Ordering properties of order statistics from heterogeneous exponentiated Weibull models

Statistics and Probability Letters

Ordering properties of order statistics from heterogeneous exponentiated Weibull models

Amarjit Kundu

^a

, Shovan Chowdhury

^b,∗

aDepartment of Mathematics, Santipur College, West Bengal, India

bQuantitative Methods and Operations Management Area, Indian Institute of Management, Kozhikode, Kerala, India

a r t i c l e i n f o

Keywords:

Order statistics Majorization

Reversed hazard rate order Likelihood ratio order

a b s t r a c t

In this paper we stochastically compare two parallel systems each having heterogeneous exponentiated Weibull components. These comparisons are made with respect to reversed hazard rate ordering and likelihood ratio ordering. Similar comparisons are also made for two systems with component lives following multiple outlier exponentiated Weibull model.

1. Introduction

Order statistics have a remarkable contribution in both theory and practice. It has a prominent role in statistics, applied probability, reliability theory, operations research, actuarial science, auction theory, hydrology and many other related and unrelated areas. Parallel and series systems are the building blocks of many complex coherent systems in reliability theory.

IfX₁:n

≤

X₂:n

≤ · · · ≤

X_n:ndenote the order statistics corresponding to the random variablesX₁

,

^X2

, . . . ,

^Xn, then the lifetime of a series system corresponds to the smallest order statisticX₁:nand that of a parallel system is represented by the largest order statisticX_n:n. Although different properties of order statistics from homogeneous populations have been studied in detail in the literature, not much work is available for order statistics from heterogeneous populations, due to its complicated nature of expressions. For properties of order statistics for independent and non-identically distributed random variables, one may refer toDavid and Nagaraja(2003). Stochastic comparisons of parallel systems of heterogeneous components with exponential, gamma, Weibull and generalized exponential (GE) lifetimes have been studied by several authors. One may refer toDykstra et al.(1997),Misra and Misra(2013),Zhao and Balakrishnan (2011, 2012),Torrado and Kochar(2015),Kundu et al.(2016) and the references there in.

The three-parameter exponentiated Weibull (EW) distribution was proposed byMudholkar and Srivastava(1993) by exponentiating the two-parameter Weibull distribution and was later analyzed extensively byMudholkar et al. (1995, 1996)andMudholkar and Hutson(1996). The EW distribution is shown to model non-monotone hazard rates, including the bathtub-shaped hazard rate, and is shown to fit many real life situations well, compared to the conventional exponential,

∗Corresponding author.

E-mail addresses:shovanc@iimk.ac.in,meetshovan@gmail.com(S. Chowdhury).

gamma or Weibull models. A random variableXis said to have EW (also known as generalized Weibull) distribution with parameters

(α, β, λ)

, written as EW

(α, β, λ)

, if the distribution function ofXis given by

F

(

^x

) =



1

−

e^−(λx)β

α

,

^x

>

⁰

, λ >

⁰

, β >

⁰

, α >

⁰

,

where

α

^and

β

are the shape parameters and

λ

is the scale parameter. Recently,Fang and Zhang(2015) have nicely com- pared two parallel systems of EW components in terms of usual stochastic order, dispersive order and likelihood ratio order.

In this paper our main aim is to compare two parallel systems in terms of reversed hazard rate order and likelihood ratio order by comparing the parameters

λ

^and

α

when the components are from two heterogeneous EW distributions as well as from the multiple outlier EW distributions. These results generalize similar results for GE distribution (taking

β =

1), generalized Rayleigh distribution (taking

β =

2), Weibull distribution (taking

α =

1) and exponential distribution (taking

β =

1 and

α =

1).

The organization of the paper is as follows. In Section2, we have given the required definitions and some useful lemmas which have been used throughout the paper. In Section3, results related to reversed hazard rate ordering between two order statisticsX_n:nandY_n:nare given under weak majorization of the parameters

α

^and

λ

. We have also shown that there exists likelihood ratio ordering betweenX₂:2andY₂:2under certain arrangements of the parameters. For the case of multiple-outlier EW model, the likelihood ratio ordering betweenX_n:nandY_n:nis also established. Finally, Section4concludes the paper.

