Hazard rate and reversed hazard rate orders on extremes of heterogeneous and dependent random variables

Accepted Manuscript Hazard rate and reversed hazard rate orders on extremes of heterogeneous and dependent random variables Chen Li, Xiaohu Li

PII: DOI: Reference:

S0167-7152(18)30352-3 https://doi.org/10.1016/j.spl.2018.11.005 STAPRO 8367

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Statistics and Probability Letters

Received date : 29 March 2018 Revised date : 2 November 2018 Accepted date : 3 November 2018 Please cite this article as: C. Li and X. Li, Hazard rate and reversed hazard rate orders on extremes of heterogeneous and dependent random variables. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.11.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.

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Hazard Rate and Reversed Hazard Rate Orders on Extremes of Heterogeneous and Dependent Random Variables Chen Li∗ School of Science Tianjin University of Commerce, Tianjin 300134, China Xiaohu Li Department of Mathematical Sciences Stevens Institute of Technology, Hoboken NJ 07030, USA November 2, 2018

Abstract We develop sufficient conditions for the hazard rate order on minimums of sample with Archimedean survival copulas and having proportional hazard rates or scales, and the reversed hazard rate order on maximums of sample with Archimedean copulas and have proportional reversed hazard rates, respectively. Also, we present applications of the hazard rate order on sample minimums in engineering reliability and actuarial risk. Keywords: Archimedean copula; Decreasing hazard rate; Proportional hazard rates; Scale model; Weak supermajorization

1

Introduction

The order statistic Xk:n denotes the k-th smallest of random variables X1 , · · · , Xn , k = 1, · · · , n. The sample extremes X1:n and Xn:n are particularly useful in many applied areas. For example, in reliability theory, Xk:n characterizes the lifetime of a (n−k +1)-out-of-n system, which works when at least n−k +1 of all the n components function normally and thus includes series system X1:n and parallel system Xn:n as special cases. In actuarial science, X1:n and Xn:n define the smallest and the largest potential losses ∗ The

corresponding author. E-mail: [email protected].

1

concerned with a policy of multiple random risks (see Barmalzan et al. (2017)). In economics, X1:n and Xn:n represent the winner’s price in the first price Dutch auction and English auction (see Fang and Li (2015)), respectively. During the past decades, researchers have done a lot on stochastic comparisons of order statistics from heterogeneous samples in the literature. For real vectors x = (x1 , · · · , xn ) and y = (y1 , · · · , yn ) in Rn , denote x(1) ≤ · · · ≤ x(n) the increasing arrangement of x1 , · · · , xn . Then, x is said to be (i) majorized m ∑n ∑n ∑j ∑j by y (denoted as x ≼ y) if i=1 xi = i=1 yi and i=1 x(i) ≥ i=1 y(i) for j ∈ In−1 , and (ii) weakly ∑j ∑j supermajorized by y (denoted as x ≼w y) if i=1 x(i) ≥ i=1 y(i) for j ∈ In , where In = {1, · · · , n}. m

It is clear that x ≼ y implies x ≼w y for any x, y ∈ Rn . For more details on the above partial orders, we refer readers to Marshall et al. (2011). For random variables X, Y with distribution functions F and

¯ = 1 − G, X is said to be smaller than Y in the (i) usual stochastic order G, denote F¯ = 1 − F and G ¯ ¯ (denoted as X ≤st Y ) if F¯ (t) ≤ G(t) for any t, (ii) hazard rate order (denoted as X ≤hr Y ) if G(t)/ F¯ (t) is increasing in t, and (iii) reversed hazard rate order (denoted as X ≤rh Y ) if G(t)/F (t) is increasing in t. For more on these stochastic orders one may refer to Shaked and Shanthikumar (2007) and Li and Li (2013). Due to the mathematical tractability, the mutual independence among concerned random samples are assumed in most of the existing references. Denote W(α, λ) the Weibull distribution with survival α function F¯ (x) = e−(λx) for x ≥ 0. In particular, for Xi ∼ W(α, λi ) with i ∈ In and Yi ∼ W(α, µi ) with

i ∈ In , both independent, Khaledi and Kochar (2006) proved that m

λ = (λ1 , · · · , λn ) ≼ (µ1 , · · · , µn ) = µ

=⇒

X1:n ≤hr Y1:n

whenever 0 < α ≤ 1.

