Loading [MathJax]/jax/output/SVG/jax.js
Research article

Ordering results of second order statistics from random and non-random number of random variables with Archimedean copulas

  • Received: 10 January 2021 Accepted: 31 March 2021 Published: 12 April 2021
  • MSC : 60E15, 60K10, 90B25

  • In this paper, we investigate stochastic comparisons of the second largest order statistics of homogeneous samples coupled by Archimedean copula, and we establish the reversed hazard rate and likelihood ratio orders, and we further generalize the corresponding results to the case of random sample size. Also, we derive some results for relative ageing between parallel systems and 2-out-of-n/2-out-of-(n+1) systems. Finally, some examples are given to illustrate the obtained results.

    Citation: Bin Lu, Rongfang Yan. Ordering results of second order statistics from random and non-random number of random variables with Archimedean copulas[J]. AIMS Mathematics, 2021, 6(6): 6390-6405. doi: 10.3934/math.2021375

    Related Papers:

    [1] Xiao Zhang, Rongfang Yan . Stochastic comparisons of extreme order statistic from dependent and heterogeneous lower-truncated Weibull variables under Archimedean copula. AIMS Mathematics, 2022, 7(4): 6852-6875. doi: 10.3934/math.2022381
    [2] Mingxia Yang . Orderings of the second-largest order statistic with modified proportional reversed hazard rate samples. AIMS Mathematics, 2025, 10(1): 311-337. doi: 10.3934/math.2025015
    [3] Bin Lu . Stochastic comparisons of second-order statistics from dependent and heterogeneous modified proportional (reversed) hazard rates scale models. AIMS Mathematics, 2024, 9(4): 8904-8919. doi: 10.3934/math.2024434
    [4] Haroon Barakat, Osama Khaled, Hadeer Ghonem . Predicting future order statistics with random sample size. AIMS Mathematics, 2021, 6(5): 5133-5147. doi: 10.3934/math.2021304
    [5] Li Zhang, Rongfang Yan . Stochastic comparisons of series and parallel systems with dependent and heterogeneous Topp-Leone generated components. AIMS Mathematics, 2021, 6(3): 2031-2047. doi: 10.3934/math.2021124
    [6] H. M. Barakat, M. H. Dwes . Asymptotic behavior of ordered random variables in mixture of two Gaussian sequences with random index. AIMS Mathematics, 2022, 7(10): 19306-19324. doi: 10.3934/math.20221060
    [7] Miaomiao Zhang, Bin Lu, Rongfang Yan . Ordering results of extreme order statistics from dependent and heterogeneous modified proportional (reversed) hazard variables. AIMS Mathematics, 2021, 6(1): 584-606. doi: 10.3934/math.2021036
    [8] Mohamed Said Mohamed, Muqrin A. Almuqrin . Properties of fractional generalized entropy in ordered variables and symmetry testing. AIMS Mathematics, 2025, 10(1): 1116-1141. doi: 10.3934/math.2025053
    [9] Snezhana Hristova, Kremena Stefanova . p-moment exponential stability of second order differential equations with exponentially distributed moments of impulses. AIMS Mathematics, 2021, 6(3): 2886-2899. doi: 10.3934/math.2021174
    [10] Xueyan Li, Chuancun Yin . Some stochastic orderings of multivariate skew-normal random vectors. AIMS Mathematics, 2023, 8(10): 23427-23441. doi: 10.3934/math.20231190
  • In this paper, we investigate stochastic comparisons of the second largest order statistics of homogeneous samples coupled by Archimedean copula, and we establish the reversed hazard rate and likelihood ratio orders, and we further generalize the corresponding results to the case of random sample size. Also, we derive some results for relative ageing between parallel systems and 2-out-of-n/2-out-of-(n+1) systems. Finally, some examples are given to illustrate the obtained results.



    Order statistics play an important role in reliability theory, auction theory, operations research, and many applied probability areas. Let Xk:n denotes the kth smallest of random variables X1,,Xn, k=1,,n. In reliability theory, Xk:n characterizes the lifetime of a (nk+1)-out-of-n system, which works if at least (nk+1) of all the n components function normally. Specifically, X1:n and Xn:n denote the lifetimes of series and parallel systems, X2:n and Xn1:n characterize the lifetime of the fail-safe system and 2-out-of-n system, respectively(see Barlow and Proschan [1]). In auction theory, X1:n and X2:n represent the final price of the first-price and second-price procurement auction, Xn1:n and Xn:n represent the final price of the first-price and second-price sealed-bid auction (see Fang and Li [2]), respectively.

    The groundbreaking work by Boland et al. [3] on the sample from i.i.d. random variables was to study ordering properties between Xi1:n and Xi:n+1. In the context of XkhrXn+1, they obtained the hazard rate order Xi1:nhrXi:n+1, and also proved that Xn+1hrXk implies that Xi:nhrXi:n+1. Raqab and Amin [4] established the likelihood ratio order between order statistics from samples of different sizes. Bapat and Kochar [5] proved the likelihood ratio ordering between order statistics from a sample with observations arranged in the likelihood ration order. For homogeneous random variables with an Archimedean (survival) copula, Li and Fang [6] derived the hazard rate, the reversed hazard rate and the likelihood ratio ordering of the extramel values and its adjacent order statistics. Subsequently, Fang and Li [7] further developed the reversed hazard rate order and the hazard rate order on sample extremes in the context of proportional reversed hazard models and proportional hazard models, respectively. For heterogenous random variables connected with an Archimedean copula, Mesfioui et al. [8] obtained the ordering properties of the maximum order statistic of the sample and its two adjacent order statistics. Barmalzan et al. [9] established the hazard rate order and reversed hazard rate order of series and parallel systems with dependent components following either modified proportional reversed hazard models or modified proportional hazard models under Archimedean copula. In fact, Pledger and Proschan [10] were the first to deal with the problem of comparing order statistics from heterogeneous exponential random variables. Subsequently, many researchers devoted themselves to stochastic comparisons of order statistics from heterogeneous independent or dependent samples, to name a few, see [11,12,13,14,15,16,17,18,19,20,21,22].