Throughout the paper, the word increasing (resp. decreasing) and nondecreasing (resp. nonincreasing) are used interchangeably, and

ℜ

denotes the set of real numbers

{

x

: −∞ <

^x

< ∞}

. We also writea^sign

=

bto mean thataandb have the same sign. For any differentiable functionk

( · )

^{, we writek′}

(

^t

)

to denote the first derivative ofk

(

^t

)

with respect tot. The random variables considered in this paper are all nonnegative. By independence of random variables we mean that they are statistically independent.

2. Preliminaries

For two absolutely continuous random variablesXandY with distribution functionsF

( · )

^andG

( · )

, density functions f

( · )

^andg

( · )

and reversed hazard rate functionsr

( · )

^ands

( · )

respectively,Xis said to be smaller thanYin (i)likelihood ratio order(denoted asX

≤

_lrY), if^g_f(⁽_t^t)₎increases int, and (ii)reversed hazard rate order(denoted asX

≤

_rhrY), if^G_F⁽_(t^t₎⁾increases intor equivalentlyr

(

^t

) ≤

s

(

^t

)

^{for all}t. For more on different stochastic orders, seeShaked and Shanthikumar(2007).

The notion of majorization (Marshall et al., 2011) is essential for the understanding of the stochastic inequalities for comparing order statistics. LetIⁿbe ann-dimensional Euclidean space whereI

⊆ ℜ

. Further, for any two real vectors x

= (

^x1

,

^x2

, . . . ,

^xn

) ∈

Iⁿandy

= (

^y1

,

^y2

, . . . ,

^yn

) ∈

Iⁿ, writex₍₁₎

≤

x₍₂₎

≤ · · · ≤

x_(n)andy₍₁₎

≤

y₍₂₎

≤ · · · ≤

y_(n)as the increasing arrangements of the components of the vectorsxandyrespectively. The following definitions may be found in Marshall et al.(2011).

Definition 2.1. (i) The vectorxis said to majorize the vectory(written asx

≽

^my) if

j



i=1

x_(i)

≤

j



i=1

y_(i)

,

^j

=

1

,

²

, . . . ,

ⁿ

−

1

,

^and

n



i=1

x_(i)

=

n



i=1

y_(i)

.

(ii) The vectorxis said to weakly supermajorize the vectory(written asx

≽

^w y) if

j



i=1

x_(i)

≤

j



i=1

y_(i)

,

^forj

=

1

,

²

, . . . ,

ⁿ

.

(iii) The vectorxis said to weakly submajorize the vectory(written asx

≽

_w y) if

n



i=j

x_(i)

≥

n



i=j

y_(i)

,

^forj

=

1

,

²

, . . . ,

ⁿ

.

Definition 2.2. A function

ψ :

Iⁿ

→ ℜ

is said to be Schur-convex (resp. Schur-concave) onIⁿifx

≽

^m y implies

ψ (

^x

) ≥ (

^resp.

≤ ) ψ (

^y

)

^{for allx}

,

^y

∈

Iⁿ.

We next present two useful lemmas which will be used in the next section to prove our main results.

Lemma 2.1. For x

, β >

^0,

φ(

^y

) =

y

 e^yxβ

−

1

−1

is decreasing in y

>

^0.

The following lemma can be found inMarshall et al.(2011,p. 87) where the parenthetical statements are not given.

Lemma 2.2. Let

ϕ :

Iⁿ

→ ℜ

. Then

(

^a1

,

^a2

, . . . ,

^an

) ≽

_w

(

^b1

,

^b2

, . . . ,

^bn

)

^implies

ϕ(

^a1

,

^a2

, . . . ,

^an

) ≥ (

^resp.

≤ ) ϕ(

^b1

,

^b2

, . . . ,

^bn

)

if, and only if,

ϕ

is increasing (resp. decreasing) and Schur-convex (resp. Schur-concave) on Iⁿ. Similarly,

(

^a1

,

^a2

, . . . ,

^an

) ≽

^w

(

^b1

,

^b2

, . . . ,

^bn

)

^implies

ϕ(

^a1

,

^a2

, . . . ,

^an

) ≥ (

^resp.