(1.1)

To study the heterogeneity in nonparametric context, statisticians introduced the following three models for sample X = (X1 , · · · , Xn ) associated with a real vector λ. (i) X follows the proportional hazard rates (PHR) model if Xi ∼ Fi = 1 − F¯ λi , for λi > 0, i ∈ In , where F¯ is the baseline survival function and λ is the frailty vector. The PHR model includes, for example, exponential and Lomax distribution with baseline survival functions e−x and (1 + αx)−1 , respectively. For more on PHR models please refer to Kumar and Klefsj¨o (1994) and references therein. (ii) X follows the proportional reversed hazard rates (PRH) model if Xi ∼ Fi = F λi for λi > 0 with i ∈ In , where F is the baseline distribution and λ is the resilience vector. The PRH model includes Fr´echet distribution F(α, λi ) and exponentiated −α

Weibull distribution ∼ EW(α, β, λi ) with baseline distribution e−x

β

and 1 − e−(αx) , respectively. For

more on PRH models, one may refer to Gupta and Gupta (2007), Li and Li (2008), and Li et al. (2010). (iii) X follows the scale model if Xi ∼ F (λi x), F is the baseline distribution and λ is the scale vector. The

2

scale model includes for example Weibull distribution W(α, λi ) and Gamma distribution Xi ∼ G(α, λi ) ∫x α with baseline distribution 1 − e−x and 0 uα−1 e−u du/Γ(α), respectively. We refer readers to Khaledi et al. (2011), Li et al. (2016) and Hazra et al. (2017, 2018) for recent advances on the scale model.

As a generalization of (1.1), Khaledi et al. (2011) showed that, for X = (X1 , · · · , Xn ) and Y = (Y1 , · · · , Yn ) following scale models with baseline hazard rate h and scale vectors λ, µ, respectively, m

λ≼µ

=⇒

X1:n ≤hr Y1:n

whenever x2 h′ (x) is decreasing in x.

(1.2)

For a comprehensive review on ordering of order statistics one may refer to Balakrishnan and Zhao (2013). For a random vector X with distribution function F , survival function F¯ , univariate marginal distributions F1 , · · · , Fn and respective survival functions F¯1 , · · · , F¯n , if there exists C(u) : [0, 1]n 7→ [0, 1] ( ) b b F¯1 (x1 ), · · · , F¯n (xn ) and C(u) : [0, 1]n 7→ [0, 1] such that F (x) = C(F1 (x1 ), · · · , Fn (xn )) and F¯ (x) = C b are called the copula and the survival copula of X, respecfor all xi with i ∈ In , then C and C tively. For a n-monotone function ψ : [0, +∞) → [0, 1] with ψ(0) = 1 and

lim ψ(x) = 0, then

x→+∞

Cψ (u) = ψ(ϕ(u1 ) + · · · + ϕ(un )) for all ui ∈ (0, 1), i ∈ In is called an Archimedean copula with generator ψ, where ϕ = ψ −1 = sup{x ∈ R : ψ(x) > u} is the right continuous inverse of ψ. Archimedean copulas cover a wide range of dependence structures including the independence copula with generator e−t . In recent studies on ordering of order statistics, statistical dependence is further taken into account through equipping the sample with a copula. For example, X is equipped with the Archimedean • survival copula with generator ψ and follows PHR model with baseline survival function F¯ and ( ∑n ) ¯ λi frailty vector λ, denoted as X ∼ PHR(F¯ , λ, ψ), i.e., X gets survival function ψ i=1 ϕ(F (xi )) ,

• copula with generator ψ and follows PRH model with baseline distribution F and resilience vector ( ∑n ) λi λ, denoted as X ∼ PRH(F, λ, ψ), i.e., X has distribution function ψ i=1 ϕ(F (xi )) , and • survival copula with generator ψ and follows the scale model with baseline survival function F¯ and ( ∑n ) scale vector λ, denoted as X ∼ S(F¯ , λ, ψ), i.e., X has survival function ψ ϕ(F¯ (λi xi )) . i=1

Navarro and Spizzichino (2010) studied stochastic order of lifetimes of series and parallel systems with component lifetimes sharing a common copula, Rezapour and Alamatsaz (2014) obtained stochastic order on the (n − k + 1)-out-of-n system lifetimes with component lifetimes linked by Archimedean survival copulas, and Li and Fang (2015) investigated the stochastic order of the maximums of PHR samples with Archimedean copulas. Particularly, for Xi ∼ W(α, λi ) with i ∈ In and Yi ∼ W(α, µi ) with i ∈ In , both

3

sharing Archimedean survival copula with generator ψ, Li and Li (2015, Corollary 4.3) announced that λ ≼w µ =⇒ X1:n ≤st Y1:n

whenever ψ is log-concave and 0 < α ≤ 1,

(1.3)

Subsequently, for X ∼ PHR(F¯ , λ, ψ), Y ∼ PHR(F¯ , µ, ψ), Fang et al. (2016, Theorem 4.1) proved that λ ≼w µ

=⇒

X1:n ≤st Y1:n

whenever ψ is log-concave,

(1.4)

and for X ∼ PRH(F, λ, ψ), Y ∼ PRH(F, µ, ψ), Fang et al. (2016, Theorem 5.1) showed that λ ≼w µ

=⇒

Xn:n ≥st Yn:n

whenever ψ is log-concave.