    The notions of relative ageing describe the rate at which one component or system is aging relative to the other. Various partial orders describing relative ageing of two life distributions have been introduced in the literature. Kalashnikov and Rachev [23] introduced a relative aging notion based on the increasing ratio of two hazard rate functions. Rezaei et al. [24] further studied the relative ageing by considering the ratio of two reversed hazard rate functions. Lai and Xie [25] showed that the parallel system with additional redundant components ageing faster in terms of the increasing hazard ratio. Li and Li [26] studied the effect of heterogeneity among independent components on the relative ageing of the series and parallel systems. Ding and Zhang [27] further investigated the effects of Arichimedean dependence and heterogeneity among components on the relative ageing of series and parallel system. For more research on relative ageing, one may refer to [28,29,30,31,32].

    In reliability theory, actuarial science and survival analysis, some observations may be lost for unavoidable reasons, and thus it may be impossible to obtain a fixed sample size. Sometimes the sample size may depend on the occurrence of some events, which makes the sample size always random. For example, if a common dose of radiation is given to a sample of animals, then the interest often is in the times that the first and the last expire (see, Consul [33]). There is quite rich literature on the sample with a random size, for example [34,35,36].

    To the best of our knowledge, Li and Fang [6] were the first to study stochastic comparison among Xn:n and Xn+1:n+1 from homogeneous random variables with an Archimedean copula. And the most existing research are focusing on the extreme order statistics, while, there are few works on stochastic comparisons among the second order statistics. In this paper, we will focus on stochastic comparisons of the second largest order statistics from the random and non-random number of homogeneous samples coupled by Archimedean copula, and investigate the impact of sample size and dependence on the the second largest order statistics. Also, some ordering results are derived for relative ageing between parallel systems and 2-out-of-n/2-out-of-(n+1) systems in terms of increasing reversed hazard ratio.

    The remainder of this paper is organized as follows: Section 2 recalls some concepts and notations used in this paper. Section 3 presents the results for the case of non-random sample size. Section 4 establishes the results for the case of random sample size. Section 5 provides the application of our main results. Section 6 summarizes our research findings.

    In this section, let us first recall some important concepts and notations related to the main results of this article.

    For random variable X with support R+=[0,+), let FX(x) be distribution function (fX(x) be densities when absolutely continuous), and denote ˉFX(x)=1FX(x) the reliability function. Let hX(x)=fX(x)/ˉFX(x) and rX(x)=fX(x)/FX(x) be the hazard rate function and reversed hazard rate function of X, respectively. The Laplace-Stieltjes transform of X is given by

    LX(x)=0extdFX(t).

    Definition 1. For two nonnegative random variables X and Y, X is said to be smaller than Y in the

    (i) stochastic order (denoted by XstY) if ˉFX(x)ˉFY(x) for all xR+;

    (ii) hazard rate order (denoted by XhrY) if ˉFY(x)/ˉFX(x) is increasing in xR+;

    (iii) reversed hazard rate order (denoted by XrhY) if FY(x)/FX(x) is increasing in xR+;

    (iv) likelihood ratio order (denoted by XlrY) if fY(x)/fX(x) is increasing in xR+;

    (v) Laplace transform ratio order (denoted by XLtrY) if LY(x)/LX(x) is decreasing in xR+.

    It is well known that the above stochastic orders have the following relations:

    Xst[Ltr]YXrhYXlrYXhrYXstY.

    For more comprehensive discussions on stochastic orders, please refer to Shaked and Shanthikumar [37], Li and Li [38] and Belzunce et al.[39].

    Next, we introduce the concept of relative ageing.

    Definition 2. X is said to be ageing faster than Y in the reversed failure rate, denoted by XbY, if rX(x)/rY(x) is decreasing in xR+.

    For more details on ageing, one may refer to Lai and Xie [40].

    Now, let us review the concept of Archimedean Copulas.

    Definition 3. For a decreasing and continuous function ψ:[0,+)[0,1] such that ψ(0)=1 and ψ(+)=0, let ϕ=ψ1 be the pseudo-inverse of ψ. Then

    Cψ(u1,...,un)=ψ(ϕ(u1)+...+ϕ(un)),ui[0,1],iIn,

    is said to be an Archimedean copula with generator ψ if (1)kψ(k)(x)0 for k=0,,n2 and (1)n2ψ(n2)(x) is decreasing and convex.

    Copula is used to describe the dependence among random variables and plays an important role in constructing joint distribution through marginal distribution as it does not contain any information of marginal distributions. Archimedean copulas are rather popular because of the mathematical tractability and the capability of capturing wide ranges of dependence. It is well known that the Archimedean family contains a great many useful copulas, including the well-known independence (product) copula, the Clayton copula and the Ali– Mikhail–Haq (AMH) copula. For detailed discussions on copulas and its applications, one may refer to Nelsen [41].

    A real-valued function g:RnR is said to be supermodular(submodular) if the following inequality

    g(x1y1,,xnyn)+g(x1y1,,xnyn)()g(x1,,xn)+g(y1,,yn)

    holds for all xi,yiR, where xy=min{x,y} and xy=max{x,y}. In particular, a function g with finite second partial derivatives on Rn is supermodular(submodular) if and only if 2g(x)/(xixj)()0 for all 1ijn and xRn.