≤ ) ϕ(

^b1

,

^b2

, . . . ,

^bn

)

if, and only if,

ϕ

is decreasing (resp. increasing) and Schur-convex (resp. Schur-concave) on Iⁿ.

3. Main results

Notation 3.1. Let us first introduce the following notations which will be used in all the upcoming theorems.

(i) D+

= { (

^x1

,

^x2

, . . . ,

^xn

) :

x₁

≥

x₂

≥ · · · ≥

x_n

>

⁰

}

. (ii)E+

= { (

^x1

,

^x2

, . . . ,

^xn

) :

0

<

^x1

≤

x₂

≤ · · · ≤

x_n

}

.

Fori

=

1

,

²

, . . . ,

^{n, letX}i

∼

EW

(α

i

, β, λ

i

)

^andYi

∼

EW

(θ

i

, β, δ

i

)

be two sets ofnindependent random variables. IfF_n:n

( · )

andG_n:n

( · )

are the distribution functions ofX_n:nandY_n:nrespectively, then

F_n:n

(

^x

) =

n



i=1



1

−

e^−(λix)β

αi

,

and

G_n:_n

(

^x

) =

n



i=1



1

−

e^−(δix)βθi

.

Again, ifr_n:n

( · )

^andsn:n

( · )

are the reversed hazard rate functions ofX_n:nandY_n:nrespectively then, r_n:n

(

^x

) =

n



i=1

α

i

βλ

^βix^β−1

e^(λix)β

−

1

,

^(3.1)

s_n:n

(

^x

) =

n



i=1

θ

i

βδ

^βi x^β−1

e^(δix)β

−

1

.

^(3.2)

The following two theorems show that under certain conditions on parameters, there exists reversed hazard rate ordering betweenX_n:nandY_n:n. Writeα

= (α

1

, α

2

, . . . , α

n

)

^,θ

= (θ

1

, θ

2

, . . . , θ

n

)

^,λ

= (λ

1

, λ

2

, . . . , λ

n

)

^andδ

= (δ

1

, δ

2

, . . . , δ

n

)

^. Theorem 3.1. For i

=

1

,

²

, . . . ,

^{n, let X}iand Y_ibe two sets of mutually independent random variables with X_i

∼

EW

(α

i

, β, λ

i

)

and Y_i

∼

EW

(α

i

, β, δ

i

)

. Further, suppose thatα

∈

_E+andλ,δ

∈

_D+. Then, for all

β >

^0,

λ

≽

^w δ^{implies X}n:n

≥

_rhrY_n:n

.

Proof. Letg_i

(

^y

) =

^{αiyβxβ−1}

e^yxβ

−1 . Differentiatingg_i

(

^y

)

with respect toy, we get g_i^′

(

^y

) = α

i

ψ(

^y

),

where, by Lemma 2.4 ofFang and Zhang(2015),

ψ(

^y

) = β

^xβ−1eyx

β

−

1

−

yx^βe^yxβ

e^yxβ

−

12

is increasing iny

≥

0, for all

β ≥

0. So, for any two real numbersa

≥

b,

ψ(

^a

) ≥ ψ(

^b

)

. Again, fromLemma 2.1it follows that

ψ(

^y

) <

^{0 for ally}

>

^{0. Now, if}α

∈

_E+, then

− α

i

ψ(

^a

) ≤ − α

i+1

ψ(

^b

)

, which implies thatg_i^′

(

^a

) ≥

g_i^′₊₁

(

^b

)

. So, by Proposition H.2 ofMarshall et al.(2011),r_n:n

(

^x

)

is Schur convex. Thus, the result follows fromLemmas 2.1and2.2.

The counterexample given below shows that without the ascending order of the components of the scale parameters and the descending order of the components of the shape parametersTheorem 3.1may not hold, even if the weak majorization betweenλ^andδis replaced by majorization order.

