chen(β, λi ) , i = 1, . . . , n and ∗ if and only if X1∗ , . . . , Xn∗ be a random sample of size n from Chen(β, λ). Then for Xk:n ≤st Xk:n 0 λ ≤ mk (n, λ) for all k = 1, 2, . . . , n. Proof:. Suppose Y1 , . . . , Yn are independent exponential r.v.’s with respective hazard rates λ1 , . . . , λn and Y1∗ , . . . , Yn∗ be a random sample of size n from the exponential distribution with common hazard rate λ. Now Theorem 4.2 of P˘alt˘anea (2011) we have for all k = 1, 2, . . . , n if and only if λ ≤ m0k (n, λ).

al

∗ Yk:n ≤st Yk:n

(5)

urn

Rest part of the proof closely follows the technique used in Theorem 3.18. The details are ommited for the sake of conciseness.  For the special case k = 2, P˘alt˘anea (2008) and Zhao et al. (2009) were able to strengthen the result contained in (5) and so ∗ Y2:n ≤hr, lr Y2:n

if and only if λ ≤ m02 (n, λ).

(6)

Jo

Zhao et al. (2009) also proved that

∗ Y2:n ≥lr Y2:n

.

if and only if λ ≥ m (n, λ).

(7)

In the next theorems we prove results analogous to (6) and (7) in the context of Chen components. Theorem 3.20. let X1 , . . . , Xn be independent r.v.’s with Xi ∼ Chen(β, λi ) , i = 1, . . . , n and X1∗ , . . . , Xn∗ be a random sample of size n from Chen(β, λ). Then the following statements are equivalent: (i) λ ≤ m02 (n, λ) ∗ (ii) X2:n ≤st X2:n ∗ (iii)X2:n ≤hr X2:n ∗ (iv) X2:n ≤lr X2:n 15

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Proof:. The theorem can be proved using similar techniques as in the proof of Theorem 3.18 and hence the details are omitted for the sake of conciseness. 

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Theorem 3.21. let X1 , . . . , Xn be independent r.v.’s with Xi ∼ Chen(β, λi ) , i = 1, . . . , n and ∗ X1∗ , . . . , Xn∗ be a random sample of size n from Chen(β, λ). Then X2:n ≥lr X2:n if and only if . λ ≥m (n, λ).

p ro

Proof:. The proof closely follows the techniques used in the proof of Theorem 3.18. Consequently, the details are omitted for the sake of brevity.  Acknowledgements

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The authors are grateful to the anonymous referee and the Editor for their valuable comments and helpful suggestions which have substantially improved the presentation of the paper. The authors are also grateful to the Council of Scientific and Industrial Research(CSIR) and University Grant Commission(UGC) for financial support. References

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