    Throughout the manuscript, all concerned random variables are assumed to be absolutely continuous and nonnegative, and the terms increasing and decreasing stand for non-decreasing and non-increasing, respectively. "sgn=" means equality of sign.

    In this section, we study the ordering results of the second largest order statistics of homogeneous sample coupled by Archimedean copula. First, we present the comparison result in the sense of the reversed hazard rate order between Xn1:n and Xn:n+1.

    Theorem 4. Suppose homogeneous random variables X1,X2,,Xn+1 having an Archimedean copula with generator ψ. If tψ(t)/ψ(t) is convex and tψ(t)/ψ(t) is decreasing, then

     Xn1:nrhXn:n+1.

    Proof. The distribution functions of Xn1:n and Xn:n+1 can be expressed as

    FXn1:n(x)=nψ((n1)ϕ(F(x)))(n1)ψ(nϕ(F(x)))

    and

    FXn:n+1(x)=(n+1)ψ(nϕ(F(x)))nψ((n+1)ϕ(F(x))),

    respectively. Let u=F(x), to obtain the desired result, it suffices to show that

    A1(u)=FXn:n+1(x)FXn1:n(x)=(n+1)ψ(nϕ(u))nψ((n+1)ϕ(u))nψ((n1)ϕ(u))(n1)ψ(nϕ(u))

    is increasing in u[0,1]. Taking the derivative of A1(u), we have

    A1(u)sgn=ϕ(u)((n+1)n(ψ(nϕ(u))ψ((n+1)ϕ(u)))(nψ((n1)ϕ(u))(n1)ψ(nϕ(u)))(n1)n(ψ((n1)ϕ(u))ψ(nϕ(u)))((n+1)ψ(nϕ(u))nψ((n+1)ϕ(u))))=nϕ(u)((n+1)(nψ(nϕ(u))ψ((n1)ϕ(u))(n1)ψ((n1)ϕ(u))ψ(nϕ(u)))+(n1)((n+1)ψ((n+1)ϕ(u))ψ(nϕ(u))(n1)ψ(nϕ(u))ψ((n+1)ϕ(u)))+n((n1)ψ((n1)ϕ(u))ψ((n+1)ϕ(u))(n+1)ψ((n+1)ϕ(u))ψ((n1)ϕ(u))))=nϕ(u)((n+1)Δ1(n1,n;u)+(n1)Δ1(n,n+1;u)nΔ1(n1,n+1;u)), (3.1)

    where, for all (i,j)

    Δ1(i,j;u)=jψ(jϕ(u))ψ(iϕ(u))iψ(iϕ(u))ψ(jϕ(u)).

    As tψ(t)/ψ(t) is decreasing implies that tψ(t)/ψ(t) decreases (c.f. Theorem 3.1 of [6]), then for i<j,

    Δ1(i,j;u)sgn=jϕ(u)ψ(jϕ(u))ψ(jϕ(u))iϕ(u)ψ(iϕ(u))ψ(iϕ(u))<0.

    Note that ψ is decreasing and tψ(t)/ψ(t) is convex, we have

    uΔ1(n1,n;u)=ψ((n1)ϕ(u))ψ(nϕ(u))[nϕ(u)ψ(nϕ(u))ψ(nϕ(u))(n1)ϕ(u)ψ((n1)ϕ(u))ψ((n1)ϕ(u))]ψ((n+1)ϕ(u))ψ(nϕ(u))[nϕ(u)ψ(nϕ(u))ψ(nϕ(u))(n1)ϕ(u)ψ((n1)ϕ(u))ψ((n1)ϕ(u))]ψ((n+1)ϕ(u))ψ(nϕ(u))[(n+1)ϕ(u)ψ((n+1)ϕ(u))ψ((n+1)ϕ(u))nϕ(u)ψ(nϕ(u))ψ(nϕ(u))]=uΔ1(n,n+1;u).

    Let l1(x,y)=I(xy)(xψ(x)ψ(y)yψ(y)ψ(x)). As tψ(t)/ψ(t) is decreasing, then for xy, we have

    2l1(x,y)xy=ψ(x)ψ(y)(xψ(x)ψ(x)yψ(y)ψ(y))0,

    which implies that l1(x,y) is submodular. Therefore, we have

    ϕ(u)(Δ1(n1,n;u)+Δ1(n,n+1;u))=l1(nu,(n1)u)+l1((n+1)u,nu)l1(nu,nu)+l1((n+1)u,(n1)u)=ϕ(u)Δ1(n1,n+1;u).

    Thus,

    (n+1)Δ1(n1,n;u)+(n1)Δ1(n,n+1;u)nΔ1(n1,n+1;u)=(n+1)(Δ1(n1,n;u)12Δ1(n1,n+1;u))+(n1)(Δ1(n,n+1;u)12Δ1(n1,n+1;u))(n1)(Δ1(n1,n;u)+Δ1(n,n+1;u)Δ1(n1,n+1;u))0.

    Then, in combination with the decreasing property of ψ, (3.1) is nonnegative, which implies that FXn:n+1(x)/FXn1:n(x) is increasing in x. Hence we complete the proof.

    In the following, we establish the likelihood ratio order of the second largest order statistics from n and n+1 homogeneous observations, respectively.

    Theorem 5. Suppose homogeneous random variables X1,X2,,Xn+1 having an Archimedean copula with generator ψ. If tψ(t)/ψ(t) is decreasing, then

     Xn1:nlrXn:n+1.