Counterexample 3.1.LetX_i

∼

EW

(α

i

, β, λ

i

)

^andYi

∼

EW

(α

i

, β, δ

i

)

^,i

=

1

,

²

,

^{3. Now, if}

(λ

1

, λ

2

, λ

3

) = (

⁵

,

⁴

,

¹

) ∈

_D+,

(δ

1

, δ

2

, δ

3

) = (

⁴

,

⁴

,

²

) ∈

_D+,

(α

1

, α

2

, α

3

) = (

²

,

⁵⁰

,

¹⁰⁰

) ∈

_E+ and

β =

2 are taken, then fromFig. 2.1it can be concluded thatX₃:3

≥

_rhrY₃:3, whileλ

≽

^m δ. But if we reverse the order of the vectorsλ^andδthen, for sameα^and

β

^, r₃:3

(

⁰

.

⁸

) −

s₃:3

(

⁰

.

⁸

) =

2

.

^{40613 andr}3:3

(

⁰

.

⁵

) −

s₃:3

(

⁰

.

⁵

) = −

22

.

6305 givingX₃:3

̸≥

_rhrY₃:3. It is to be mentioned here that while plotting the curve, the substitutionx

= −

lnyhas been used so thatr₃:3

( −

lny

) −

s₃:3

( −

lny

) =

a

(

^y

)

^{, say.}

Theorem 3.1shows the ordering betweenX_n:nandY_n:nwhenλ^majorizesδkeeping the other parameters same. Now the question arises—what will happen ifα^majorizesθwhile the scale parametersλ^{, and}δare equal? The theorem given below answers this question.

Fig. 2.1. Graph ofa(^y)^.

Theorem 3.2. For i

=

1

,

²

, . . . ,

^{n, let X}iand Y_ibe two sets of mutually independent random variables with X_i

∼

EW

(α

i

, β, λ

i

)

and Y_i

∼

EW

(θ

i

, β, λ

i

)

^{. If} α

≽

^mθ^and

(i) α,θ

∈

_D+andλ

∈

_E+, then X_n:n

≥

_rhrY_n:n; (ii) α,θ,λ

∈

_{D+, then Xn:n}

≤

_rhrYn:n.

Proof. Assumingg

(

^xi

) =

x_i,u_i

=

^λ

β iβ^xβ−1 e(λi x)^β−1

and

ϕ(α) =

n



i=1

u_ig

(α

i

),

and by noting the fact thatu_i

∈

_D+

(

^ui

∈

_E+

)

^whenever

λ

i

∈

_E+

(λ

i

∈

_D+

)

^(byLemma 2.1) and using Theorem 3.1 and Theorem 3.2 ofKundu et al.(2016) it can be proved thatr_n:n

(

^x

)

is Schur convex under condition (i), and Schur concave under condition (ii). This proves the result.

The following theorem follows fromLemma 2.2andTheorem 3.2.

Theorem 3.3. For i

=

1

,

²

, . . . ,

^{n, let X}iand Y_ibe two sets of mutually independent random variables with X_i

∼

EW

(α

i

, β, λ

i

)

and Y_i

∼

EW

(θ

i

, β, λ

i

)

^.

(i) If α,θ

∈

_D+andλ

∈

_E+, thenα

≽

_wθ^{implies X}n:n

≥

_rhrY_n:n. (ii) If α,θ,λ

∈

_D+, thenα

≽

^w θ^{implies X}n:n

≤

_rhrY_n:n.

The following theorem shows that, based on the majorization order of the parameters, the ratio of the reversed hazard rate functions ofX₂:2andY₂:2is monotone.

Theorem 3.4. For i

=

1

,

^{2, let X}iand Y_ibe two sets of mutually independent random variables with X_i

∼

EW

(α

i

, β, λ

i

)

^and Y_i

∼

EW

(α

i

, β, δ

i

)

^{. If} α

∈

_E+,λ,δ

∈

_D+andλ

≽

^mδ, then, for0

< β ≤

1,

r₂:2

(

^x

)

s₂:2

(

^x

)

is increasing in x

.