(1.5)

Lately, for X ∼ S(F¯ , λ, ψ) and Y ∼ S(F¯ , µ, ψ) with the baseline having decreasing hazard rate (DHR), Li et al. (2016) got the following generalization of (1.3), λ ≼w µ =⇒ X1:n ≤st Y1:n

whenever ψ is log-concave.

(1.6)

To the best of our knowledge, except for Li and Li (2018) there is no ongoing research on hazard rate and reversed hazard rate orders on sample extremes from random variables with Archimedean copulas. Due to the popular application in reliability and actuarial risks, it is of both practical and theoretical interest to probe into them through a detailed study. The remaining part of this manuscript is rolled out as follows: Section 2 recalls some concerned notions, including definitions of aging properties, stochastic orders, majorization and copula etc., and introduces several lemmas. In Section 3 we develop the hazard rate and the reversed hazard rate orders for sample extremes from heterogeneous random variables coupled by Archimedean copulas. Also, some applications of the main results in actuarial risk and reliability are presented in Section 4. For convenience, from now on we denote R = (−∞, +∞), R+ = (0, +∞), real vectors λ = (λ1 , · · · , λn ) and µ = (µ1 , · · · , µn ), and random vectors X = (X1 , · · · , Xn ) and Y = (Y1 , · · · , Yn ). Let F¯ = 1 − F for distribution F and denote f , h and r the corresponding density, hazard rate and the reversed hazard rate functions. Denote ei = (0, · · · , 0, 1, 0, · · · , 0) for i ∈ In . Throughout this note, all random variables | {z } | {z } i

n−i

are assumed to be nonnegative and absolutely continuous, and the terms increasing and decreasing mean nondecreasing and nonincreasing, respectively.

4

2

Some preliminaries

For ease of reference, here we present several technical lemmas to be utilized in the coming sections. Lemma 2.1 If H(x) ∈ [0, 1] and ψ is log-concave, then ϕ(H λ (x)) is increasing and concave in λ > 0. Proof: It is straightforward and hence omitted for brevity. m

A real function ~ defined on A ⊆ Rn is said to be Schur-convex (Schur-concave) on A if x ≼ y on A implies ~(x) ≤ (≥) ~(y). Clearly, ~ is Schur-concave on A if and only if −~ is Schur-convex. For more details on Schur-convexity and Schur-concavity we refer readers to Marshall et al. (2011). ∑ n ψ′ ( n u ) ∑ ψ(ui ) ln ψ(ui ) ∑ni=1 i is decreasing in ψ( i=1 ui ) ψ ′ (ui ) i=1 log-concave and ψ ψln′ ψ is increasing and concave.

Lemma 2.2 ℓ1 (u) = w.r.t. u if ψ is

us for each s ∈ In and Schur-convex

Proof: Clearly, ℓ1 (u) is symmetric. For s, t ∈ In with s ̸= t and ui ∈ [0, 1] with i ∈ In , ( ∑n ) n ( ∑n ) ∑ ψ(ui ) ln ψ(ui ) ψ ′ ∂ℓ1 (u) ∂ ψ′ ∂ ψ(us ) ln ψ(us ) i=1 ui i=1 ui ( ∑n ) · ) · = + ( ∑n , ′ (u ) ∂us ∂us ψ ψ ∂u ψ ′ (us ) u ψ u i s i i i=1 i=1 i=1 ( ∑n ) ( ∑n ) ui ui ∂ ψ′ ∂ ψ′ ( ∑ni=1 ) = ( ∑ni=1 ) . ∂us ψ ∂ut ψ i=1 ui i=1 ui Since ψ is log-concave, ψ ′ /ψ is decreasing and have

∂ ψ(us ) ln ψ(us ) ∂us ψ ′ (us )

that

∂ℓ1 (u) ∂us

′ ∑n ∂ ψ (∑ i=1 ui ) ∂us ψ( n i=1 ui )

(2.1) (2.2)

≤ 0. Due to the increasing (ψ ln ψ)/ψ ′ , we

≥ 0. Note that ψ ′ (x)/ψ(x) ≤ 0 and [ψ(x) ln ψ(x)]/ψ ′ (x) ≥ 0. By (2.1) we conclude

≤ 0. Then, ℓ1 (u) is decreasing in us for any s ∈ In . Also, since (ψ ln ψ)/ψ ′ is concave, it

holds that, for us , ut ∈ [0, 1] with s, t ∈ In and s ̸= t, (us − ut ){∂[ψ(us ) ln ψ(us )/ψ ′ (us )]/∂us − ∂[ψ(ut ) ln ψ(ut )/ψ ′ (ut )]/∂ut } ≤ 0.