    Proof. The density functions of Xn1:n and Xn:n+1 can be expressed as

    fXn1:n(x)=n(n1)ϕ(F(x))f(x)(ψ((n1)ϕ(F(x)))ψ(nϕ(F(x))))

    and

    fXn:n+1(x)=n(n+1)ϕ(F(x))f(x)(ψ(nϕ(F(x)))ψ((n+1)ϕ(F(x)))),

    respectively. Let u=F(x), it is sufficient to show that

    A2(u)sgn=fXn1:n(x)fXn:n+1(x)=ψ((n1)ϕ(u))ψ(nϕ(u))ψ(nϕ(u))ψ((n+1)ϕ(u))

    is decreasing in u[0,1]. Taking the derivative of A2(u), we have

    A2(u)sgn=ϕ(u)((n1)ψ((n1)ϕ(u))ψ(nϕ(u))nψ(nϕ(u))ψ((n1)ϕ(u)))+(n+1)ψ((n+1)ϕ(u))ψ((n1)ϕ(u))(n1)ψ((n1)ϕ(u))ψ((n+1)ϕ(u)))+nψ(nϕ(u))ψ((n1)ϕ(u))(n+1)ψ((n+1)ϕ(u))ψ(nϕ(u)))=ϕ(u)(Δ2(n1,n;u)+Δ2(n,n+1;u)Δ2(n1,n+1;u)), (3.2)

    where, for all (i,j)

    Δ2(i,j;u)=iψ(iϕ(u))ψ(jϕ(u))jψ(jϕ(u))ψ(iϕ(u)).

    Let l2(x,y)=I(xy)(xψ(x)ψ(y)yψ(y)ψ(x)). As tψ(t)/ψ(t) is decreasing, then for xy, we have

    2l2(x,y)xy=ψ(x)ψ(y)(xψ(x)ψ(x)yψ(y)ψ(y))0,

    which implies that l2(x,y) is supermodular. Then,

    ϕ(u)(Δ2(n1,n;u)+Δ2(n,n+1;u))=l2((n1)u,nu)+l2(nu,(n+1)u)l2((n1)u,(n+1)u)+l2(nu,nu)=ϕ(u)Δ2(n1,n+1;u).

    That is,

    Δ2(n1,n;u)+Δ2(n,n+1;u)Δ2(n1,n+1;u)0.

    Thus, (3.2) is non-positive, which implies that fXn1:n(x)/fXn:n+1(x) is decreasing in x. Then the proof is completed.

    The results of Theorem 4 and Theorem 5 are based on Theorem 4(iii) and (iv) in Navarro [17], we present a sufficient condition for the reversed hazard rate and likelihood ratio orders between 2-out-of-(n+1) and 2-out-of-n system. Theorem 4 and Theorem 5 state that 2-out-of-(n+1) system is more reliable than 2-out-of-n system in the sense of the reversed hazard rate and likelihood ratio orders. Now, we present the following example to illustrate the above results.

    Example 6. Consider the Clayton copula with generator ψ(t)=(t+1)1, we have

    (tψ(t)ψ(t))=2(1+t)30,(tψ(t)ψ(t))=2(1+t)20,(tψ(t)ψ(t))=3(1+t)30.

    Thus, the generator satisfies all conditions of Theorem 4 and Theorem 5. Assume that Xi has common exponential distribution function ex, i=1,2,3,4. To display the whole of survival curves of X2:3 and Y3:4 on [0,), we perform the transformation (x+1)1:[0,)[0,1]. Then, as seen in Figure 1(a),  X2:3rhX3:4, and Figure 1(b) confirms that  X2:3lrX3:4.

    Figure 1.  The curves of ˉF(X2:3+1)1(x)/ˉF(X3:4+1)1(x) and f(X2:3+1)1(x)/f(X3:4+1)1(x).

    In the following, in the context of system consist of dependent and homogeneous components, we build the comparison results for the relative ageing between parallel system and 2-out-of-n/2-out-of-(n+1) system. First, we present the relative ageing between parallel and 2-out-of-n systems with respect to the increasing reversed hazard ratio.

    Theorem 7. Suppose homogeneous random variables X1,X2,,Xn having an Archimedean copula with generator ψ. If both t(ψ(t)/ψ(t)ψ(t)/ψ(t)) and tψ(t)/ψ(t) are decreasing in t0, then

     Xn1:nbXn:n.

    Proof. The reversed hazard rate functions of Xn1:n and Xn:n can be expressed as

    rXn1:n(x)=n(n1)ϕ(F(x))f(x)ψ((n1)ϕ(F(x)))ψ(nϕ(F(x)))nψ((n1)ϕ(F(x)))(n1)ψ(nϕ(F(x)))

    and

    rXn:n(x)=nϕ(F(x))f(x)ψ(nϕ(F(x)))ψ(nϕ(F(x))).