Proof. Letu

(

^x

) =

^xβ

e^xβ−1. So, r₂:2

(

^x

)

s₂:2

(

^x

) =

α1βλ^β₁xβ−1 e(^λ1x)^β−1

+

^α2βλ

β 2xβ−1 e(^λ2x)^β−1 α1βδ₁^βxβ−1

e(δ1x)^β−1

+

^α2βδ

β 2xβ−1 e(δ2x)^β−1

= α

1u

(λ

1x

) + α

2u

(λ

2x

)

α

1u

(δ

1x

) + α

2u

(δ

2x

) = η(

^x

)(

^say

).

Now,u

(

^x

)

, by Lemma 3.1 ofTorrado and Kochar(2015), is decreasing and convex inx

≥

0, for all 0

< β ≤

1. Again, differentiatingu

(

^x

)

with respect tox, we get

u^′

(

^x

) = β

xu

(

^x

)v(

^x

),

where

v(

^x

) =

1

−

^x

β 1

−

e^−xβ

.

By Lemma 3.2 ofTorrado and Kochar(2015),

v(

^x

)

is decreasing inx. Again, differentiating

η(

^x

)

with respect tox, we get

η

^′

(

^x

)

^sign

=



α

1

λ

1u^′

(λ

1x

) + α

2

λ

2u^′

(λ

2x

)



(α

1u

(δ

1x

) + α

2u

(δ

2x

))

−



α

1

δ

1u^′

(δ

1x

) + α

2

δ

2u^′

(δ

2x

)



(α

1u

(λ

1x

) + α

2u

(λ

2x

)) .

Thus, it is clear that

η(

^x

)

is increasing inx

≥

0 if

Ψ

(λ

1

, λ

2

) = α

1

λ

1u^′

(λ

1x

) + α

2

λ

2u^′

(λ

2x

) α

1u

(λ

1x

) + α

2u

(λ

2x

) = β

x

α

1u

(λ

1x

)v(λ

1x

) + α

2u

(λ

2x

)v(λ

2x

) α

1u

(λ

1x

) + α

2u

(λ

2x

)

is Schur-convex in

(λ

1

, λ

2

)

^{. Writing}

w(

^x

) =

u

(

^x

)v

^′

(

^x

)( ≤

0

)

^{, we get}

∂

Ψ

∂λ

1

− ∂

Ψ

∂λ

2 sign

=



(α

1

α

2

(v(λ

1x

) − v(λ

2x

)))



u^′

(λ

1x

)

^u

(λ

2x

) +

u^′

(λ

2x

)

^u

(λ

1x

)



+ ((α

1u

(λ

1x

) + α

2u

(λ

2x

))) (α

1

w(λ

1x

) − α

2

w(λ

2x

))

^]

.

^(3.3) Again, by Lemma 3.4 ofTorrado and Kochar(2015),

w(

^x

) = β

^e2β−1exβ

x^β

+

1

−

e^xβ



e^xβ

−

13

is increasing inx

≥

0 for all 0

< β ≤

1.

Now, asλ

∈

_D+and

v(

^x

)

is decreasing inx, it can be written that

v(λ

1x

) − v(λ

2x

) ≤

0 for all

λ

1

≥ λ

2. Thus, by noting the fact thatu

(

^x

)

is decreasing inx, it can be shown that the first term of(3.3)is nonnegative. Again, forα

∈

_E+,

α

1

w(λ

1x

) ≥ α

2

w(λ

2x

)

. Hence, for allx

≥

0, the second term of(3.3)is also nonnegative. Thus, by Lemma 3.1 ofKundu et al.

(2016),Ψ is Schur convex inλ^{. Thus,}λ

≽

^mδ^gives

η

^′

(

^x

) ≥

0, proving the result.

Kundu et al.(2016) have shown that, for generalized exponential distribution, although there does not exist lr ordering betweenX₃:3andY₃:3whenδis majorized byλ, the result is true forn

=

2.Corollary 3.1shows that the same can be concluded for EW distribution.

Corollary 3.1. For i

=

1

,

^{2, let X}iand Y_ibe two sets of mutually independent random variables with X_i

∼

EW

(α

i

, β, λ

i

)

^and Y_i

∼

EW

(α

i

, β, δ

i

)

^{. If} α

∈

_E+,λ,δ

∈

_D+and0

< β ≤

1, then

λ

≽

^mδ^{implies X}2:2

≥

_lrY₂:2

.