(2.3)

In combination with (2.1), (2.2) and (2.3), we have, for s, t ∈ In , s ̸= t, and us , ut ∈ [0, 1], ( ∑n )[ ] ] [ ψ′ ∂ ψ(us ) ln ψ(us ) ∂ ψ(ut ) ln ψ(ut ) ∂ℓ1 (u) ∂ℓ1 (u) i=1 ui ( ) ∑n − = (us − ut ) − ≥ 0. (us − ut ) ∂us ∂ut ∂us ψ ′ (us ) ∂ut ψ ′ (ut ) ψ i=1 ui

Thus, as per Theorem 3.A.4 of Marshall et al. (2011), ℓ1 (u) is Schur-convex w.r.t. u. Lemma 2.3 If λ ≼w µ, ψ is log-concave and

ψ ln ψ ψ′

is increasing and concave, then

( ) ( ) ℓ1 ϕ(H λ1 (x)), · · · , ϕ(H λn (x)) ≤ ℓ1 ϕ(H µ1 (x)), · · · , ϕ(H µn (x)) , for any H(x) ∈ [0, 1].

5

(2.4)

Proof: By log-concave ψ and Lemma 2.1, ϕ(H λ (x)) is increasing and concave in λ > 0. As per Marshall et ( ) ( ) al. (2011, Theorem 5.A.2), λ ≼w µ implies ϕ(H λ1 (x)), · · ·, ϕ(H λn (x)) ≼w ϕ(H µ1 (x)), · · ·, ϕ(H µn (x)) .

Also,

ψ ln ψ ψ′

is increasing and concave. From Lemma 2.2 it follows that ℓ1 (u) is decreasing in us for any

s ∈ In and Schur-convex. Thus, in accordance with Theorem 3.A.8 of Marshall et al. (2011), λ ≼w µ further implies (2.4). Lemma 2.4 If ψ is log-concave and F is DHR, then, f (λx)/ψ ′ (ϕ(F¯ (λx))) is increasing in λ > 0. Proof: Since ϕ and F¯ are both decreasing, ϕ(F¯ (λx)) is increasing in λ > 0. Also, F is DHR, then f (λx)/F¯ (λx) is nonnegative and decreasing in λ > 0. By the log-concave ψ, we have ψ/ψ ′ is non-positive and increasing, and thus

3

f (λx) ψ ′ (ϕ(F¯ (λx)))

=

f (λx) F¯ (λx)

·

ψ(ϕ(F¯ (λx))) ψ ′ (ϕ(F¯ (λx)))

is increasing in λ.

Main results

Now, we are ready to present the hazard rate order and the reversed hazard rate order on the maximum and minimum of samples with heterogeneous and statistically dependent observations. Theorem 3.1 For X ∼ PHR(F¯ , λ, ψ) and Y ∼ PHR(F¯ , µ, ψ), if ψ is log-concave and

ψ ln ψ ψ′

is increasing

and concave, then λ ≼w µ implies X1:n ≤hr Y1:n . ∑n Proof: X1:n has survival function F¯1:n (x) = ψ( i=1 ϕ(F¯ λi (x))) and thus hazard rate, for x ≥ 0, ( ∑n ( λ )) n ¯ i ∑ λi F¯ λi (x)h(x) ( ) ψ′ h(x) i=1 ϕ F (x) ( )) ( ( )) = h1:n (x) = ( ∑n ℓ1 ϕ(F¯ λ1 (x)), · · · , ϕ(F¯ λn (x)) . λ ′ λ ¯ ¯ ¯ i i ln F (x) ψ ψ ϕ F (x) i=1 ϕ F (x) i=1

(3.1)

( ) ˜ 1:n (x) = [h(x)/ ln F¯ (x)]ℓ1 ϕ(F¯ µ1 (x)), · · · , ϕ(F¯ µn (x)) for x ≥ 0. Similarly, Y1:n has the hazard rate h Since λ ≼w µ, ψ is log-concave and

ψ ln ψ ψ′

is increasing and concave, by Lemma 2.3 we have

( ) ( ) ℓ1 ϕ(F¯ λ1 (x)), · · · , ϕ(F¯ λn (x)) ≤ ℓ1 ϕ(F¯ µ1 (x)), · · · , ϕ(F¯ µn (x)) .

(3.2)

˜ 1:n (x) for all x. That is X1:n ≤hr Y1:n . Based on (3.1), (3.2) and ln F¯ (x) ≤ 0, we have h1:n (x) ≥ h As an extension of (1.4), Theorem 3.1 improves the usual stochastic order on the sample minimum to the hazard rate order at the cost of imposing increasing and concave

ψ ln ψ ψ′

on the log-concave generator ψ,

which is fulfilled by most members of Archimedean family. For example, independence copula, Gumbel1

Hougaard copula, the 3-dimensional Archimedean copula generated by ψ(x) = (0.5(ex + 1))− θ with √ 1/θ θ ∈ (0, 13 ] and the 3-dimensional Archimedean copula with generator ψ(x) = e1−(1+x) for θ ∈ [ 2/2, 1].