    Let u=F(x), it is sufficient to show that

    A3(u)=rXn1:n(x)rXn:n(x)sgn=ψ((n1)ϕ(u))ψ(nϕ(u))1nψ((n1)ϕ(u))ψ(nϕ(u))(n1)

    is decreasing in u[0,1]. Taking the derivative of A3(u), we have

    A3(u)sgn=ϕ(u)[(n1)ψ((n1)ϕ(u))ψ((n1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u))](nψ((n1)ϕ(u))ψ(nϕ(u))(n1))ψ((n1)ϕ(u))ψ(nϕ(u))ϕ(u)[(n1)ψ((n1)ϕ(u))ψ((n1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u))](nψ((n1)ϕ(u))ψ(nϕ(u))n)ψ((n1)ϕ(u))ψ(nϕ(u))=ϕ(u)[((n1)ψ((n1)ϕ(u))ψ((n1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u)))(nψ((n1)ϕ(u))ψ(nϕ(u))(n1))ψ((n1)ϕ(u))ψ(nϕ(u))((n1)ψ((n1)ϕ(u))ψ((n1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u)))(nψ((n1)ϕ(u))ψ(nϕ(u))n)ψ((n1)ϕ(u))ψ(nϕ(u))]=ϕ(u)Δ3(u),

    where

    Δ3(u)=((n1)ψ((n1)ϕ(u))ψ((n1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u)))(nψ((n1)ϕ(u))ψ(nϕ(u))(n1))ψ((n1)ϕ(u))ψ(nϕ(u))((n1)ψ((n1)ϕ(u))ψ((n1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u)))(nψ((n1)ϕ(u))ψ(nϕ(u))n)ψ((n1)ϕ(u))ψ(nϕ(u)).

    As ϕ(u)0, we just need to show that Δ3(u) is nonnegative. Note that

    (nψ((n1)ϕ(u))ψ(nϕ(u))(n1))ψ((n1)ϕ(u))ψ(nϕ(u))(nψ((n1)ϕ(u))ψ(nϕ(u))n)ψ((n1)ϕ(u))ψ(nϕ(u))=nψ((n1)ϕ(u))ψ(nϕ(u))(n1)ψ((n1)ϕ(u))ψ(nϕ(u))=[nϕ(u)ψ(nϕ(u))ψ(nϕ(u))(n1)ϕ(u)ψ((n1)ϕ(u))ψ((n1)ϕ(u))]ψ((n1)ϕ(u))ϕ(u)ψ(nϕ(u))0,

    where the last inequality is due to the assumption that tψ(t)/ψ(t) is decreasing in t0 and ψ(x)0. Thus

    Δ3(u)[((n1)ψ((n1)ϕ(u))ψ((n1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u)))((n1)ψ((n1)ϕ(u))ψ((n1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u)))]×(nψ((n1)ϕ(u))ψ(nϕ(u))n)ψ((n1)ϕ(u))ψ(nϕ(u))sgn=((n1)ψ((n1)ϕ(u))ψ((n1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u)))((n1)ψ((n1)ϕ(u))ψ((n1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u)))=(n1)ϕ(u)(ψ((n1)ϕ(u))ψ((n1)ϕ(u))ψ((n1)ϕ(u))ψ((n1)ϕ(u)))nϕ(u)(ψ(nϕ(u))ψ(nϕ(u))ψ(nϕ(u))ψ(nϕ(u)))0,

    where the last inequality is by the assumption that t(ψ(t)/ψ(t)ψ(t)/ψ(t)) is decreasing in t0. Hence, we complete the proof.

    Next, we establish the relative ageing between parallel system and 2-out-of-(n+1) system in the sense of increasing reversed hazard ratio.

    Theorem 8. Suppose homogeneous random variables X1,X2,,Xn+1 having an Archimedean copula with generator ψ. If both t(ψ(t)/ψ(t)ψ(t)/ψ(t)) and tψ(t)/ψ(t) are decreasing in t0, then

     Xn:n+1bXn:n.

    Proof. The reversed hazard rate functions of Xn:n+1 and Xn:n can be expressed as

    rXn:n+1(x)=n(n+1)ϕ(F(x))f(x)ψ(nϕ(F(x)))ψ((n+1)ϕ(F(x)))(n+1)ψ(nϕ(F(x)))nψ((n+1)ϕ(F(x)))

    and

    rXn:n(x)=nϕ(F(x))f(x)ψ(nϕ(F(x)))ψ(nϕ(F(x))),

    respectively. Let u=F(x), it just need to show that

    A4(u)=rXn:n+1(x)rXn:n(x)sgn=ψ((n+1)ϕ(u))ψ(nϕ(u))1nψ((n+1)ϕ(u))ψ(nϕ(u))(n+1)

    is decreasing in u[0,1]. Taking the derivative of A4(u), we have

    A4(u)sgn=ϕ(u)[((n+1)ψ((n+1)ϕ(u))ψ((n+1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u)))(nψ((n+1)ϕ(u))ψ(nϕ(u))(n+1))ψ((n+1)ϕ(u))ψ(nϕ(u))((n+1)ψ((n+1)ϕ(u))ψ((n1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u)))(nψ((n+1)ϕ(u))ψ(nϕ(u))n)ψ((n+1)ϕ(u))ψ(nϕ(u))]=ϕ(u)Δ4(u),

    where

    Δ4(u)=((n+1)ψ((n+1)ϕ(u))ψ((n+1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u)))(nψ((n+1)ϕ(u))ψ(nϕ(u))(n+1))ψ((n+1)ϕ(u))ψ(nϕ(u))((n+1)ψ((n+1)ϕ(u))ψ((n+1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u)))(nψ((n+1)ϕ(u))ψ(nϕ(u))n)ψ((n+1)ϕ(u))ψ(nϕ(u)).