Proof. Iff₂:2

( · )

^andg2:2

( · )

denote the density functions ofX₂:2andY₂:2respectively, then f₂:2

(

^x

)

g₂:2

(

^x

) =

^F2:2

(

^x

)

G₂:2

(

^x

) .

^r2:2

(

^x

)

s₂:2

(

^x

)

is increasing inx

.

This is because the first ratio is increasing inxbyTheorem 3.1and the second ratio is increasing inxbyTheorem 3.4.

The next theorem shows that, ifα^majorizesθ^andλis kept fixed, then depending upon certain conditions onα,θ^andλ^,

r2:2(x)

s2:2(x)will be monotone.

Theorem 3.5. For i

=

1

,

²

, . . . ,

^{n, let X}iand Y_ibe two sets of mutually independent random variables with X_i

∼

EW

(α

i

, β, λ

i

)

and Y_i

∼

EW

(θ

i

, β, λ

i

)

. Further, suppose thatα

≽

^mθ^{. Then}

(i) ^r_s^2:2(x)

2:2(x)is increasing in x if α,θ

∈

_D+andλ

∈

_E+. (ii) ^r_s^2:2(x)

2:2(x)is decreasing in x if α,θ,λ

∈

_D+. Proof. FollowingTheorem 3.4, we have

r₂:2

(

^x

)

s₂:2

(

^x

) = α

1u

(λ

1x

) + α

2u

(λ

2x

)

θ

1u

(λ

1x

) + θ

2u

(λ

2x

) = η

1

(

^x

)(

^say

),

whereu

(

^x

)

is as defined inTheorem 3.4. Differentiating

η

1

(

^x

)

with respect tox, we get

η

^′1

(

^x

)

^sign

=



α

1

λ

1u^′

(λ

1x

) + α

2

λ

2u^′

(λ

2x

)



(θ

1u

(λ

1x

) + θ

2u

(λ

2x

))

−



θ

1

λ

1u^′

(λ

1x

) + θ

2

λ

2u^′

(λ

2x

)



(α

1u

(λ

1x

) + α

2u

(λ

2x

)) .

Fig. 2.2. Graph ofb(^y)^.

To prove that

η

^′1

(

^x

) ≥ ( ≤ )

^{0 in}x, we have to show that Ψ

(α

1

, α

2

) = β

x

α

1u

(λ

1x

)v(λ

1x

) + α

2u

(λ

2x

)v(λ

2x

) α

1u

(λ

1x

) + α

2u

(λ

2x

)

is Schur-convex (Schur-concave) in

(α

1

, α

2

)

^with

v(

^x

)

as defined earlier. Now,

∂

Ψ

∂α

1

− ∂

Ψ

∂α

2

sign

= (α

1

+ α

2

) (v(λ

1x

) − v(λ

2x

))

^u

(λ

1x

)

^u

(λ

2x

).

Thus, ifλ

∈

_E+then_∂α^∂Ψ

1

−

_∂λ^∂Ψ

2

≥

0. Now, using Lemma 3.1 ofKundu et al.(2016) and noting the fact thatα

∈

_D+, it can be proved thatΨis Schur-convex inα. Again, ifλ

∈

_D+we see that_∂α^∂Ψ

1

−

_∂λ^∂Ψ

2

≤

0. So, ifα

∈

_D+then by Lemma 3.1 ofKundu et al.(2016) it can be concluded thatΨ is Schur-concave inα^{. So, if}α,θ

∈

_D+andλ

∈

_E+, then^r_s^2:2(x)

2:2(x) is increasing inxand ifα,θ,λ

∈

_D+, then^r_s^2:2(x)

2:2(x)is decreasing inx.

The following counterexample shows thatTheorem 3.5does not hold forn

≥

3.

Counterexample 3.2. LetX_i

∼

EW

(α

i

, β, λ

i

)

^andYi

∼

EW

(θ

i

, β, λ

i

)

^,i

=

1

,

²

,

^{3, where}

(λ

1

, λ

2

, λ

3

) = (

⁵⁰⁰

,

⁰

.