6

As for the proportional reversed hazard rates model, we obtain the reversed hazard rate order on the sample maximum in the same context on the generator function. Theorem 3.2 For X ∼ PRH(F, λ, ψ) and Y ∼ PRH(F, µ, ψ), if ψ is log-concave and

ψ ln ψ ψ′

is increasing

and concave, then λ ≼w µ implies Xn:n ≥rh Yn:n . ∑n Proof: Xn:n has distribution Fn:n (x) = ψ( i=1 ϕ(F λi (x))) and thus the reversed hazard rate, for x ≥ 0, ( ∑n ( λ )) n i ∑ λi F λi (x)r(x) ( ) ψ′ r(x) i=1 ϕ F (x) ( )) ( ( )) = rn:n (x) = ( ∑n ℓ1 ϕ(F λ1 (x)), · · · , ϕ(F λn (x)) . λ ′ λ i ln F (x) ψ ψ ϕ F i (x) i=1 ϕ F (x) i=1

(3.3)

( ) Similarly, Yn:n has the reversed hazard rate r˜n:n (x) = [r(x)/ ln F (x)]ℓ1 ϕ(F µ1 (x)), · · · , ϕ(F µn (x)) , for x ≥ 0. Since λ ≼w µ, ψ is log-concave and

ψ ln ψ ψ′

is increasing and concave, by Lemma 2.3 we have

( ) ( ) ℓ1 ϕ(F λ1 (x)), · · · , ϕ(F λn (x)) ≤ ℓ1 ϕ(F µ1 (x)), · · · , ϕ(F µn (x)) .

(3.4)

Based on (3.3), (3.4) and ln F (x) ≤ 0, we have rn:n (x) ≥ r˜n:n (x) for x ≥ 0. That is, Xn:n ≥rh Yn:n . The stochastic order on the sample maximum in (1.5) is upgraded to the reversed hazard order in Theorem 3.2 at the cost of appending the increasing and concave

ψ ln ψ ψ′ .

A distribution F with hazard rate h is said to be of increasing proportional hazard rate (denoted as IPHR) if xh(x) is increasing. Readers may refer to Marshall and Olkin (2007) and Righter et al. (2009) for more on IPHR. For the scale model we build the hazard rate order on the sample minimum. In the following, we assume the differentiable baseline hazard rate for technical convenience. Theorem 3.3 For X ∼ S(F¯ , λ, ψ) and Y ∼ S(F¯ , µ, ψ) with log-concave ψ, λ ≼w µ implies X1:n ≤hr Y1:n if (i) −ψ ′ /ψ is log-convex, and (ii) F is DHR and IPHR with concave xh(x) and decreasing xh2 (x). Proof: For ease of reference, let us list the following facts. Q1: ψ(x) ≥ 0 and ψ ′ (x) ≤ 0 for any x ≥ 0. Q2: The log-concave ψ implies the decreasing ψ ′ /ψ and hence ψ ′′ (x)ψ(x) ≤ [ψ ′ (x)]2 for any x ≥ 0. Q3: The log-convex −ψ ′ /ψ implies that {ψ ′′ (x)ψ(x) − [ψ ′ (x)]2 }/[ψ(x)ψ ′ (x)] ≥ 0 increases in x ≥ 0. Q4: Since F is IPHR with concave xh(x), then h(x) + xh′ (x) is nonnegative and decreasing in x ≥ 0. )) ( ∑n ( ¯ and thus the hazard rate Obviously, X1:n has survival function F¯1:n (x) = ψ i=1 ϕ F (λi x) )) n )) ( ∑n ( ( ( ¯ ∑ ψ′ ψ ϕ F¯ (λi x) i=1 ϕ F (λi x) )) )) = ℓ2 (λ, x), ( h1:n (x) = ( ∑n λi h(λi x) ′ ( ( ¯ ¯ ψ ψ ϕ F (λi x) i=1 ϕ F (λi x) i=1 7

for x ≥ 0.

˜ 1:n (x) = ℓ2 (µ, x) for x ≥ 0. Denote Likewise, Y1:n gets the hazard rate h Jψ (x) = A(λ(s,t) , x) =

( ∑n ) ¯ ψ i=1 ϕ(F (λi x)) ( ), ∑ B(λ, x) = ′ n ¯ ψ i=1 ϕ(F (λi x)) n (∑ ) ϕ(F¯ (λi x)) B(λ, x). C(λ, x) = Jψ

ψ ′′ (x)ψ(x) − [ψ ′ (x)]2 , [ψ(x)]2 ∑

λi h(λi x)B(λi ei , x),

i=1

i̸=s,t

Note that, for s, t ∈ In with s ̸= t and ui ∈ [0, 1] with i ∈ In , (∑ ) n n −xf (λs x) ∑ ψ(ϕ(F¯ (λi x))) ¯ J λ h(λ x) ϕ( F (λ x)) ψ i i i ψ ′ (ϕ(F¯ (λs x))) i=1 ψ ′ (ϕ(F¯ (λi x))) i=1 ( ∑n ) ) ψ(ϕ(F¯ (λs x))) ϕ(F¯ (λi x)) ( ψ′ ) h(λs x) + λs xh′ (λs x) ′ + ( ∑ni=1 ¯ ψ (ϕ(F¯ (λs x))) ψ ϕ(F (λi x)) ( ∑ni=1 ) ′ ψ ϕ(F¯ (λi x)) xf (λs x) 2 ¯ ) λs h(λs x) ( ) + ( ∑ni=1 ¯ ′ ¯ (λs x)) Jψ (ϕ(F (λs x)))[B(λs es , x)] ψ ϕ( F (λ x)) ψ ϕ( F i i=1 [ ] −xf (λs x)A(λ(s,t) , x) 2 2 = − xλs h (λs x)[B(λs es , x)] − λt xh(λs x)h(λt x)B(λs es , x)B(λt et , x) ψ ′ (ϕ(F¯ (λs x))) (∑ ) n λs h(λs x) xf (λs x) 2 ¯ ·Jψ ϕ(F¯ (λi x)) + ′ ¯ (λs x))) Jψ (ϕ(F (λs x)))[B(λs es , x)] B(λ, x) ψ (ϕ( F i=1 ( ) + h(λs x) + λs xh′ (λs x) B(λs es , x)/B(λ, x).

∂ℓ2 (λ, x) ∂λs

=

Owing to Q3, it holds that C(λ, x) ≥ C(λs es , x). By Q1, Q2, Q4 and C(λ, x) ≥ C(λs es , x) we have ∂ℓ2 (λ, x) ∂λs

=

[

] (∑ ) n −xf (λs x)A(λ(s,t) , x) ¯ − λt xh(λs x)h(λt x)B(λs es , x)B(λt et , x) Jψ ϕ(F (λi x)) ψ ′ (ϕ(F¯ (λs x))) i=1

+[h(λs x) + λs xh′ (λs x)]

B(λs es , x) B(λ, x)

+λs xh2 (λs x)[C(λs es , x) − C(λ, x)] ≥

0,

[B(λs es , x)]2 B(λ, x)

(3.5)

for all x ≥ 0.

Therefore, ℓ2 (λ, x) is increasing in λi for any i ∈ In . Also, by (3.5) we have, for s, t ∈ In with s ̸= t,

=

[ ] (λs − λt ) ∂ℓ2 (λ, x)/∂λs − ∂ℓ2 (λ, x)/∂λt (∑ ) n [ ] Jψ ϕ(F¯ (λi x)) (λs − λt )xA(λ(s,t) , x) f (λt x)/ψ ′ (ϕ(F¯ (λt x))) − f (λs x)/ψ ′ (ϕ(F¯ (λs x))) i=1

+Jψ

(∑ n i=1

) ¯ ϕ(F (λi x)) xh(λs x)h(λt x)(λs − λt )2 B(λs es , x)B(λt et , x)

) ( ) ] (λs − λt ) [( h(λs x) + λs xh′ (λs x) B(λs es , x) − h(λt x) + λt xh′ (λt x) B(λt et , x) + B(λ, x) ( ) (λs − λt ) [ + λt xh2 (λt x)[B(λt et , x)]2 C(λ, x) − C(λt et , x) B(λ, x) 8

( )] −λs xh2 (λs x)[B(λs es , x)]2 C(λ, x) − C(λs es , x) [ ] (λs − λt )xC(λ, x)A(λ(s,t) , x) f (λs x)/ψ ′ (ϕ(F¯ (λs x))) − f (λt x)/ψ ′ (ϕ(F¯ (λt x)))

sgn

=

−(λs − λt )2 xh(λs x)h(λt x)C(λ, x)B(λs es , x)B(λt et , x) [ ] ( ) ] −(λs − λt ) h(λs x) + λs xh′ (λs x) B(λs es , x) − h(λt x) + λt xh′ (λt x) B(λt et , x) [ ( ) +(λs − λt ) λs xh2 (λs x)[B(λs es , x)]2 C(λ, x) − C(λs es , x) ( )] −λt xh2 (λt x)[B(λt et , x)]2 C(λ, x) − C(λt et , x) ,

(3.6)

sgn

where ‘ = ’ means both sides have the same sign. Since ψ is log-concave and F is DHR, by Lemma 2.4 we have (λs − λt )[f (λs x)/ψ ′ (ϕ(F¯ (λs x))) − f (λt x)/ψ ′ (ϕ(F¯ (λt x)))] ≥ 0. As a result, based on Q1 and Q3 we have, for x, t ∈ In with s ̸= t, [ ] (λs − λt )C(λ, x)A(λ(s,t) , x) f (λs x)/ψ ′ (ϕ(F¯ (λs x))) − f (λt x)/ψ ′ (ϕ(F¯ (λt x)))

−(λs − λt )2 xh(λs x)h(λt x)B(λs es , x)B(λt et , x)C(λ, x)

≤ 0,

(3.7)

≤ 0.

(3.8)

Note that Q4 implies h(λt x) + λt xh′ (λt x) ≥ h(λs x) + λs xh′ (λs x) ≥ 0 for λs ≥ λt . Both ϕ and F¯ are decreasing, then ϕ(F¯ (λx)) is increasing in λ. Also, Q2 implies that ψ(ϕ(F¯ (λx)))/ψ ′ (ϕ(F¯ (λx))) ≤ 0 is increasing in λ. Thus, B(λt et , x) ≤ B(λs es , x) ≤ 0 for λs ≥ λt . So, it holds that, for λs ≥ λt , [( ) ( ) )] −(λs − λt ) h(λs x) + λs xh′ (λs x) B(λs es , x) − h(λt x) + λt xh′ (λt x) B(λt et , x) ≤ 0.

(3.9)

Similarly, (3.9) can also be verified for λs ≤ λt and hence it holds actually for any λs , λt . On the other hand, since xh2 (x) ≥ 0 is decreasing and [B(λx)]2 ≥ 0 is decreasing in λ, λxh2 (λx)B(λx) is decreasing in λ. Thus, λt xh2 (λt x)[B(λt et x)]2 ≥ λs xh2 (λs x)[B(λs es x)]2 ≥ 0 for λs ≥ λt . By Q3, for λs ≥ λt we have C(λ, x) − C(λt et , x) ≥ C(λ, x) − C(λs es , x) ≥ 0, and likewise it is verified that [ ( ) (λs −λt ) λs xh2 (λs x)[B(λs es , x)]2 C(λ, x)−C(λs es , x) ( )] −λt xh2 (λt x)[B(λt et , x)]2 C(λ, x)−C(λt et , x) ≤ 0, for any λs , λt .

(3.10)

By (3.7)-(3.10), we conclude that (3.6) is nonnegative and hence ℓ2 (λ, x) is Schur-concave w.r.t. λ. So, −ℓ2 (λ, x) is decreasing and Schur-convex w.r.t. λ. As per Theorem 3.A.8 of Marshall et al. (2011), ˜ 1:n (x) for all x. That is, X1:n ≤hr Y1:n . λ ≼w µ implies −h1:n (x) = −ℓ2 (λ) ≤ −ℓ2 (µ) = −h The above Theorem 3.3 complements (1.2) (i.e., Theorem 2.1(i) of Khaledi et al. (2011)) and (1.6) (i.e., Theorem 4.2 of Li et al. (2016)). In addition, as mentioned in the proof of Khaledi et al. (2011,

9

Theorem 2.1) the decreasing xh′ (x) + h(x) is equivalent to the decreasing x2 h′ (x). Since the concave xh(x) is equivalent to the decreasing xh′ (x) + h(x), then the concave xh(x) assumed in Theorem 3.3 is equivalent to the decreasing x2 h′ (x) assumed in (1.2). To close this section, we illustrate the assumption of Theorem 3.3 through one example. Example 3.4 According to Li and Li (2017), both the independence copula and the Gumbel-Hougaard √ x copula (n = 3) with ψ(x) = e(1−e )/θ , θ ∈ (0, 0.5(3 − 5)] are examples of survival copulas with logconcave ψ and log-convex −ψ ′ /ψ, and thus satisfy the assumption on the generator in Theorem 3.3. On the other hand, let Xi ∼ W(α, λi ) for i ∈ I3 and Yi ∼ W(α, µi ) for i ∈ I3 , then both X and Y follow α the scale model with common baseline survival function F¯ (x) = e−x for x ≥ 0. It is easy to verify that

h(x) = αxα−1 is decreasing, xh(x) is increasing and concave and xh2 (x) is decreasing for 0 < α ≤ 0.5, and hence it fulfills the requirement on the baseline distribution in Theorem 3.3.

3.0 2.5 2.0 1.5 1.0 0.5 0.2

0.4

0.6

0.8

1.0

y

˜ 1:n (tan(πy/2)) (solid) Figure 1: h1:n (tan(πy/2)) (dashed) and h Let α = 0.25, θ = 0.25, λ = (0.1, 0.2, 0.3) ≼w (0.01, 0.25, 0.28) = µ. As is seen in Figure 1, it holds ˜ 1:n (tan(πy/2)) for y ∈ [0, 1], which implies h1:n (x) ≥ h ˜ 1:n (x) for x ≥ 0. That is, that h1:n (tan(πy/2)) ≥ h X1:n ≤hr Y1:n . This coincides with the result of Theorem 3.3.

4

Some applications

For probability vectors p = (p1 , · · · , pn ) and q = (q1 , · · · , qn ), assume Ip = (Ip1 , · · · , Ipn ) ∼ B(p) and Iq = (Iq1 , · · · , Iqn ) ∼ B(q) two vectors of mutually independent Bernoulli random variables with E[Ipi ] = pi and E[Iqi ] = qi for i ∈ In . For nonnegative X and Y denote V1:n = min{X1 Ip1 , · · · , Xn Ipn } and W1:n = min{Y1 Iq1 , · · · , Yn Iqn }. In actuarial science, Xi ’s represent claim sizes of risks covered by a policy and Ipi ’s are the indicators of their occurrence, and thus V1:n defines the smallest claim amounts of the policy. In reliability, Xi ’s are component lifetimes and Ipi ’s are status of their starters, and then V1:n is the series system lifetime of the components equipped with starters.

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The Gumbel copula C(u) = exp{−[(− ln u1 )θ +· · ·+(− ln un )θ ]1/θ } for θ ≥ 1 and ui ∈ [0, 1], i ∈ In gets ( ( 1/θ 1/θ ) 1/θ ) generator ψ(x) = e−x . For X ∼ PHR F¯ , λ, e−x , Y ∼ PHR F¯ , µ, e−x , Ip ∼ B(p) independent

of X and Iq ∼ B(q) independent of Y , Li and Li (2018, Theorem 3.1) proved that V1:n ≥hr W1:n iff ∑n ∑n ∏n ∏n θ θ i=1 λi ≤ i=1 µi and i=1 pi ≥ i=1 qi . By Theorem 3.1 we get Corollary 4.1.

Corollary 4.1 For X ∼ PHR(F¯ , λ, ψ) independent of Ip ∼ B(p) and Y ∼ PHR(F¯ , µ, ψ) independent ∏n ∏n of Iq ∼ B(q), λ ≼w µ along with i=1 pi ≤ i=1 qi imply V1:n ≤hr W1:n if ψ is log-concave and ψ ψln′ ψ is

increasing and concave.

∏n Proof: For all x ≥ 0, P(V1:n > x) = P(X1 > x, · · · , Xn > x)P(Ipj = 1, j ∈ In ) = P(X1:n > x) i=1 pi . ∏n Likewise, P(W1:n > x) = P(Y1:n > x) i=1 qi . Since V1:n and W1:n both have positive probability at 0, P(W1:n > x)/P(V1:n > x) is increasing iff

P(Y1:n >0) P(X1:n >0)

·

increases in x ≥ 0. Since the former is equivalent to

∏n i=1 qi ∏n i=1 pi

Theorem 3.1, we complete the proof. The random vector Ip gets the entropy − log

∏n

∏n

i=1

i=1

=

∏n i=1 qi ∏n i=1 pi

pi ≤

∏n

≥ 1 and P(Y1:n > x)/P(X1:n > x)

i=1 qi

and the latter is guaranteed by

pi . As per Corollary 4.1, a series system of PHR

component lifetimes with more majorized frailty vector and less uncertain vector of occurrence attains a smaller hazard rate in the context of Archimedean copula with log-concave generator ψ and increasing and concave

ψ ln ψ ψ′ .

Also, Gumbel copula has a log-convex generator ψ(x) = e−x

1/θ

. Thus Corollary 4.1

serves as one nice supplement of Theorem 3.1 of Li and Li (2018). In the context of (i) X ∼ S(F¯ , λ, e−x ) and Y ∼ S(F¯ , µ, e−x ), (ii) Ip ∼ B(p) independent of X and Iq ∼ B(q) independent of Y , Barmalzan et al. (2017, Theorem 5) built that λ ≼w µ along with ∏n ∏n i=1 qi imply V1:n ≤hr W1:n if xh(x) is increasing and concave. Accordingly, for a scale vector i=1 pi ≤

of claim amounts making the scale less majorized and increasing the uncertainty of vector of occurrence leads to a larger hazard rate for the minimum of all claim amounts. ∏n ∏n It should be remarked here that i=1 pi ≤ i=1 qi was overlooked in Barmalzan et al. (2017, Theorem

5). As an immediate consequence of Theorem 3.3, we can also obtain one natural generalization of Theorem 5 in Barmalzan et al. (2017). The proof is similar to that of Corollary 4.1 and hence is omitted. Corollary 4.2 For X ∼ S(F¯ , λ, ψ) independent of Ip ∼ B(p) and Y ∼ S(F¯ , µ, ψ) independent of ∏n ∏n Iq ∼ B(q), λ ≼w µ along with i=1 pi ≤ i=1 qi imply V1:n ≤hr W1:n if (i) ψ is log-concave and −ψ ′ /ψ is log-convex, and (ii) F is DHR and IPHR with concave xh(x) and decreasing xh2 (x).

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Acknowledgement The authors would like to thank the two reviewers for pertinent suggestions on the earlier version of this manuscript. Chen Li’s research is supported by Scientific Research Foundation of Tianjin University of Commerce (R160106), Science and Technology Development Foundation of Tianjin (2017KJ176) and The 3rd level of Tianjin 131 Innovative Talent Training Project. Also she expresses her thanks to Stevens Institute of Technology for providing a nice work space during her visit.

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