    As ϕ(u)0, it only need to show that Δ4(u) is nonnegative. Note that

    (nψ((n+1)ϕ(u))ψ(nϕ(u))(n+1))ψ((n+1)ϕ(u))ψ(nϕ(u))(nψ((n+1)ϕ(u))ψ(nϕ(u))n)ψ((n+1)ϕ(u))ψ(nϕ(u))=nψ((n+1)ϕ(u))ψ(nϕ(u))(n+1)ψ((n+1)ϕ(u))ψ(nϕ(u))=[nϕ(u)ψ(nϕ(u))ψ(nϕ(u))(n+1)ϕ(u)ψ((n+1)ϕ(u))ψ((n+1)ϕ(u))]ψ((n+1)ϕ(u))ϕ(u)ψ(nϕ(u))0,

    where the last inequality is due to the assumption that tψ(t)/ψ(t) is decreasing in t0 and ψ(x)0. Thus

    Δ4(u)[((n+1)ψ((n+1)ϕ(u))ψ((n+1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u)))((n+1)ψ((n+1)ϕ(u))ψ((n+1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u)))]×(nψ((n+1)ϕ(u))ψ(nϕ(u))n)ψ((n+1)ϕ(u))ψ(nϕ(u))sgn=((n+1)ψ((n+1)ϕ(u))ψ((n1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u)))+((n+1)ψ((n+1)ϕ(u))ψ((n+1)ϕ(u))nψ(nϕ(u))ψ(nϕ(u)))sgn=nϕ(u)(ψ(nϕ(u))ψ(nϕ(u))ψ(nϕ(u))ψ(nϕ(u)))(n+1)ϕ(u)(ψ((n+1)ϕ(u))ψ((n+1)ϕ(u))ψ((n+1)ϕ(u))ψ((n+1)ϕ(u)))0,

    where the last inequality is by the assumption that t(ψ(t)/ψ(t)ψ(t)/ψ(t)) is decreasing in t0. Hence, we complete the proof.

    Theorem 7 and Theorem 8 state that 2-out-of-n system and 2-out-of-(n+1) system ageing faster than parallel system consisting of n components in the sense of the increasing reversed hazard ratio, respectively. The next example demonstrates the above results.

    Example 9. Consider the Clayton copula with generator ψ(t)=(t+1)1, we have

    t(ψ(t)ψ(t)ψ(t)ψ(t))=1+11+t,(tψ(t)ψ(t))=1(1+t)2.

    Thus, the generator satisfies the conditions of Theorem 7 and Theorem 8. Assume that Xi has common exponential distribution function ex, i=1,2,3,4,5. The curves of h(X3:4+1)1(x)/h(X4:4+1)1(x) and h(X4:5+1)1(x)/h(X4:4+1)1(x) are plotted in Figure 2(a) and Figure 2(b), respectively, from which we can see that  X3:4bX4:4 and  X4:5bX4:4.

    Figure 2.  The curves of h(X3:4+1)1(x)/h(X4:4+1)1(x) and h(X4:5+1)1(x)/h(X4:4+1)1(x).

    In this section, we study the ordering results of the second largest order statistics from random number of homogeneous samples coupled by Archimedean copula. First, we present the comparison result in the sense of the reversed hazard rate order between XN11:N1 and XN21:N2.

    Theorem 10. Suppose homogeneous random variables X1,X2, having an Archimedean copula with generator ψ, and let N1(2) and N2(2) be positive integer-valued random variables which are independent of Xi, i=1,2,. If tψ(t)/ψ(t) is convex and tψ(t)/ψ(t) is decreasing, and N1LtrN2, then

    XN11:N1rhXN21:N2.

    Proof. The distribution functions of XNj1:Nj can be expressed as

    FXNj1:Nj(x)=P(XNj1:Njx)=P(XNj1:Njx|Nj=n)P(Nj=n)=P(Xn1:nx)P(Nj=n)=n=2FXn1:n(x)P(Nj=n)=LNj(logFXn1:n(x)),j=1,2.

    As N1LtrN2 implies that LN2(x)/LN1(x) is decreasing in x>0. Note that logFXn1:n(x) is decreasing in x>0, thus

    LN2(logFXn1:n(x))LN1(logFXn1:n(x))

    is increasing in x>0. Then we have XN11:N1rhXN21:N2. Which finishes the proof.

    Shaked and Wong [34] have also shown that N1rhN2 implies N1LtrN2, thus we have the following corollary.

    Corollary 11. Suppose homogeneous random variables X1,X2, having an Archimedean copula with generator ψ, and let N1(2) and N2(2) be positive integer-valued random variable which are independent of Xi, i=1,2,. If tψ(t)/ψ(t) is convex and tψ(t)/ψ(t) is decreasing, and N1rhN2, then

    XN11:N1rhXN21:N2.

    The following theorem establishes the likelihood ratio order between XN11:N1 and XN21:N2.

    Theorem 12. Suppose homogeneous random variables X1,X2, having an Archimedean copula with generator ψ, and let N1(2) and N2(2) be positive integer-valued random variables which are independent of Xi, i=1,2,. If tψ(t)/ψ(t) is decreasing, and N1lrN2, then

    XN11:N1lrXN21:N2.

    Proof. The density function of XNj1:Nj can be expressed as

    fXNj1:Nj(x)=n=2fXn1:n(x)P(Nj=n),j=1,2.

    As N1lrN2 implies that

    P(N1=n)P(N2=n)P(N1=n+1)P(N2=n+1),

    thus P(Nj=n) is TP2 in n2 and j(j=1,2). According to Theorem 5, for x1x2, we have

    fXn1:n(x2)fXn:n+1(x2)fXn1:n(x1)fXn:n+1(x1),

    that is, fXn1:n(x) is TP2 in n2 and x. Then, by the Theorem 5.1 of Karlin [42], fXNj1:Nj(x) is TP2 in x and j(j=1,2). Therefore, XN11:N1lrXN21:N2. Thus we complete the proof.

    From Corollary 11 and Theorem 12, we obtain the following two corollaries.