⁰¹

,

⁰

.

⁰⁰¹

)

∈

_D+,

(α

1

, α

2

, α

3

) = (

¹

,

⁵

,

⁶

) ∈

_E+,

(θ

1

, θ

2

, θ

3

) = (

²

,

⁵

,

⁵

) ∈

_E+and

β =

0

.

4. Clearly,

(α

1

, α

2

, α

3

) ≽

^m

(θ

1

, θ

2

, θ

3

)

^{. By} substitutingx

= −

lny, we see fromFig. 2.2that^r_s^3:3(x)

3:3(x)

=

b

(

^y

)

is not monotone.

Corollary 3.2. For i

=

1

,

^{2, let X}iand Y_ibe two sets of mutually independent random variables with X_i

∼

EW

(α

i

, β, λ

i

)

^and Y_i

∼

EW

(θ

i

, β, λ

i

)

. Further, suppose thatα

≽

^mθ^{. Then}

(i) X₂:2

≥

_lrY₂:2if α,θ

∈

_D+andλ

∈

_E+. (ii) X₂:2

≤

_lrY2:2if α,θ,λ

∈

_D+.

Proof. UsingTheorems 3.2and3.5and following similar argument as in the proof ofCorollary 3.1, the results follow.

AlthoughTheorem 3.4does not hold forn

≥

3, the following theorem shows that in case of multiple-outlier model the result is true for any positive integern.

Theorem 3.6. For i

=

1

,

²

, . . . ,

^{n, let X}i and Y_i be two sets of mutually independent random variables each following the multiple-outlier EW model such that X_i

∼

EW

(α, β, λ)

^{and Y}i

∼

EW

(α, β, δ)

^{for i}

=

1

,

²

, . . . ,

ⁿ1, X_i

∼

EW

(α

^∗

, β, λ

^∗

)

and Y_i

∼

EW

(α

^∗

, β, δ

^∗

)

^{for i}

=

n₁

+

1

,

ⁿ1

+

2

, . . . ,

ⁿ1

+

n₂

( =

n

)

^{. For0}

< β ≤

1, if

(λ, λ, . . . , λ,

  

n1

λ

^∗

, λ

^∗

, . . . , λ

^∗

  

n2

) ≽

^m

(δ, δ, . . . , δ,

  

n1

δ

^∗

, δ

^∗

, . . . , δ

^∗

  

n2

)

and

α ≤ α

^∗

, λ ≥ λ

^∗,

δ ≥ δ

^{∗, then X}n:n

≥

_lrY_n:n. Proof. Using(3.1)and(3.2)we get,

r_n:n

(

^x

)

s_n:n

(

^x

) =

n



i=1

αiβλ^β_ixβ−1 e(λi x)^β−1 n



i=1 αiβδ_i^βxβ−1

e(δi x)^β−1

=

n



i=1

α

iu

(λ

ix

)

n



i=1

α

iu

(δ

ix

)

= η

2

(

^x

)(

^say

),

where

λ

i

= λ

^,

δ

i

= δ

^,

α

i

= α

^fori

=

1

,

²

, . . . ,

ⁿ1and

λ

i

= λ

^∗,

δ

i

= δ

^∗,

α

i

= α

^∗fori

=

n₁

+

1

,

ⁿ1

+

2

, . . . ,

ⁿ1

+

n₂

=

nand u

(

^x

)

is as defined earlier. So, in view ofTheorem 3.1we need only to prove that

η

2

(

^x

)

is increasing inx. Note that

η

^′2

(

^x

)

^sign

=

 _n



i=1

α

i

λ

iu^′

(λ

ix

)

  _n



i=1

α

iu

(δ

ix

)



−

 _n



i=1

α

i

δ

iu^′

(δ

ix

)

  _n



i=1

α

iu

(λ

ix

)



.

Thus,

η

2

(

^x

)

is increasing inx, if

Ψ

(λ,

^x

) =

n



i=1

α

i

λ

iu^′

(λ

ix

)

n



i=1

α

iu

(λ

ix

)

= β

x

n



i=1

α

iu

(λ

ix

)v(λ

ix

)

n



i=1

α

iu

(λ

ix

)

is Schur-convex inλ^{, where}

v(

^x

)

is as defined earlier. Further,

∂

Ψ

∂λ

i

= α

i



w(λ

ix

)



n₁

α

^u

(λ

^x

) +

n₂

α

^∗u

(λ

^∗x

)



+

n₁

α

^u

(λ

^x

)

^u′

(λ

ix

) (v(λ

ix

) − v(λ

^x

)) +

n₂

α

^∗u

(λ

^∗x

)

^u′

(λ

ix

)



v(λ

ix

) − v(λ

^∗x

)



· β

x

_n



i=1

α

iu

(λ

ix

)

2

,

where

w(

^x

)

is as defined inTheorem 3.4.

Now, three cases may arise:

Case(i) 1

≤

i

<

^j

≤

n₁. Here

α

i

= α

j

= α

^and

λ

i

= λ

j

= λ

^{, so that}

∂

Ψ

∂λ

i

= α



w(λ

^x

)



n₁

α

^u

(λ

^x

) +

n₂

α

^∗u

(λ

^∗x

)



+

n₂

α

^∗u

(λ

^∗x

)

^u′

(λ

^x

)



v(λ

^x

) − v(λ

^∗x

)



· β

x

_n



i=1

α

iu

(λ

ix

)

2

= ∂

Ψ

∂λ

j

,

giving^∂_∂λ^Ψ

i

−

^∂_∂λ^Ψ

j

=

0.

Case(ii) Ifn₁

+

1

≤

i

<

^j

≤

n,i.e.if

α

i

= α

j

= α

^∗and

λ

i

= λ

j

= λ

^{∗, then}

∂

Ψ

∂λ

i

= α

^∗

w(λ

^∗x

)



n₁

α

^u

(λ

^x

) +

n₂

α

^∗u

(λ

^∗x

)



+

n₁

α

^u

(λ

^x

)

^u′

(λ

^∗x

)



v(λ

^∗x

) − v(λ

^x

)



· β

x

_n



i=1

α

iu

(λ

ix

)

2

= ∂

Ψ

∂λ

j

,

giving^∂_∂λ^Ψ

i

−

^∂_∂λ^Ψ

j

=

0.

Case(iii) If 1

≤

i

≤

n₁andn₁

+

1

≤

j

≤

n, then

α

i

= α

^,

λ

i

= λ

^and

α

j

= α

^∗,

λ

j

= λ

^{∗. So,}

∂

Ψ

∂λ

i

− ∂

Ψ

∂λ

j

=



n₁

α

^u

(λ

^x

) +

n₂

α

^∗u

(λ

^∗x

)

 

αw(λ

^x

) − α

^∗

w(λ

^∗x

)



+



v(λ

^x

) − v(λ

^∗x

)

 

n₁

α

^u

(λ

^x

)

^u′

(λ

^∗x

) +

n₂

α

^∗u

(λ

^∗x

)

^u′

(λ

^x

)



· β

x

_n



i=1

α

iu

(λ

ix

)

2

.

Now, if

λ ≥ λ

^∗and

α ≤ α

^∗, we have, for allx

≥

_0,

αw(λ

^x

) − α

^∗

w(λ

^∗x

) ≥

0

.

Again, as

v(

^x

)

^andu

(

^x

)

are decreasing inx, we have_∂λ^∂Ψ

i

−

^∂_∂λ^Ψ

j

≥

0 for allx. Hence, by Lemma 3.1 ofKundu et al.(2016),Ψ ^is Schur-convex.

Theorem 3.7. For i

=

1

,

²

, . . . ,

^{n, let X}iand Y_ibe two sets of independent random variables each following the multiple-outlier EW model such that X_i

∼

EW

(α, β, λ)

^{and Y}i

∼

EW

(θ, β, λ)

^{for i}

=

1

,

²

, . . . ,

ⁿ1, X_i

∼

EW

(α

^∗

, β, λ

^∗

)

^{and Y}i

∼

EW