    Corollary 13. Suppose homogeneous random variables X1,X2, having an Archimedean copula with generator ψ, and let N(2) be positive integer-valued random variable which is independent of Xi, i=1,2,. If tψ(t)/ψ(t) is convex and tψ(t)/ψ(t) is decreasing, and nk=2P(Nn)/n1k=2P(Nn1) is decreasing in n2, then

    XN1:NrhXN:N+1.

    Corollary 14. Suppose homogeneous random variables X1,X2, having an Archimedean copula with generator ψ, and let N(2) be positive integer-valued random variable which is independent of Xi, i=1,2,. If tψ(t)/ψ(t) is decreasing, and P(N=n)/P(N=n1) is decreasing in n2, then

    XN1:NlrXN:N+1.

    In the following, we present an example which satisfies the condition P(N=n)/P(N=n1) is decreasing in n2 in Corollary 14.

    Example 15. Suppose positive integer-valued random variable N follows distribution with density function

    P(N=k)=λk2(k2)!eλ,k=2,3,,

    where λ>0. It is easy to check that

    P(N=n)P(N=n1)=λn2

    is decreasing in n2.

    In reliability theory, the k-out-of-n system as the popular fault tolerant system has been widely applied in industrial engineering and system security. Specifically, X1:n and Xn:n denote the lifetimes of series and parallel systems, X2:n and Xn1:n characterize the lifetime of the fail-safe system and 2-out-of-n system, respectively. Theorem 4 states that adding a more component to 2-out-of-n system as a redundancy will lead to a more reliable system in the sense of the reversed hazard rate order.

    The second-price sealed-bid auction is of important theoretical and practical interest in auction theory. There are several bidders competing to buy a good, bidders hand in their bids to the auctioneers simultaneously without the knowledge of their rivals' bids. The bidder with the highest bid wins the object and pays the second highest bid in the English auction. Theorem 4 states that attracting one more bidder makes the final price of second-price sealed-bid auction stochastically higher in terms of the reversed hazard rate order.

    In this paper, in the context of system consisting of dependent and homogeneous components, we investigate the problem of stochastic comparisons of the second largest order statistics, and we build the reversed hazard rate and likelihood ratio orders, and we further generalize the corresponding results to the case of random sample size. We also derive some results for relative ageing between parallel systems and 2-out-of-n/2-out-of-(n+1) systems in terms of the increasing reversed hazard rate order. And we present two applications of the main results. The hazard rate and likelihood ratio orders for the second smallest order statistics can be obtained in a similar method, also, relative ageing between series systems and (n1)-out-of-n/n-out-of-(n+1) systems in terms of the increasing hazard rate order.

    This research was supported by National Natural Science Foundation of China (11861058).

    The authors declare no conflict of interest.



    [1] R. E. Barlow, F. Proschan, Statistical theory of reliability and life testing: probability models, Holt Rinehart & Winston of Canada Ltd, 1975.
    [2] R. Fang, X. H. Li, Advertising a second-price auction, J. Math. Econ., 61 (2015), 246–252. doi: 10.1016/j.jmateco.2015.04.003
    [3] P. J. Boland, E. El-Neweihi, F. Proschan, Applications of the hazard rate ordering in reliability and order statistics, J. Appl. Prob., 31 (1994), 180–192. doi: 10.2307/3215245
    [4] M. Z. Raqab, W. A. Amin, Some ordering results on order statistics and record values, Iapqr Trans., 21 (1996), 1–8.
    [5] R. B. Bapat, S. C. Kochar, On likelihood-ratio ordering of order statistics, Linear Algebra Appl., 199 (1994), 281–291. doi: 10.1016/0024-3795(94)90353-0
    [6] X. H. Li, R. Fang, Ordering properties of order statistics from random variables of archimedean copulas with applications, J. Multivariate Anal., 133 (2015), 304–320. doi: 10.1016/j.jmva.2014.09.016
    [7] R. Fang, X. H. Li, Ordering extremes of interdependent random variables, Commun Stat.-Theory M., 47 (2018), 4187–4201. doi: 10.1080/03610926.2017.1371754
    [8] M. Mesfioui, M. Kayid, S. Izadkhah, Stochastic comparisons of order statistics from heterogeneous random variables with archimedean copula, Metrika, 80 (2017), 749–766. doi: 10.1007/s00184-017-0626-z
    [9] G. Barmalzan, N. Balakrishnan, S. M. Ayat, A. Akrami, Orderings of extremes dependent modified proportional hazard and modified proportional reversed hazard variables under archimedean copula, Commun Stat.-Theory M., 2020, 1–22.
    [10] G. Pledger, F. Proschan, Comparisons of order statistics and of spacings from heterogeneous distributions, In: Optimizing methods in statistics, Proceedings of a Symposium Held at the Center for Tomorrow, New York: Academic Press, 1971, 89–113.
    [11] B. E. Khaledi, S. Kochar, Some new results on stochastic comparisons of parallel systems, J. Appl. Probab., 37 (2000), 1123–1128. doi: 10.1239/jap/1014843091
    [12] N. Torrado, On magnitude orderings between smallest order statistics from heterogeneous beta distributions, J. Math. Anal. Appl., 426 (2015), 824–838. doi: 10.1016/j.jmaa.2015.02.003
    [13] N. Torrado, Tail behaviour of consecutive 2-within-m-out-of-n systems with nonidentical components, Appl. Math. Model., 39 (2015), 4586–4592. doi: 10.1016/j.apm.2014.12.042
    [14] A. Arriaza, M. A. Sordo, A. Súarez-Llorens, Comparing residual lives and inactivity times by transform stochastic orders, IEEE T. Reliab., 66 (2017), 366–372. doi: 10.1109/TR.2017.2679158
    [15] M. V. Koutras, I. S. Triantafyllou, S. Eryilmaz, Stochastic comparisons between lifetimes of reliability systems with exchangeable components, Methodol. Comput. Appl. Probab., 18 (2016), 1081–1095. doi: 10.1007/s11009-014-9433-4
    [16] J. Navarro, Stochastic comparisons of coherent systems, Metrika, 81 (2018), 465–482. doi: 10.1007/s00184-018-0650-7
    [17] J. Navarro, Y. del Águila, Stochastic comparisons of distorted distributions, coherent systems and mixtures with ordered components, Metrika, 80 (2017), 627–648. doi: 10.1007/s00184-017-0619-y
    [18] R. Fang, C. Li, X. H. Li, Ordering results on extremes of scaled random variables with dependence and proportional hazards, Statistics, 52 (2018), 458–478. doi: 10.1080/02331888.2018.1425998
    [19] J. Navarro, N. Torrado, Y. del Águila, Comparisons between largest order statistics from multiple-outlier models with dependence, Methodol. Comput. Appl. Probab., 20 (2018), 411–433. doi: 10.1007/s11009-017-9562-7
    [20] M. M. Zhang, B. Lu, R. F. Yan, Ordering results of extreme order statistics from dependent and heterogeneous modified proportional (reversed) hazard variables, AIMS Mathematics, 6 (2020), 584–606.
    [21] J. Navarro, F. Durante, J. Fernández-Sánchez, Connecting copula properties with reliability properties of coherent systems, Appl. Stoch. Model. Bus., 2020, DOI: 10.1002/asmb.2579.
    [22] J. Navarro, J. Mulero, Comparisons of coherent systems under the time-transformed exponential model, TEST, 29 (2020), 255–281. doi: 10.1007/s11749-019-00656-4
    [23] V. V. Kalashnikov, S. T. Rachev, Characterization of queueing models and their stability, Adv. Appl. Prob., 17 (1985), 868–886. doi: 10.2307/1427091
    [24] M. Rezaei, B. Gholizadeh, S. Izadkhah, On relative reversed hazard rate order, Commun. Stat.-Theory M., 44 (2015), 300–308. doi: 10.1080/03610926.2012.745559
    [25] C. D. Lai, M. Xie, Relative ageing for two parallel systems and related problems, Math. Comput. Model., 38 (2003), 1339–1345. doi: 10.1016/S0895-7177(03)90136-1
    [26] C. Li, X. H. Li, Relative ageing of series and parallel systems with statistically independent and heterogeneous component lifetimes, IEEE T. Reliab., 65 (2016), 1014–1021. doi: 10.1109/TR.2015.2512226
    [27] W. Y. Ding, Y. Y. Zhang, Relative ageing of series and parallel systems: Effects of dependence and heterogeneity among components, Oper. Res. Lett., 46 (2018), 219–224. doi: 10.1016/j.orl.2018.01.005
    [28] D. Sengupta, J. V. Deshpande, Some results on the relative ageing of two life distributions, J. Appl. Prob., 31 (1994), 991–1003. doi: 10.1017/S0021900200099514
    [29] N. Misra, J. Francis, Relative ageing of (n-k+1)-out-of-n systems, Stat. Probabil. Lett., 106 (2015), 272–280. doi: 10.1016/j.spl.2015.07.013
    [30] N. Misra, J. Francis, Relative aging of (n-k+1)-out-of-n systems based on cumulative hazard and cumulative reversed hazard functions, Nav. Res. Log., 65 (2018), 566–575. doi: 10.1002/nav.21822
    [31] N. K. Hazra, N. Misra, On relative ageing of coherent systems with dependent identically distributed components, Adv. Appl. Probab., 52 (2020), 348–376. doi: 10.1017/apr.2019.63
    [32] N. K. Hazra, N. Misra, On relative aging comparison of coherent systems with identically distributed components, Probab. Eng. Inform. Sc., 2020, 1–15.
    [33] P. C. Consul, On the distributions of order statistics for a random sample size, Stat. Neerl., 38 (1984), 249–256. doi: 10.1111/j.1467-9574.1984.tb01115.x
    [34] M. Shaked, T. Wong, Stochastic orders based on ratios of Laplace transforms, J. Appl. Prob., 34 (1997), 404–419. doi: 10.2307/3215380
    [35] M. Shaked, T. Wong, Stochastic comparisons of random minima and maxima, J. Appl. Prob., 34 (1997), 420–425. doi: 10.2307/3215381
    [36] X. H. Li, M. J. Zuo, Preservation of stochastic orders for random minima and maxima, with applications, Nav. Res. Log., 51 (2004), 332–344. doi: 10.1002/nav.10122
    [37] M. Shaked, G. Shanthikumar, Stochastic orders, New York: Springer, 2007.
    [38] H. J. Li, X. H. Li, Stochastic orders in reliability and risk, New York: Springer, 2013.
    [39] F. Belzunce, C. Martinez-Riquelme, J. Mulero, An introduction to stochastic orders, London: Elsevier Academic, 2015.
    [40] C. D. Lai, M. Xie, Stochastic ageing and dependence for reliability, New York: Springer, 2006.
    [41] R. B. Nelsen, An introduction to copulas, New York: Springer, 2006.
    [42] S. Karlin, Total positivity, Vol I, Stanford: Stanford University Press, 1968.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2419) PDF downloads(123) Cited by(0)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog