# Orderings of Extreme Claim Amounts from Heterogeneous and Dependent Weibull-G Insurance Portfolios

Authors: Ling Liu; Rongfang Yan
Journal: Journal of Mathematics
 (2022)

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## Abstract

In the context of insurance, the smallest and largest claim amounts become increasingly important in the context of insurance analysis, which provides crucial and key information for setting an annual premium. We first give some sufficient conditions for comparing the largest claim amounts arising from two sets of dependent heterogeneous Weibull-G insurance portfolios with respect to the usual stochastic order under Archimedean copula dependence. Second, we establish some sufficient conditions to compare the smallest claim amounts in the sense of the usual stochastic and hazard rate orders when the claim severities are independent or dependent heterogeneous Weibull-G insurance portfolios. Finally, to illustrate the key theoretical insights, some numerical examples are offered.


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## Body

## 1. Introduction

In the context of insurance, annual premium is the amount paid by the policyholder to purchase the insurance policy in order to transfer risk. In determining the proper annual premium, the stochastic behavior of the extreme claim amounts is critical. In need, the largest claim amount is contained in the LCR (largest claim reinsurance) and the ECOMOR (excedent du cout moyen relatif) reinsurance treaties, which are widely used in pricing insurance portfolios of the policyholder (cf. Ammeter. [1] and Hansjörg Albrecher and Teugels [2]). Consider an insurance portfolio consisting of n business. Let X1,X2,…,Xn be a set of nonnegative random variables with Xi denoting the ith claim size in given insurance portfolio, for i=1,2,…,n and let I1,I2,…,In be a group of Bernoulli random variables, independent of the Xi′s, with Ii indicating whether the claim Xi occurs or not and such that EIi=pi for i=1,2,…,n. Thus, the risk vector of the insurance portfolio can be expressed as I1X1,I2X2,…,InXn. The past decades have witnessed the rich developments in the investigation of stochastic properties of different quantities for the individual risk model. For example, Zhang et al. [3, 4] investigated how the heterogeneity among occurrence probabilities and claim severities affects the aggregate claim numbers and aggregate claim amount for an insurance portfolio. Zhang et al. [5] established sufficient conditions for comparing extreme claim amounts arising from two sets of heterogeneous insurance portfolios according to various stochastic orders.Many scholars have been studying insurance and reinsurance in recent decades, particularly the largest and smallest claim amounts. For example, Barmalzan and Payandeh Najafabadi [6] investigated the ordering properties of the smallest claim amounts from a set of Weibull heterogeneous portfolios in the sense of the convex transform order and the right spread order. Barmalzan et al. [7] discussed the likelihood ratio order and dispersive order between the smallest claim amounts from two sets of independent heterogeneous Weibull claims. Barmalzan et al. [8] investigated the smallest claim amounts in terms of the usual stochastic and hazard rate orders, as well as established the usual stochastic order between the largest claim amounts for a general scale model. Balakrishnan et al. [9] gave the stochastic orderings between the largest claim amounts and ranges from two sets of independent heterogeneous portfolio risks. Zhang et al. [5] established sufficient conditions to stochastically compare the largest and smallest claim amounts arising from two sets of heterogeneous insurance portfolios in the sense of the usual stochastic, hazard rate, likelihood ratio, right-spread, and convex transform orders. Naqvi et al. [10] investigated the effects of heterogeneity among occurrence probabilities and claim severities on the aggregate claim amount when the claim severities are right tail weakly stochastic arrangement increasing for the location-scale family. Barmalzan et al. [11] built stochastic comparisons for two largest claim amounts from two different sets of random variables with different or identical parameters. Das et al. [12] compared the smallest claim amounts arising from two heterogeneous exponential location-scale claims with respect to the usual stochastic, hazard rate and likelihood orderings. Nadeb et al. [13] obtained some stochastic comparisons results between two extreme claim amounts from two heterogeneous portfolios for the transmuted-G model. Recently, Zhang [14] investigated the ordering properties of the largest claim amounts with heterogeneous (and homogeneous) insurance portfolios in terms of the convex transform order and the star order.Alzaatreh et al. [15] proposed a new method to generate continuous distributions: let X be a random variable with distribution Fx, T be a continuous random variable with support a,b−∞≤a<b≤∞ and density function ut, respectively. For a function wx: 0,1⟶a,b, if wFx is differentiable and monotonically nondecreasing, then limx⟶−∞wFx=a, limx⟶∞wFx=b. Then,(1)HTXx=∫awFxutdt,x∈ℝ,is a continuous distribution. Choose T the Weibull random variable, that is,(2)ut=αβtβ−1e−αtβ,t∈ℝ+;α,β>0,and wFγx=Fγx/1−Fγx, and X follows a scale family of distribution, i.e., X∼Fγxγ>0, the distribution function in (1) is just the Weibull-G distribution defined by Bourguignon et al. [16]. For convenience, denote the Weibull-G distribution by W-Gα,β,γ. The Weibull-G distribution can be expressed as(3)HWGx=1−exp−αFγxF¯γxβ,x∈ℝ+;α,β,γ∈ℝ+,where α and β are the scale and shape parameters, respectively, and Fγx is a baseline distribution function with scale parameter γ. As described in Bourguignon et al. [16]; (3) can deal with general situations in modeling data with various shapes of the hazard rate function.Cossette et al. [17] proposed two constructions, which allow dependence between the risks of an insurance portfolio in the individual risk model. Sarabia et al. [18] studied the aggregation of dependent risks in mixtures of exponential distributions. Zhang, Li and Cheung [3] discussed the effects of orderings and heterogeneity among scale parameters on the aggregate claim amount when both claim occurrence probabilities and claim severities are dependent. Torrado and Navarro [19] investigated the extreme claim amounts in dependent individual risk models. Inspired the these outstanding works, we will investigate the stochastic behaviors of the largest and smallest claim amounts with dependent insurance portfolios, and provide some quite significantly perceptions and insights into determining the annual premium for an insurant, we try to establish the stochastic comparisons of the largest and smallest claim amounts from two different sets of the dependent and heterogeneous Weibull-G random variables under Archimedean copula.The reminder of this paper is proceed as follows. In Section2, we first introduce some key definitions and notation pertinent to stochastic orders, multivariate majorization, vector majorization, related orders and copulas. Section 3 present the usually stochastic order of the independent largest claim amounts from two sets of independent heterogeneous portfolio risks with chain majorized parameters. Section 4, we develop some stochastic comparisons of the smallest claim amounts in two cases: (i) when claim severities are independent, and then (ii) when claim severities are Archimedean copula dependent. Conclusions and discussions are given in Section 6.


## 2. Preliminaries

In this section, we recall some basic definitions and lemmas, which are useful to obtain the main result in subsequent developments. Throughout this paper, the terms increasing and decreasing are used for monotone non-decreasing and monotone nonincreasing, respectively, and “=sign” means that both sides of the equality have the same sign. We denote ℝ=−∞,+∞, ℝ+=0,+∞, and h−1 the inverse function of strictly monotone function h, and f′, f″ and fk the first, second and k th derivatives of a differentiable function f, respectively. All random variables mentioned are assumed to be absolutely continuous.


### 2.1. Stochastic Orders

For a nonnegative random variableX, denote the distribution function, survival function, density function, hazard rate function and reversed hazard rate function of X by FXx, F¯Xx, fXx, hXx and r˜Xx, respectively. Then, some fundamental stochastic orders are follows.Definition 1.
(Definition1.A.1 on Page 3, 1.B.1 on Page 16, and 1.B.6 on Page 36 of Shaked and Shanthikumar [20]). X is said to be larger than Y in the(i)
Usual stochastic order (denoted byX≥stY) if F¯Xx≥F¯Yx for all x∈ℝ+(ii)
Hazard rate order (denoted byX≥hrY) if and only if F¯Xx/F¯Yx is increasing in x∈ℝ+, or equivalently, hXx≤hYx for all x∈ℝ+
The hazard rate order is generally known to imply the usual stochastic order, while the opposite is not true. Interested readers may refer to Müller and Stoyan [21] and Shaked and Shanthikumar [20] for comprehensive discussions on various stochastic orders and their applications in different contexts. Besides, the hazard function also known as Mills’s ratio (cf. Škanata [22] in the context of actuarial risk science, and the hazard function are seen as the fundamental feature that may be used to describe probabilistic component of risk and may be used to describe time evolution of the probabilistic component of risk. For more discussion of the hazard function in actuarial risk, one can refer to the work of Škanata [22].


### 2.2. Majorizations

The concept of the majorization order is a key tool in establishing various inequalities arising from actuarial science, reliability theory and other applied probability fields. Letx1:n≤x2:n≤…≤xn:n be the increasing arrangement of the components of the vector x=x1,x2,…,xn, then we have the following definition.Definition 2.
(Definition A.1 on Page 8 and A.2 on Page 42 of Marshall et al. [23]). A vector x=x1,x2,…,xn∈ℝn is said to(i)
Majorize another vectory=y1,y2,…,yn∈ℝn, denoted by x≽my, if ∑i=1jxi:n≤∑i=1jyi:n for j=1,2,…,n−1, and ∑i=1nxi=∑i=1nyi(ii)
Weakly submajorize another vectory∈ℝn, denoted by x≽wy, if ∑i=jnxi:n≥∑i=jnyi:n for every j=1,2,…,n
It is clear thatx≽my implies x≽wy. For an elaborate discussion on the theory of the majorization order and their applications, one may refer to Marshall et al. [23] and Kundu et al. [24].Definition 3.
(Definition3.A.1 on Page 80 of Marshall et al. [23]). We say that a real-valued function ϕ, defined on a set A⊆ℝn, is Schur-convex (Schur-concave) on A, if and only if x≽my implies ϕx≥≤ϕy, for any x,y∈A.Lemma 1.
(Theorem A.4 on Page 84 of Marshall et al. [23]). Consider the real-valued continuously differentiable function ϕ on Jn, where J⊆ℝ is an open interval. Then, ϕ is Schur-convex (Schur-concave) on Jn if and only if ϕ is symmetric on Jn, and for all i≠j and all z∈Jn,(4)zi−zj∂ϕz∂zi−∂ϕz∂zj≥≤0,where ∂ϕz/∂zi denotes the partial derivative of ϕ with respect to its ith argument.Lemma 2.
(Definition A.4 on Page 15 Marshall et al. [23]). Consider a real-valued function φ, defined on a set A⊆ℝn. Then, x≽wy, implies φx≥≤φy if and only if φ is increasing (decreasing) and schur-convex (schur-concave) on A.Lemma 3.
(Definition B.2 on Page 90 Marshall et al. [23]). Consider the composition function ψx=ϕgx1,gx2,…,gxn, where ϕ:ℝn↦ℝ and g:ℝ↦ℝ are two real-valued functions. If ϕ is decreasing and Schur-convex in x, and g is concave, then ψx is Schur-convex in x.
Now, we describe the notion of the majorization on matrices. A square matrixΠn, of order n, is said to be a permutation matrix if each row and column has a single entry as 1, and all other entries are zero. It is obvious that a permutation matrix can be constructed by interchanging rows (or columns) of the n×n identity matrix In. The T-transform matrix has the form(5)Tw=wIn+1−wΠn,0≤w≤1,whereIn is the identity matrix and Πn is a permutation matrix that only interchanges two co-ordinates. It should be mentioned here that all permutation matrices used in the T-transform matrix are matrices that interchange only two columns of the identity matrix In. For two T-transform matrices Tw1=w1In+1−w1Π1 and Tw2=w2In+1−w2Π2, they are said to be have the same structure if Π1=Π2. It is well-known that a finite product of T-transform matrices with the same structure is also a T-transform matrix, while it is not necessarily a T-transform matrix for different structures(see Balakrishnan et al. [25]. Below, we introduce the concept of the multivariate majorization.Definition 4.
(Definition6 on Page 4 of Marshall et al. [23]). Consider two m×n matrices U=uij and S=sij, where i=1,2,…,m and j=1,2,…,n. Then, a matrix U is larger than matrix S in the chain majorization, denoted by U≫S, if there exists a finite set of n×n T-transform matrices Tw1,Tw2,…,Twn such that S=UTw1Tw2…Twn.
Interested readers may refer to Marshall et al. [23] for an elaborate discussion on the theory of vector and matrix majorizations and their applications in various contexts. For simplicity, let us denote(6)ℳn=x,y=x1…xny1…yn:xi,yi>0andxi−xjyi−yj≥0,i,j=1,2,…,n,D+=x1,x2,…,xn∈ℝn:x1≥x2≥…≥xn,ℐ+=x1,x2,…,xn∈ℝn:x1≤x2≤…≤xn.
Next, we present some results that are required for established on the multivariate chain majorizations. First, we present to lemmas that are useful for comparison results when heterogeneity in two parameters.Lemma 4 (Theorem2 on Page 4 of Balakrishnan et al. [25]).
Consider the differentiable functionϕ:R+4⟶ℝ+. Then,(7)ϕU≤ϕVfor allU,V,such thatU∈ℳ2andU≫V,if and only if(i)
ϕU=ϕUΠ for all permutation matrices Π(ii)
∑i=12uik−uijϕikU−ϕijU≤0 for all j,k=1,2, where ϕijU=∂ϕU/∂uijLemma 5.
(Theorem3 on Page 4 of Balakrishnan et al. [25]). Consider the differentiable function ϕ:R+2⟶R+, and let the function φn:R+2n⟶R+, be defined as φn=∏i=1nϕu1i,u2i. If φ2 satisfies (3), then φnU≥φnV, where U∈Sn and V=UT.


### 2.3. Archimedean Copula

Now, let us review the concept of the copula. LetX=X1,X2,…,Xn be a random vector. The joint distribution and the survival functions of X are denoted by H and H¯, respectively. Further, we assume that FX1,FX2,…,FXn and F¯X1,F¯X2,…,F¯Xn are respectively the marginal distribution functions and the survival functions of X1,X2,…,Xn. When the random variables X1,X2,…,Xn are dependent, the joint cumulative distribution function can be represented in terms of the copula C as(8)Hx1,x2,…,xn=CFX1,FX2,…,FXn.Further, the joint survival function can also be represented in terms of the survival copula C^, which is given by(9)H¯x1,x2,…,xn=C^F¯X1,F¯X2,…,F¯Xn.It is noted that the copula is a multivariate distribution function of a random vector having uniform marginal distributions in the interval0,1.Definition 5.
(Expression 4.6.1 on Page 151 of Nelsen [26]). For a decreasing and continuous function ψ:ℝ+↦0,1 such that ψ0=1, ψ+∞=0. Denote ϕ=ψ−1 the pseudo-inverse of ψx, and ψkx the kth derivative of the ψx (k=0,1,…,n−2). If −1kψkx≥0 for all 0≤k≤n−2, and −1n−2ψn−2x is decreasing and convex, then(10)Cψu1,u2,…,un=ψ∑i=1nϕui,ui∈0,1,is called an Archimedean copula with the generator ψ.
Archimedean copulas are widely used to model the dependence among variables due to their mathematical tractability as well as their ability to model a wide range of dependence structures.Definition 6 (Li and Fang [27]).
A functionf is said to be supper-additive if fx+y≥fx+fy for all x and y in the domain of f.
Based on Lemma A.1 in Li and Fang [27]; it is known that, for two n-dimensional Archimedean copulas Cψ1u and Cψ2u with respective generators ψ1 and ψ2 and pseudo-inverses ϕ1 and ϕ2, if ϕ2ψ1 is supper-additive, then Cψ1u≤Cψ2u for all u∈0,1n. For more detailed discussions on copulas and their properties, one may refer to Nelsen [26] and McNeil and Nešlehová [28].


## 2.1. Stochastic Orders

For a nonnegative random variableX, denote the distribution function, survival function, density function, hazard rate function and reversed hazard rate function of X by FXx, F¯Xx, fXx, hXx and r˜Xx, respectively. Then, some fundamental stochastic orders are follows.Definition 1.
(Definition1.A.1 on Page 3, 1.B.1 on Page 16, and 1.B.6 on Page 36 of Shaked and Shanthikumar [20]). X is said to be larger than Y in the(i)
Usual stochastic order (denoted byX≥stY) if F¯Xx≥F¯Yx for all x∈ℝ+(ii)
Hazard rate order (denoted byX≥hrY) if and only if F¯Xx/F¯Yx is increasing in x∈ℝ+, or equivalently, hXx≤hYx for all x∈ℝ+
The hazard rate order is generally known to imply the usual stochastic order, while the opposite is not true. Interested readers may refer to Müller and Stoyan [21] and Shaked and Shanthikumar [20] for comprehensive discussions on various stochastic orders and their applications in different contexts. Besides, the hazard function also known as Mills’s ratio (cf. Škanata [22] in the context of actuarial risk science, and the hazard function are seen as the fundamental feature that may be used to describe probabilistic component of risk and may be used to describe time evolution of the probabilistic component of risk. For more discussion of the hazard function in actuarial risk, one can refer to the work of Škanata [22].


## 2.2. Majorizations

The concept of the majorization order is a key tool in establishing various inequalities arising from actuarial science, reliability theory and other applied probability fields. Letx1:n≤x2:n≤…≤xn:n be the increasing arrangement of the components of the vector x=x1,x2,…,xn, then we have the following definition.Definition 2.
(Definition A.1 on Page 8 and A.2 on Page 42 of Marshall et al. [23]). A vector x=x1,x2,…,xn∈ℝn is said to(i)
Majorize another vectory=y1,y2,…,yn∈ℝn, denoted by x≽my, if ∑i=1jxi:n≤∑i=1jyi:n for j=1,2,…,n−1, and ∑i=1nxi=∑i=1nyi(ii)
Weakly submajorize another vectory∈ℝn, denoted by x≽wy, if ∑i=jnxi:n≥∑i=jnyi:n for every j=1,2,…,n
It is clear thatx≽my implies x≽wy. For an elaborate discussion on the theory of the majorization order and their applications, one may refer to Marshall et al. [23] and Kundu et al. [24].Definition 3.
(Definition3.A.1 on Page 80 of Marshall et al. [23]). We say that a real-valued function ϕ, defined on a set A⊆ℝn, is Schur-convex (Schur-concave) on A, if and only if x≽my implies ϕx≥≤ϕy, for any x,y∈A.Lemma 1.
(Theorem A.4 on Page 84 of Marshall et al. [23]). Consider the real-valued continuously differentiable function ϕ on Jn, where J⊆ℝ is an open interval. Then, ϕ is Schur-convex (Schur-concave) on Jn if and only if ϕ is symmetric on Jn, and for all i≠j and all z∈Jn,(4)zi−zj∂ϕz∂zi−∂ϕz∂zj≥≤0,where ∂ϕz/∂zi denotes the partial derivative of ϕ with respect to its ith argument.Lemma 2.
(Definition A.4 on Page 15 Marshall et al. [23]). Consider a real-valued function φ, defined on a set A⊆ℝn. Then, x≽wy, implies φx≥≤φy if and only if φ is increasing (decreasing) and schur-convex (schur-concave) on A.Lemma 3.
(Definition B.2 on Page 90 Marshall et al. [23]). Consider the composition function ψx=ϕgx1,gx2,…,gxn, where ϕ:ℝn↦ℝ and g:ℝ↦ℝ are two real-valued functions. If ϕ is decreasing and Schur-convex in x, and g is concave, then ψx is Schur-convex in x.
Now, we describe the notion of the majorization on matrices. A square matrixΠn, of order n, is said to be a permutation matrix if each row and column has a single entry as 1, and all other entries are zero. It is obvious that a permutation matrix can be constructed by interchanging rows (or columns) of the n×n identity matrix In. The T-transform matrix has the form(5)Tw=wIn+1−wΠn,0≤w≤1,whereIn is the identity matrix and Πn is a permutation matrix that only interchanges two co-ordinates. It should be mentioned here that all permutation matrices used in the T-transform matrix are matrices that interchange only two columns of the identity matrix In. For two T-transform matrices Tw1=w1In+1−w1Π1 and Tw2=w2In+1−w2Π2, they are said to be have the same structure if Π1=Π2. It is well-known that a finite product of T-transform matrices with the same structure is also a T-transform matrix, while it is not necessarily a T-transform matrix for different structures(see Balakrishnan et al. [25]. Below, we introduce the concept of the multivariate majorization.Definition 4.
(Definition6 on Page 4 of Marshall et al. [23]). Consider two m×n matrices U=uij and S=sij, where i=1,2,…,m and j=1,2,…,n. Then, a matrix U is larger than matrix S in the chain majorization, denoted by U≫S, if there exists a finite set of n×n T-transform matrices Tw1,Tw2,…,Twn such that S=UTw1Tw2…Twn.
Interested readers may refer to Marshall et al. [23] for an elaborate discussion on the theory of vector and matrix majorizations and their applications in various contexts. For simplicity, let us denote(6)ℳn=x,y=x1…xny1…yn:xi,yi>0andxi−xjyi−yj≥0,i,j=1,2,…,n,D+=x1,x2,…,xn∈ℝn:x1≥x2≥…≥xn,ℐ+=x1,x2,…,xn∈ℝn:x1≤x2≤…≤xn.
Next, we present some results that are required for established on the multivariate chain majorizations. First, we present to lemmas that are useful for comparison results when heterogeneity in two parameters.Lemma 4 (Theorem2 on Page 4 of Balakrishnan et al. [25]).
Consider the differentiable functionϕ:R+4⟶ℝ+. Then,(7)ϕU≤ϕVfor allU,V,such thatU∈ℳ2andU≫V,if and only if(i)
ϕU=ϕUΠ for all permutation matrices Π(ii)
∑i=12uik−uijϕikU−ϕijU≤0 for all j,k=1,2, where ϕijU=∂ϕU/∂uijLemma 5.
(Theorem3 on Page 4 of Balakrishnan et al. [25]). Consider the differentiable function ϕ:R+2⟶R+, and let the function φn:R+2n⟶R+, be defined as φn=∏i=1nϕu1i,u2i. If φ2 satisfies (3), then φnU≥φnV, where U∈Sn and V=UT.


## 2.3. Archimedean Copula

Now, let us review the concept of the copula. LetX=X1,X2,…,Xn be a random vector. The joint distribution and the survival functions of X are denoted by H and H¯, respectively. Further, we assume that FX1,FX2,…,FXn and F¯X1,F¯X2,…,F¯Xn are respectively the marginal distribution functions and the survival functions of X1,X2,…,Xn. When the random variables X1,X2,…,Xn are dependent, the joint cumulative distribution function can be represented in terms of the copula C as(8)Hx1,x2,…,xn=CFX1,FX2,…,FXn.Further, the joint survival function can also be represented in terms of the survival copula C^, which is given by(9)H¯x1,x2,…,xn=C^F¯X1,F¯X2,…,F¯Xn.It is noted that the copula is a multivariate distribution function of a random vector having uniform marginal distributions in the interval0,1.Definition 5.
(Expression 4.6.1 on Page 151 of Nelsen [26]). For a decreasing and continuous function ψ:ℝ+↦0,1 such that ψ0=1, ψ+∞=0. Denote ϕ=ψ−1 the pseudo-inverse of ψx, and ψkx the kth derivative of the ψx (k=0,1,…,n−2). If −1kψkx≥0 for all 0≤k≤n−2, and −1n−2ψn−2x is decreasing and convex, then(10)Cψu1,u2,…,un=ψ∑i=1nϕui,ui∈0,1,is called an Archimedean copula with the generator ψ.
Archimedean copulas are widely used to model the dependence among variables due to their mathematical tractability as well as their ability to model a wide range of dependence structures.Definition 6 (Li and Fang [27]).
A functionf is said to be supper-additive if fx+y≥fx+fy for all x and y in the domain of f.
Based on Lemma A.1 in Li and Fang [27]; it is known that, for two n-dimensional Archimedean copulas Cψ1u and Cψ2u with respective generators ψ1 and ψ2 and pseudo-inverses ϕ1 and ϕ2, if ϕ2ψ1 is supper-additive, then Cψ1u≤Cψ2u for all u∈0,1n. For more detailed discussions on copulas and their properties, one may refer to Nelsen [26] and McNeil and Nešlehová [28].


## 3. On Usual Stochastic Ordering of Largest Claim Amounts

Consider two sets of heterogeneous and independent non-negative random observations from Weibull-G claim severities. Suppose thatωγx=Fγx/F¯γx. In this section, we establish the comparison results between the largest claim amounts arising from these sets in the sense of the usual stochastic order. Assume X1,X2,…,Xn are nonnegative random variables and I1,I2,…,In are Bernoulli random variables. Let Yi=IiXi,i=1,2,…,n, then the survival function of Yi can be expressed as(11)F¯Yix=ℙYi>x=ℙXiIi>x∣Ii=1pi+ℙXiIi>x∣Ii=01−pi=piF¯Xix,x∈ℝ+,i=1,2,…,n.


### 3.1. Insurance Portfolios of Independent Risks

Throughout the paper, we useYn:n and Yn:n∗ to denote the largest claim amounts from the insurance portfolios I1X1,I2X2,…,InXn and I1∗X1∗,I2∗X2∗,…,In∗Xn∗, respectively.The following Theorem1 provides some sufficient conditions for comparing the largest claim amounts in terms of the usual stochastic order.Theorem 1.
SupposeX1,X2X1∗,X2∗ are independent non-negative random variables with Xi∼W-Gαi,β,γXi∗∼W−Gαi∗,β,γ, i=1,2. Let Ip1,Ip2Ip1∗,Ip2∗ are independent Bernoulli random variables and independent of Xi′sXi∗′s, with EIpi=piEIpi∗=pi∗, i=1,2. Assume that h:0,1⟶0,∞ is a differentiable and strictly decreasing convex function. Then, for hp,α∈ℳ2, it holds that(12)hp1hp2α1α2≫hp1∗hp2∗α1∗α2∗⇒Y2:2≥stY2:2∗.Proof.
Without loss of generality, let us assume thatu1=hp1≥hp2=u2>0, u1∗=hp1∗≥hp2∗=u2∗>0, then α1≥α2>0 and α1∗≥α2∗>0 .Denotes the inverse function of h by h−1, and set pi=h−1ui (i=1,2), the distribution function of the largest claim amounts Y2:2 and Y2:2∗ can be expressed as(13)FY2:2x=∏i=121−h−1uie−αiwβγx,(14)FY2:2∗x=∏i=121−h−1ui∗e−αi∗wβγx,respectively. To establish the desired result, we need to check the conditions (i) and (ii) of Lemma 4. Clearly, for fixed x∈ℝ+, FY2:2x is the permutation invariant on ℳ2, which confirms the condition (i) of Lemma 4. On the other hand, for fixed x∈ℝ+ and 1=i,j=2, let(15)φu,α=u1−u2∂FY2:2x∂u1−∂FY2:2x∂u2+α1−α2∂FY2:2x∂α1−∂FY2:2x∂α2,then we have(16)∂FY2:2x∂ui=−FY2:2xdh−1uiduie−αiwβγx1−h−1uie−αiwβγx,(17)∂FY2:2x∂αi=FY2:2xh−1uiwβγxe−αiwβγx1−h−1uie−αiwβγx.
For anyu1≥u2, α1≥α2, equation (15) can be rewritten as(18)φu,α=u1−u2FY2:2xmu2,α2−mu1,α1+α1−α2FY2:2xnu1,α1−nu2,α2,where(19)mu,α=dh−1udue−αwβγx1−h−1ue−αwβγx,nu,α=h−1ue−αwβγxwβγx1−h−1ue−αwβγx.
Owing toh−1u is also strictly decreasing convex, we obtain(20)∂mu,α∂u=sign1−h−1ue−αwβγxd2h−1udu2e−αwβγx+dh−1udu2e−2αwβγx≥0,(21)∂mu,α∂α=−wβγxh−1ue−αwβγx1−h−1ue−αwβγx+e−2αwβγxdh−1udu=sign−e−αwβγxwβγxdh−1udu≥0.
It follows that(22)mu1,α1≥mu2,α1≥mu2,α2,which implies(23)mu2,α2−mu1,α1≤0.
Similarly, we have(24)nu2,α2−nu1,α1≤0,and thus φu,α≤0, which confirms the condition (ii) of Lemma 4. The desired result is proved.Remark 1.
Theorem 1 shows that the more the heterogeneity among the distributions of claim severities and occurrence probabilities, the larger the largest claim amounts, in the sense of the usual stochastic order.
TakingT-transform matrix as T=1/21/21/21/2 in Theorem 1, we immediately obtain a lower bound on the survival function of the largest claim amounts.Corollary 1.
Under the setup of Theorem 1,1−1−h−1h¯pe−α¯wβγx2 is a low bound of F¯Y2:2x, that is, for any x∈ℝ+, we have(25)F¯Y2:2x≥1−1−h−1h¯pe−α¯wβγx2.
Next, we give a numerical example to illustrate the effectiveness of Theorem1 and Corollary 1.Example 1.
Suppose the baseline random variable follows Weibull0.02,2 distribution. Suppose X1,X2X1∗,X2∗ are independent random variables with Xi∼W−G0.02,2,αi,3, Xi∗∼W−G0.02,2,αi∗,3, i=1,2. Let Ip1,Ip2Ip1∗,Ip2∗, are independent Bernoulli random variables, independent of Xi′s, with EIpi0=pi,i=1,2. To plot the whole survival curves of Y2:2 and Y2:2∗ in 0,∞, we perform the transformation e−x:0,∞⟶0,1. Then, it is obvious that Y2:2≥stY2:2∗ is equivalent with e−Y2:2≤ste−Y2:2∗.(i)
Supposeα1,α2=3,1, α1∗,α2∗=1.72,2.28, p1,p2=e−2,e−1 and p1∗,p2∗=e−1.36,e−1.64. Taking hp=−lnp and considering the following T-transform matrix,(26)T0.36=0.361001+0.640110,we then havehp1∗hp2∗α1∗α2∗=hp1hp2α1α2T0.36; thus,(27)hp1hp2α1α2≫hp1∗hp2∗α1∗α2∗.
It is easy to check that the conditions of Theorem 1 are all satisfied. As is seen in Figure1(a), the graph of F¯Y2:2x−F¯Y2:2∗x is always over the zero plane, that is, the difference functions F¯Y2:2x−F¯Y2:2∗x is always non-negative for all x=−lnu and u∈0,1, which coincides with the result of Theorem 1.(i)
Consider the following T-transform matrix, with other conditions unchanged,(28)T0.5=0.51001+0.50110.It is easy to observe thatα¯,α¯=2,2 and h¯p=3.5, where α¯=1/α1+α2. From Figure 1(b) which it can be see that the difference functions F¯Y2:2∗x−F¯Y2:2∗∗x is often non-negative, for all x=−lnu and u∈0,1, which verifies the result of Corollary 1.
The next Theorem2 will generalize the result of Theorem 1 to the largest claim amounts among n heterogeneous portfolio risks.Figure 1
(a) Plot of the difference functionsF¯Y2:2x−F¯Y2:2∗x. (b) Plot of the difference functions F¯Y2:2∗x−F¯Y2:2∗∗x, for all x=−lnu and u∈0,1.
(a)(b)Theorem 2.
SupposeX1,X2,…,XnX1∗,X2∗,…,Xn∗ are independent nonnegative random variables with Xi∼W−Gαi,β,γXi∗∼W−Gαi∗,β,γ, i=1,2,…,n. Let , Ip1,Ip2…,IpnIp1∗,Ip2∗,…,Ipn∗ are independent Bernoulli random variables, independent of Xi′sXi∗′s, with EIpi=piEIpi∗=pi∗, i=1,2,…,n. Then, under the same settings as in Theorem 1, for hp,α∈ℳ2, it holds that(29)hp1∗hp2∗⋯hpn∗α1∗α2∗⋯αn∗=hp1hp2⋯hpnα1α2⋯αnT⇒Yn:n≥stYn:n∗.Proof.
Settingφnp,α=FYn:nx and ϕp,α=1−pe−αwβγx, for any x≥0, we then have φnp,α=∏i=1nϕpi,αi. According to Theorem 1, φ2 satisfies in (7) of Lemma 4. Now, the required result follows immediately from Lemma 5.Remark 2.
Note that the distribution in Theorem2 and Theorem 1 are the special case of the proportional hazard rate model, hence, our theoretical findings verify the case of the proportional hazard rate model. Besides, Theorem 2 result indicates that more heterogeneity among some specified transformed occurrence probabilities and scale parameter α in the sense of the chain majorization order leads to the greater survival function of the largest claim amount.
The following Theorem3 gives a sufficient condition under which Y2:2≥stY2:2∗ holds when the baseline distribution follows exponential. We only provide the result of n=2 case.Theorem 3.
SupposeX1,X2X1∗,X2∗ are two independent non-negative random variables with Xi∼W−Expα,β,γiXi∗∼W−Expα,β,γi∗,i=1,2. Let Ip1,Ip2Ip1∗,Ip2∗ are independent Bernoulli random variables, independent of Xi′Xi∗′, with EIpi=piEIpi∗=pi∗,i=1,2. Assume that h:0,1⟶0,∞ is differentiable and strictly decreasing convex. Then, for any hp,γ∈ℳ2, we have(30)hp1hp2γ1γ2≫hp1∗hp2∗γ1∗γ2∗⇒Y2:2≥stY2:2∗.Proof.
Note that the baseline distributionFx=1−e−γx,x>0, which implies wFx=eγx−1. Set pi=h−1ui, for i=1,2. Then, the distribution function of the largest claim amounts Y2:2 is given by(31)FY2:2x=∏i=121−h−1uie−αeγix−1β.Clearly, for fixed x≥0, FY2:2x is the permutation invariant on ℳ2, which confirms the condition (i) of Lemma 4. On the other hand, we need prove the condition (ii) of Lemma 4, for any x≥0 and 1=i≠j=2, set(32)φu,γ=u1−u2∂FY2:2x∂u1−∂FY2:2x∂u2+γ1−γ2∂FY2:2x∂γ1−∂FY2:2x∂γ2,where u=u1,u2 and γ=γ1,γ2. Then, we have(33)∂FY2:2x∂ui=−FY2:2xdh−1uiduie−αeγix−1β1−h−1uie−αeγix−1β,∂FY2:2x∂γi=αβxFY2:2xh−1uieγixeγix−1β−1eαeγix−1β−1.
Similar to the proof of Theorem1, equation (32) can be represented as(34)φu,γ=u1−u2FY2:2xku2,γ2−ku1,γ1+αβxγ1−γ2FY2:2xlu1,γ1−lu2,γ2,where(35)ku,γ=dh−1udu1eαeγx−1β−h−1u,lu,γ=h−1ueγxeγx−1β−1eαeγx−1β−1,
Without loss of generality, assumeu1≥u2, γ1≥γ2. Owing to h−1 is a strictly decreasing convex function, we have(36)∂ku,γ∂u=signd2h−1udu2e−αwβγx1−h−1ue−αwβγx+dh−1udu2e−2αwβγx≥0.(37)∂ku,γ∂γ=sign−eαeγx−1βeγx−1β−1eγxxdh−1udu≥0.
Further, we have(38)ku1,γ1≥ku2,γ1≥ku2,γ2,which implies that the first side of (32) is nonpositive. Besides, we obtain(39)∂lu,γ∂u=dh−1udueγxeγx−1β−1eαeγx−1β−1≤0,(40)∂lu,γ∂γ=sign−αβeγx−1β−βeγx+1eαeγx−1β+βeγx−1=−φ1eγx,where(41)φ1t=αβt−1β−βt+1eαt−1β+βt−1,t≥1.
Whent>1, we conclude(42)dφ1tdt=βt−1eαt−1βαt−1β+α2βt−12β−1t+1+t−1,=signeαt−1βαt−1β+α2βt−12β−1t+1+t−1≕φ2t.
For anyt>1, it follows that(43)dφ2tdt=1−eαt−1β+eαt−1βαt−1β1+αβt−1β−11+2β+1t+βtt−1β+βt−1≥1−eαt−1β+αt−1βeαt−1β≕φ3αt−1β,where φ3z=1−ez+zez,z≥0. It follows from dφ3z/dz=zez≥0 and φ30=0 that dφ2t/dt≥0. Combining φ21=0 with φ11=0, we conclude that φ1t>0, which implies ∂lu,γ/∂γ≤0. Basing on u1≥u2 and γ1≥γ2, it follows that(44)lu1,γ1≤lu2,γ1≤lu2,γ2,which confirms the second side of (32) is also non-positive. Upon combining (38) with (44), we have φu,γ≤0; hence, condition (ii) of Lemma 4 is verified. The proof of desired result is completed.Remark 3.
According to Theorem3, more heterogeneity among some specified transformed occurrence probabilities and scale parameter γ in terms of the chain majorization order causes the greater survival function of the largest claim amount with two independent insurance portfolios.
The proof of the following Theorem4 is similar to that of Theorem 2 and thus omitted here.Theorem 4.
SupposeX1,X2,…,XnX1∗,X2∗,…,Xn∗ are independent nonnegative random variables with Xi∼W−Expα,β,γiXi∗∼W−Gα,β,γi∗, i=1,2,…,n. Let , Ip1,Ip2…,IpnIp1∗,Ip2∗,…,Ipn∗ are independent Bernoulli random variables, independent of Xi′sXi∗′s, with EIpi=piEIpi∗=pi∗, i=1,2,…,n. Then, under the same settings as in Theorem 1, for hp,γ∈ℳ2, it holds that(45)hp1∗hp2∗⋯hpn∗γ1∗γ2∗⋯γn∗=hp1hp2⋯hpnγ1γ2⋯γnT⇒Yn:n≥stYn:n∗.Remark 4.
Note that the distribution in Theorem3 and Theorem 4 are the special case of the scaled model in Barmalzan et al. [8]; hence, our established results check the setting of the scaled model. Meanwhile, according to Theorem 4, more heterogeneity among the scale parameter γ and the specified transformed occurrence probabilities with respect to the chain majorization order obtains the greater survival function of the largest claim amount from multiple independent insurance portfolios.
Naturally, one can question Theorem Theorems3 and 4 whether holds for other baseline function, unfortunately, the following numerical example gives a negative answer.Example 2.
Suppose the baseline random variable follows Frechet0,γ,1 distribution. Suppose X1,X2X1∗,X2∗ are independent random variables with Xi∼W−G0,γi,4,0.6,2, Xi∗∼W−G0,γi∗,4,0.6,2, i=1,2, β=2. Let Ip1,Ip2Ip1∗,Ip2∗, are independent Bernoulli random variables, independent of Xi′s, with EIpi=pi,i=1,2. To plot the whole survival curves of Y2:2 and Y2:2∗ in 0,∞. Suppose γ1,γ2=0.2,0.5, γ1∗,γ2∗=0.425,0.275, p1,p2=e−2,e−1 and p1∗,p2∗=e−1.25,e−1.75. Taking hp=−lnp and considering the following T-transform matrix,(46)T0.25=0.251001+0.640110.
We then havehp1∗hp2∗γ1∗γ2∗=hp1hp2γ1γ2T0.36; thus, a matrix hp1hp2γ1γ2 is larger than matrix hp1∗hp2∗γ1∗γ2∗ in the chain majorization, that is, it is easy to check that the conditions of Theorem 3 are all satisfied. As is seen in Figure 2, the survival functions F¯Y2:2x and F¯Y2:2∗x are each crossing, which states Theorem 3 is not necessarily hold for other baseline function.Figure 2
Plots of survival functionsF¯Y2:2x and F¯Y2:2∗x.


### 3.2. Insurance Portfolios of Dependent Risks

In this section, we use the usual stochastic order to compare the largest claim amountsYn:n and Yn:n∗ arising from two sets of dependent heterogeneous insurance portfolios when occurrence probabilities are dependent and claim severities are also dependent.To proceed, let us define0=0,0,…,0, 1=1,1,…,1, pλ=ℙI=λ and(47)Αk=λ|λi=0or1,i=1,2,…,n,λ1+λ2+⋯+λn=k,k=0,1,…,n.For all1≤i≠j≤n and k=1,2,…,n−1, denote(48)Αki,j0,0=λ∈Αk|λi=0,λj=0,Αki,j1,1=λ∈Αk|λi=1,λj=1.Obviously,Α1i,j1,1=Αn−1i,j0,0=Φ and(49)Αk=Αki,j1,0∪Αki,j0,1∪Αki,j1,1∪Αki,j0,0.With the help of the above notations, Cai and Wei [29] presented characterizations of stochastic arrangement increasing(SAI) Bernoulli random vector, which is very useful to model the positive dependence among occurrence probabilities.Lemma 6.
Cai and Wei [29]). A multivariate Bernoulli random vector I is SAI if and only if pτijλ≤pλ, for any λ∈Αki,j0,1,1≤i<j≤n and k=1,2,…,n−1.Theorem 5.
SupposeI is SAI, Xi∼W−Gαi,β,γi and the associated Archimedean copula is with ψ1ϕ1=ψ1−1 and Xi∗∼W−Gαi∗,β,γi, and the associated Archimedean copula with generator ψ2ϕ2=ψ2−1, for all i=1,2,…,n and αi,αi∗,γi∈ℝ+, γ,∈D+. Assume the following conditions hold:(i)
α1∗,α2∗,…,αn∗<¯mα1,α2,…,αn for α,α∗∈D+(ii)
ϕ2°ψ1 is super additive
Then, we haveYn:n≥stYn:n∗.Proof.
We need to show thatFYn:nx≤FYn:n∗x, for all x∈ℝ+. By the nature of majorization order, it is sufficient prove the desired result when αi∗,αj∗≼mαi,αj, for some pair 1≤i≤j≤n, and αs=αs∗ for s≠i,j. Let(50)FYsi,jx=FY1X,…,FYi−1X,FYi+1X,…,FYj−1X,FYj+1X,…,FYnX.
The distribution function ofYn:n is given by(51)FYn:nx=ℙmaxIp1X1,Ip2X2,…,IpnXn≤x=∑k=0n∑λ∈ΑkℙIp1X1≤x,…,IpnXn≤x∣I=λpλ=p0+p1maxX1,…,Xn≤x+∑k=1n−1∑λ∈ΑkpλℙmaxIp1X1,…,IpnXn≤x=p0+p1ψ1∑i=1nϕ11−e−αiwγixβ+∑k=1n−1∑λ∈Αki,j1,0pλℙλiXi≤x,λjXj≤x,λsXs≤x,s≠i,j+∑k=1n−1∑λ∈Αki,j0,1pλℙλiXi≤x,λjXj≤x,λsXs≤x,s≠i,j+∑k=1n−1∑λ∈Αki,j0,0pλℙλiXi≤x,λjXj≤x,λsXs≤x,s≠i,j+∑k=1n−1∑λ∈Αki,j1,1pλℙλiXi≤x,λjXj≤x,λsXs≤x,s≠i,j=p0+p1ψ1∑i=1nϕ11−e−αiwγixβ+∑k=1n−1∑λ∈Αki,j1,0pλψ1ϕ11+ϕ11−e−αiwβγix+∑s≠i,jnϕ11−pse−αswβγsx+∑k=1n−1∑λ∈Αki,j0,1pλψ1ϕ11+ϕ11−e−αjwβγjx+∑s≠i,jnϕ11−pse−αswβγsx+∑k=1n−1∑λ∈Αki,j0,0pλψ1ϕ11+ϕ11+∑s≠i,jnϕ11−pse−αswβγsx+∑k=1n−1∑λ∈Αki,j1,1pλψ1ϕ11−e−αjwβγjx+ϕ11−e−αiwβγix+∑s≠i,jnϕ11−pse−αswβγsx.
Similarly,(52)FYn:n∗x=p0+p1ψ2∑i=1nϕ21−e−αi∗wγixβ+∑k=1n−1∑λ∈Αki,j1,0pλψ2ϕ21+ϕ21−e−αi∗wβγix+∑s≠i,jnϕ21−pse−αs∗wβγsx+∑k=1n−1∑λ∈Αki,j0,1pλψ2ϕ21+ϕ21−e−αj∗wβγjx+∑s≠i,jnϕ21−pse−αs∗wβγsx+∑k=1n−1∑λ∈Αki,j0,0pλψ2ϕ21+ϕ21+∑s≠i,jnϕ21−pse−αs∗wβγsx+∑k=1n−1∑λ∈Αki,j1,1pλψ2ϕ21−e−αj∗wβγjx+ϕ21−e−αi∗wβγix+∑s≠i,jnϕ21−pse−αs∗wβγsx.
From the convexity and Schur-convexity of−ψ1∑i=1nϕ1ui and the concavity of Fx;α=1−e−αwβγx in α, we know that −ψ1∑i=1nϕ1ui is Schur-convex in α by applying Lemma 3. Hence, according to the super additivity of ϕ2°ψ1 and α∗≺¯mα, we have(53)ψ2∑i=1nϕ21−e−αi∗wγixβ≥ψ1∑i=1nϕ11−e−αi∗wγixβ≥ψ1∑i=1nϕ11−e−αiwγixβ,x∈ℝ+.
According toϕ2°ψ1 is superadditive and αi∗,αj∗≺¯mαi,αj, for any x∈ℝ+, we have(54)ψ2ϕ21+ϕ21+∑s≠i,jnϕ21−pse−αs∗wβγsx≥ψ1ϕ11+ϕ11+∑s≠i,jnϕ11−pse−αswβγsx,(55)ψ2ϕ21−e−αj∗wβγjx+ϕ21−e−αi∗wβγix+∑s≠i,jnϕ21−pse−αs∗wβγsx≥ψ1ϕ11−e−αjwβγjx+ϕ11−e−αiwβγix+∑s≠i,jnϕ11−pse−αswβγsx.
On the other hand, the concavities of bothψ1∑i=1nϕ1ui and Fx;α imply that ψ∑i=1nϕFx;αi is also concave in α, which leads to the fact that(56)ψ1ϕ11+ϕ11−e−αjwβγjx+∑s≠i,jnϕ11−pse−αswβγsx+ψ1ϕ11−e−αiwβγix+ϕ11+∑s≠i,jnϕ11−pse−αswβγsxis Schur-concave in α by proposition I:3.C.1 on Page 80 of Marshall et al. [23]; for any x∈ℝ+. Hence, it follows from the superadditivity of ϕ2°ψ1 and αi∗,αj∗≺¯mαi,αj, for all x∈ℝ+,(57)ψ2ϕ21+ϕ21−e−αj∗wβγjx+∑s≠i,jnϕ21−pse−αswβγsx+ψ2ϕ21+ϕ21−e−αi∗wβγix+∑s≠i,jnϕ21−pse−αswβγsx≥ψ1ϕ11+ϕ11−e−αj∗wβγjx+∑s≠i,jnϕ11−pse−αswβγsx+ψ1ϕ11+ϕ11−e−αj∗wβγjx+∑s≠i,jnϕ11−pse−αswβγsx≥ψ1ϕ11+ϕ11−e−αjwβγjx+∑s≠i,jnϕ11−pse−αswβγsx.
Observe that(58)FYn:n∗x−FYn:nx=p1ψ2∑i=1nϕ21−e−αi∗wγixβ−ψ1∑i=1nϕ11−e−αiwγixβ+∑k=1n−1∑λ∈Αki,j1,0pλψ2ϕ21+ϕ2Fx;αi∗+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ11+ϕ1Fx;αi+∑s≠i,jnϕ21−pse−αs∗wβγsx+∑k=1n−1∑λ∈Αki,j0,1pλψ2ϕ21+ϕ2Fx;αj∗+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ11+ϕ1Fx;αj+∑s≠i,jnϕ11−pse−αswβγsx+∑k=1n−1∑λ∈Αki,j0,0pλψ2ϕ21+ϕ21+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ11+ϕ11+∑s≠i,jnϕ11−pse−αs∗wβγsx+∑k=1n−1∑λ∈Αki,j1,1pλψ2ϕ2Fx;αj∗+ϕ2Fx;αi∗+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ1Fx;αj+ϕ1Fx;αi+∑s≠i,jnϕ11−pse−αs∗wβγsx≥∑k=1n−1∑λ∈Αki,j1,0pλψ2ϕ21+ϕ2Fx;αi∗+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ11+ϕ1Fx;αi+∑s≠i,jnϕ21−pse−αs∗wβγsx+∑k=1n−1∑λ∈Αki,j0,1pλψ2ϕ21+ϕ2Fx;αj∗+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ11+ϕ1Fx;αj+∑s≠i,jnϕ11−pse−αswβγsx≥∑k=1n−1∑λ∈Αki,j0,1pτi,jλψ2ϕ21+ϕ2Fx;αi∗+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ11+ϕ1Fx;αi+∑s≠i,jnϕ21−pse−αs∗wβγsx+∑k=1n−1∑λ∈Αki,j0,1pτi,jλψ2ϕ21+ϕ2Fx;αj∗+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ11+ϕ1Fx;αj+∑s≠i,jnϕ11−pse−αswβγsx=∑k=1n−1∑λ∈Αki,j0,1pτi,jλψ2ϕ21+ϕ2Fx;αi∗+∑s≠i,jnϕ21−pse−αs∗wβγsx+ψ2ϕ21+ϕ2Fx;αj∗+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ11+ϕ1Fx;αi+∑s≠i,jnϕ21−pse−αs∗wβγsx+ψ1ϕ11+ϕ1Fx;αj+∑s≠i,jnϕ11−pse−αswβγsx≥0,where the first inequality is from (53)–(55). The second inequality is due to Lemma 6 and the last inequality is based on (57). Hence, the desired result follows.Remark 5.
In the light of Theorem5, we know that more heterogeneity among the distribution functions of claim severities leads to a larger value of the largest claim amount in terms of the usual stochastic ordering when occurrences probabilities are positively dependent via SAI and claim severities are also dependent via a general copula that is concave and Schur-concave.
Next, we give a numerical example to illustrate the result of Theorem5.Example 3.
SupposeX1,X2,X3X1∗,X2∗,X3∗ are three non-negative random variables, let Fx=1−e−γx, and generators ψ1x=η1x+1−1/η1, η1>0, ψ2x=e1/η21−ex, 0<η2≤1. Set α∗=0.6,0.5,0.2≺¯w0.5,0.45,0.35=α, β∈0.8,1.6, η1=1.6, η2=0.8, γ=0.12,0.06,0.04, p0,0,0=0.16, p1,0,0=0.05, p0,1,0=0.17, p0,0,1=0.2, p1,1,0=0.02, p1,0,1=0.1, p0,1,1=0.18, and p1,1,1=0.15. It is easy to verify that the conditions of Theorem 5 are all satisfied. The survival function of Y3:3 can be expressed as(59)F¯Y3:3x=1−p0,0,0+p1,1,1ψ1ϕ11−e−α11eγ1x−1β+ϕ11−e−α12eγ2x−1β+ϕ11−e−α13eγ3x−1β+p1,0,0ψ1ϕ11−e−α11eγ1x−1β+2ϕ11+p0,1,0ψ1ϕ11−e−α12eγ2x−1β+ϕ11+ϕ11+p0,0,1ψ1ϕ11+ϕ11+ϕ11−e−α13eγ3x−1β+p1,1,0ψ1ϕ11−e−α11eγ1x−1β+ϕ11−e−α12eγ2x−1β+ϕ11+p1,0,1ψ1ϕ11−e−α11eγ1x−1β+ϕ11+ϕ11−e−α13eγ3x−1β+p0,1,1ψ1ϕ11+ϕ11−e−α12eγ2x−1β+ϕ11−e−α13eγ3x−1β.
The survival functions ofY3:3 and Y3:3∗, denoted as F¯Y3:3x and F¯Y3:3∗x are plotted in Figure 3, respectively, for all x=−lnu and u∈0,1, from which we see that survival function F¯e−Y3:3x is always larger than that of F¯e−Y3:3∗x. Thus, the effectiveness of Theorem 5 is validated.
The proof of the following Theorem6 is similar to that of Theorem 5, and thus omitted here.Figure 3
Plot of the difference functionsF¯e−Y3:3x−F¯e−Y3:3∗x, for all x=−lnu and u∈0,1.Theorem 6.
SupposeI is SAI, Xi∼W−Gαi,β,γi and the associated Archimedean copula is with ψ1ϕ1=ψ1−1 and Xi∗∼W−Gαi,β,γi∗, and the associated Archimedean copula with generator ψ2ϕ2=ψ2−1, for all i=1,2,…,n and γi,γi∗,αi∈ℝ+, αi,∈D+. Assume the following conditions hold:(i)
γ1∗,γ2∗,…,γn∗≺¯mγ1,γ2,…,γn. for γ,γ∗∈D+(ii)
ϕ2°ψ1 is super additive
Then, we haveYn:n≥stYn:n∗.Remark 6.
Theorem6 means more heterogeneity among the distribution functions of claim severities leads to a larger value of the largest claim amount among scale parameter γ in terms of the usual stochastic ordering when occurrences probabilities are positively dependent via SAI and claim severities are also dependent via a Archimedean copula.


## 3.1. Insurance Portfolios of Independent Risks

Throughout the paper, we useYn:n and Yn:n∗ to denote the largest claim amounts from the insurance portfolios I1X1,I2X2,…,InXn and I1∗X1∗,I2∗X2∗,…,In∗Xn∗, respectively.The following Theorem1 provides some sufficient conditions for comparing the largest claim amounts in terms of the usual stochastic order.Theorem 1.
SupposeX1,X2X1∗,X2∗ are independent non-negative random variables with Xi∼W-Gαi,β,γXi∗∼W−Gαi∗,β,γ, i=1,2. Let Ip1,Ip2Ip1∗,Ip2∗ are independent Bernoulli random variables and independent of Xi′sXi∗′s, with EIpi=piEIpi∗=pi∗, i=1,2. Assume that h:0,1⟶0,∞ is a differentiable and strictly decreasing convex function. Then, for hp,α∈ℳ2, it holds that(12)hp1hp2α1α2≫hp1∗hp2∗α1∗α2∗⇒Y2:2≥stY2:2∗.Proof.
Without loss of generality, let us assume thatu1=hp1≥hp2=u2>0, u1∗=hp1∗≥hp2∗=u2∗>0, then α1≥α2>0 and α1∗≥α2∗>0 .Denotes the inverse function of h by h−1, and set pi=h−1ui (i=1,2), the distribution function of the largest claim amounts Y2:2 and Y2:2∗ can be expressed as(13)FY2:2x=∏i=121−h−1uie−αiwβγx,(14)FY2:2∗x=∏i=121−h−1ui∗e−αi∗wβγx,respectively. To establish the desired result, we need to check the conditions (i) and (ii) of Lemma 4. Clearly, for fixed x∈ℝ+, FY2:2x is the permutation invariant on ℳ2, which confirms the condition (i) of Lemma 4. On the other hand, for fixed x∈ℝ+ and 1=i,j=2, let(15)φu,α=u1−u2∂FY2:2x∂u1−∂FY2:2x∂u2+α1−α2∂FY2:2x∂α1−∂FY2:2x∂α2,then we have(16)∂FY2:2x∂ui=−FY2:2xdh−1uiduie−αiwβγx1−h−1uie−αiwβγx,(17)∂FY2:2x∂αi=FY2:2xh−1uiwβγxe−αiwβγx1−h−1uie−αiwβγx.
For anyu1≥u2, α1≥α2, equation (15) can be rewritten as(18)φu,α=u1−u2FY2:2xmu2,α2−mu1,α1+α1−α2FY2:2xnu1,α1−nu2,α2,where(19)mu,α=dh−1udue−αwβγx1−h−1ue−αwβγx,nu,α=h−1ue−αwβγxwβγx1−h−1ue−αwβγx.
Owing toh−1u is also strictly decreasing convex, we obtain(20)∂mu,α∂u=sign1−h−1ue−αwβγxd2h−1udu2e−αwβγx+dh−1udu2e−2αwβγx≥0,(21)∂mu,α∂α=−wβγxh−1ue−αwβγx1−h−1ue−αwβγx+e−2αwβγxdh−1udu=sign−e−αwβγxwβγxdh−1udu≥0.
It follows that(22)mu1,α1≥mu2,α1≥mu2,α2,which implies(23)mu2,α2−mu1,α1≤0.
Similarly, we have(24)nu2,α2−nu1,α1≤0,and thus φu,α≤0, which confirms the condition (ii) of Lemma 4. The desired result is proved.Remark 1.
Theorem 1 shows that the more the heterogeneity among the distributions of claim severities and occurrence probabilities, the larger the largest claim amounts, in the sense of the usual stochastic order.
TakingT-transform matrix as T=1/21/21/21/2 in Theorem 1, we immediately obtain a lower bound on the survival function of the largest claim amounts.Corollary 1.
Under the setup of Theorem 1,1−1−h−1h¯pe−α¯wβγx2 is a low bound of F¯Y2:2x, that is, for any x∈ℝ+, we have(25)F¯Y2:2x≥1−1−h−1h¯pe−α¯wβγx2.
Next, we give a numerical example to illustrate the effectiveness of Theorem1 and Corollary 1.Example 1.
Suppose the baseline random variable follows Weibull0.02,2 distribution. Suppose X1,X2X1∗,X2∗ are independent random variables with Xi∼W−G0.02,2,αi,3, Xi∗∼W−G0.02,2,αi∗,3, i=1,2. Let Ip1,Ip2Ip1∗,Ip2∗, are independent Bernoulli random variables, independent of Xi′s, with EIpi0=pi,i=1,2. To plot the whole survival curves of Y2:2 and Y2:2∗ in 0,∞, we perform the transformation e−x:0,∞⟶0,1. Then, it is obvious that Y2:2≥stY2:2∗ is equivalent with e−Y2:2≤ste−Y2:2∗.(i)
Supposeα1,α2=3,1, α1∗,α2∗=1.72,2.28, p1,p2=e−2,e−1 and p1∗,p2∗=e−1.36,e−1.64. Taking hp=−lnp and considering the following T-transform matrix,(26)T0.36=0.361001+0.640110,we then havehp1∗hp2∗α1∗α2∗=hp1hp2α1α2T0.36; thus,(27)hp1hp2α1α2≫hp1∗hp2∗α1∗α2∗.
It is easy to check that the conditions of Theorem 1 are all satisfied. As is seen in Figure1(a), the graph of F¯Y2:2x−F¯Y2:2∗x is always over the zero plane, that is, the difference functions F¯Y2:2x−F¯Y2:2∗x is always non-negative for all x=−lnu and u∈0,1, which coincides with the result of Theorem 1.(i)
Consider the following T-transform matrix, with other conditions unchanged,(28)T0.5=0.51001+0.50110.It is easy to observe thatα¯,α¯=2,2 and h¯p=3.5, where α¯=1/α1+α2. From Figure 1(b) which it can be see that the difference functions F¯Y2:2∗x−F¯Y2:2∗∗x is often non-negative, for all x=−lnu and u∈0,1, which verifies the result of Corollary 1.
The next Theorem2 will generalize the result of Theorem 1 to the largest claim amounts among n heterogeneous portfolio risks.Figure 1
(a) Plot of the difference functionsF¯Y2:2x−F¯Y2:2∗x. (b) Plot of the difference functions F¯Y2:2∗x−F¯Y2:2∗∗x, for all x=−lnu and u∈0,1.
(a)(b)Theorem 2.
SupposeX1,X2,…,XnX1∗,X2∗,…,Xn∗ are independent nonnegative random variables with Xi∼W−Gαi,β,γXi∗∼W−Gαi∗,β,γ, i=1,2,…,n. Let , Ip1,Ip2…,IpnIp1∗,Ip2∗,…,Ipn∗ are independent Bernoulli random variables, independent of Xi′sXi∗′s, with EIpi=piEIpi∗=pi∗, i=1,2,…,n. Then, under the same settings as in Theorem 1, for hp,α∈ℳ2, it holds that(29)hp1∗hp2∗⋯hpn∗α1∗α2∗⋯αn∗=hp1hp2⋯hpnα1α2⋯αnT⇒Yn:n≥stYn:n∗.Proof.
Settingφnp,α=FYn:nx and ϕp,α=1−pe−αwβγx, for any x≥0, we then have φnp,α=∏i=1nϕpi,αi. According to Theorem 1, φ2 satisfies in (7) of Lemma 4. Now, the required result follows immediately from Lemma 5.Remark 2.
Note that the distribution in Theorem2 and Theorem 1 are the special case of the proportional hazard rate model, hence, our theoretical findings verify the case of the proportional hazard rate model. Besides, Theorem 2 result indicates that more heterogeneity among some specified transformed occurrence probabilities and scale parameter α in the sense of the chain majorization order leads to the greater survival function of the largest claim amount.
The following Theorem3 gives a sufficient condition under which Y2:2≥stY2:2∗ holds when the baseline distribution follows exponential. We only provide the result of n=2 case.Theorem 3.
SupposeX1,X2X1∗,X2∗ are two independent non-negative random variables with Xi∼W−Expα,β,γiXi∗∼W−Expα,β,γi∗,i=1,2. Let Ip1,Ip2Ip1∗,Ip2∗ are independent Bernoulli random variables, independent of Xi′Xi∗′, with EIpi=piEIpi∗=pi∗,i=1,2. Assume that h:0,1⟶0,∞ is differentiable and strictly decreasing convex. Then, for any hp,γ∈ℳ2, we have(30)hp1hp2γ1γ2≫hp1∗hp2∗γ1∗γ2∗⇒Y2:2≥stY2:2∗.Proof.
Note that the baseline distributionFx=1−e−γx,x>0, which implies wFx=eγx−1. Set pi=h−1ui, for i=1,2. Then, the distribution function of the largest claim amounts Y2:2 is given by(31)FY2:2x=∏i=121−h−1uie−αeγix−1β.Clearly, for fixed x≥0, FY2:2x is the permutation invariant on ℳ2, which confirms the condition (i) of Lemma 4. On the other hand, we need prove the condition (ii) of Lemma 4, for any x≥0 and 1=i≠j=2, set(32)φu,γ=u1−u2∂FY2:2x∂u1−∂FY2:2x∂u2+γ1−γ2∂FY2:2x∂γ1−∂FY2:2x∂γ2,where u=u1,u2 and γ=γ1,γ2. Then, we have(33)∂FY2:2x∂ui=−FY2:2xdh−1uiduie−αeγix−1β1−h−1uie−αeγix−1β,∂FY2:2x∂γi=αβxFY2:2xh−1uieγixeγix−1β−1eαeγix−1β−1.
Similar to the proof of Theorem1, equation (32) can be represented as(34)φu,γ=u1−u2FY2:2xku2,γ2−ku1,γ1+αβxγ1−γ2FY2:2xlu1,γ1−lu2,γ2,where(35)ku,γ=dh−1udu1eαeγx−1β−h−1u,lu,γ=h−1ueγxeγx−1β−1eαeγx−1β−1,
Without loss of generality, assumeu1≥u2, γ1≥γ2. Owing to h−1 is a strictly decreasing convex function, we have(36)∂ku,γ∂u=signd2h−1udu2e−αwβγx1−h−1ue−αwβγx+dh−1udu2e−2αwβγx≥0.(37)∂ku,γ∂γ=sign−eαeγx−1βeγx−1β−1eγxxdh−1udu≥0.
Further, we have(38)ku1,γ1≥ku2,γ1≥ku2,γ2,which implies that the first side of (32) is nonpositive. Besides, we obtain(39)∂lu,γ∂u=dh−1udueγxeγx−1β−1eαeγx−1β−1≤0,(40)∂lu,γ∂γ=sign−αβeγx−1β−βeγx+1eαeγx−1β+βeγx−1=−φ1eγx,where(41)φ1t=αβt−1β−βt+1eαt−1β+βt−1,t≥1.
Whent>1, we conclude(42)dφ1tdt=βt−1eαt−1βαt−1β+α2βt−12β−1t+1+t−1,=signeαt−1βαt−1β+α2βt−12β−1t+1+t−1≕φ2t.
For anyt>1, it follows that(43)dφ2tdt=1−eαt−1β+eαt−1βαt−1β1+αβt−1β−11+2β+1t+βtt−1β+βt−1≥1−eαt−1β+αt−1βeαt−1β≕φ3αt−1β,where φ3z=1−ez+zez,z≥0. It follows from dφ3z/dz=zez≥0 and φ30=0 that dφ2t/dt≥0. Combining φ21=0 with φ11=0, we conclude that φ1t>0, which implies ∂lu,γ/∂γ≤0. Basing on u1≥u2 and γ1≥γ2, it follows that(44)lu1,γ1≤lu2,γ1≤lu2,γ2,which confirms the second side of (32) is also non-positive. Upon combining (38) with (44), we have φu,γ≤0; hence, condition (ii) of Lemma 4 is verified. The proof of desired result is completed.Remark 3.
According to Theorem3, more heterogeneity among some specified transformed occurrence probabilities and scale parameter γ in terms of the chain majorization order causes the greater survival function of the largest claim amount with two independent insurance portfolios.
The proof of the following Theorem4 is similar to that of Theorem 2 and thus omitted here.Theorem 4.
SupposeX1,X2,…,XnX1∗,X2∗,…,Xn∗ are independent nonnegative random variables with Xi∼W−Expα,β,γiXi∗∼W−Gα,β,γi∗, i=1,2,…,n. Let , Ip1,Ip2…,IpnIp1∗,Ip2∗,…,Ipn∗ are independent Bernoulli random variables, independent of Xi′sXi∗′s, with EIpi=piEIpi∗=pi∗, i=1,2,…,n. Then, under the same settings as in Theorem 1, for hp,γ∈ℳ2, it holds that(45)hp1∗hp2∗⋯hpn∗γ1∗γ2∗⋯γn∗=hp1hp2⋯hpnγ1γ2⋯γnT⇒Yn:n≥stYn:n∗.Remark 4.
Note that the distribution in Theorem3 and Theorem 4 are the special case of the scaled model in Barmalzan et al. [8]; hence, our established results check the setting of the scaled model. Meanwhile, according to Theorem 4, more heterogeneity among the scale parameter γ and the specified transformed occurrence probabilities with respect to the chain majorization order obtains the greater survival function of the largest claim amount from multiple independent insurance portfolios.
Naturally, one can question Theorem Theorems3 and 4 whether holds for other baseline function, unfortunately, the following numerical example gives a negative answer.Example 2.
Suppose the baseline random variable follows Frechet0,γ,1 distribution. Suppose X1,X2X1∗,X2∗ are independent random variables with Xi∼W−G0,γi,4,0.6,2, Xi∗∼W−G0,γi∗,4,0.6,2, i=1,2, β=2. Let Ip1,Ip2Ip1∗,Ip2∗, are independent Bernoulli random variables, independent of Xi′s, with EIpi=pi,i=1,2. To plot the whole survival curves of Y2:2 and Y2:2∗ in 0,∞. Suppose γ1,γ2=0.2,0.5, γ1∗,γ2∗=0.425,0.275, p1,p2=e−2,e−1 and p1∗,p2∗=e−1.25,e−1.75. Taking hp=−lnp and considering the following T-transform matrix,(46)T0.25=0.251001+0.640110.
We then havehp1∗hp2∗γ1∗γ2∗=hp1hp2γ1γ2T0.36; thus, a matrix hp1hp2γ1γ2 is larger than matrix hp1∗hp2∗γ1∗γ2∗ in the chain majorization, that is, it is easy to check that the conditions of Theorem 3 are all satisfied. As is seen in Figure 2, the survival functions F¯Y2:2x and F¯Y2:2∗x are each crossing, which states Theorem 3 is not necessarily hold for other baseline function.Figure 2
Plots of survival functionsF¯Y2:2x and F¯Y2:2∗x.


## 3.2. Insurance Portfolios of Dependent Risks

In this section, we use the usual stochastic order to compare the largest claim amountsYn:n and Yn:n∗ arising from two sets of dependent heterogeneous insurance portfolios when occurrence probabilities are dependent and claim severities are also dependent.To proceed, let us define0=0,0,…,0, 1=1,1,…,1, pλ=ℙI=λ and(47)Αk=λ|λi=0or1,i=1,2,…,n,λ1+λ2+⋯+λn=k,k=0,1,…,n.For all1≤i≠j≤n and k=1,2,…,n−1, denote(48)Αki,j0,0=λ∈Αk|λi=0,λj=0,Αki,j1,1=λ∈Αk|λi=1,λj=1.Obviously,Α1i,j1,1=Αn−1i,j0,0=Φ and(49)Αk=Αki,j1,0∪Αki,j0,1∪Αki,j1,1∪Αki,j0,0.With the help of the above notations, Cai and Wei [29] presented characterizations of stochastic arrangement increasing(SAI) Bernoulli random vector, which is very useful to model the positive dependence among occurrence probabilities.Lemma 6.
Cai and Wei [29]). A multivariate Bernoulli random vector I is SAI if and only if pτijλ≤pλ, for any λ∈Αki,j0,1,1≤i<j≤n and k=1,2,…,n−1.Theorem 5.
SupposeI is SAI, Xi∼W−Gαi,β,γi and the associated Archimedean copula is with ψ1ϕ1=ψ1−1 and Xi∗∼W−Gαi∗,β,γi, and the associated Archimedean copula with generator ψ2ϕ2=ψ2−1, for all i=1,2,…,n and αi,αi∗,γi∈ℝ+, γ,∈D+. Assume the following conditions hold:(i)
α1∗,α2∗,…,αn∗<¯mα1,α2,…,αn for α,α∗∈D+(ii)
ϕ2°ψ1 is super additive
Then, we haveYn:n≥stYn:n∗.Proof.
We need to show thatFYn:nx≤FYn:n∗x, for all x∈ℝ+. By the nature of majorization order, it is sufficient prove the desired result when αi∗,αj∗≼mαi,αj, for some pair 1≤i≤j≤n, and αs=αs∗ for s≠i,j. Let(50)FYsi,jx=FY1X,…,FYi−1X,FYi+1X,…,FYj−1X,FYj+1X,…,FYnX.
The distribution function ofYn:n is given by(51)FYn:nx=ℙmaxIp1X1,Ip2X2,…,IpnXn≤x=∑k=0n∑λ∈ΑkℙIp1X1≤x,…,IpnXn≤x∣I=λpλ=p0+p1maxX1,…,Xn≤x+∑k=1n−1∑λ∈ΑkpλℙmaxIp1X1,…,IpnXn≤x=p0+p1ψ1∑i=1nϕ11−e−αiwγixβ+∑k=1n−1∑λ∈Αki,j1,0pλℙλiXi≤x,λjXj≤x,λsXs≤x,s≠i,j+∑k=1n−1∑λ∈Αki,j0,1pλℙλiXi≤x,λjXj≤x,λsXs≤x,s≠i,j+∑k=1n−1∑λ∈Αki,j0,0pλℙλiXi≤x,λjXj≤x,λsXs≤x,s≠i,j+∑k=1n−1∑λ∈Αki,j1,1pλℙλiXi≤x,λjXj≤x,λsXs≤x,s≠i,j=p0+p1ψ1∑i=1nϕ11−e−αiwγixβ+∑k=1n−1∑λ∈Αki,j1,0pλψ1ϕ11+ϕ11−e−αiwβγix+∑s≠i,jnϕ11−pse−αswβγsx+∑k=1n−1∑λ∈Αki,j0,1pλψ1ϕ11+ϕ11−e−αjwβγjx+∑s≠i,jnϕ11−pse−αswβγsx+∑k=1n−1∑λ∈Αki,j0,0pλψ1ϕ11+ϕ11+∑s≠i,jnϕ11−pse−αswβγsx+∑k=1n−1∑λ∈Αki,j1,1pλψ1ϕ11−e−αjwβγjx+ϕ11−e−αiwβγix+∑s≠i,jnϕ11−pse−αswβγsx.
Similarly,(52)FYn:n∗x=p0+p1ψ2∑i=1nϕ21−e−αi∗wγixβ+∑k=1n−1∑λ∈Αki,j1,0pλψ2ϕ21+ϕ21−e−αi∗wβγix+∑s≠i,jnϕ21−pse−αs∗wβγsx+∑k=1n−1∑λ∈Αki,j0,1pλψ2ϕ21+ϕ21−e−αj∗wβγjx+∑s≠i,jnϕ21−pse−αs∗wβγsx+∑k=1n−1∑λ∈Αki,j0,0pλψ2ϕ21+ϕ21+∑s≠i,jnϕ21−pse−αs∗wβγsx+∑k=1n−1∑λ∈Αki,j1,1pλψ2ϕ21−e−αj∗wβγjx+ϕ21−e−αi∗wβγix+∑s≠i,jnϕ21−pse−αs∗wβγsx.
From the convexity and Schur-convexity of−ψ1∑i=1nϕ1ui and the concavity of Fx;α=1−e−αwβγx in α, we know that −ψ1∑i=1nϕ1ui is Schur-convex in α by applying Lemma 3. Hence, according to the super additivity of ϕ2°ψ1 and α∗≺¯mα, we have(53)ψ2∑i=1nϕ21−e−αi∗wγixβ≥ψ1∑i=1nϕ11−e−αi∗wγixβ≥ψ1∑i=1nϕ11−e−αiwγixβ,x∈ℝ+.
According toϕ2°ψ1 is superadditive and αi∗,αj∗≺¯mαi,αj, for any x∈ℝ+, we have(54)ψ2ϕ21+ϕ21+∑s≠i,jnϕ21−pse−αs∗wβγsx≥ψ1ϕ11+ϕ11+∑s≠i,jnϕ11−pse−αswβγsx,(55)ψ2ϕ21−e−αj∗wβγjx+ϕ21−e−αi∗wβγix+∑s≠i,jnϕ21−pse−αs∗wβγsx≥ψ1ϕ11−e−αjwβγjx+ϕ11−e−αiwβγix+∑s≠i,jnϕ11−pse−αswβγsx.
On the other hand, the concavities of bothψ1∑i=1nϕ1ui and Fx;α imply that ψ∑i=1nϕFx;αi is also concave in α, which leads to the fact that(56)ψ1ϕ11+ϕ11−e−αjwβγjx+∑s≠i,jnϕ11−pse−αswβγsx+ψ1ϕ11−e−αiwβγix+ϕ11+∑s≠i,jnϕ11−pse−αswβγsxis Schur-concave in α by proposition I:3.C.1 on Page 80 of Marshall et al. [23]; for any x∈ℝ+. Hence, it follows from the superadditivity of ϕ2°ψ1 and αi∗,αj∗≺¯mαi,αj, for all x∈ℝ+,(57)ψ2ϕ21+ϕ21−e−αj∗wβγjx+∑s≠i,jnϕ21−pse−αswβγsx+ψ2ϕ21+ϕ21−e−αi∗wβγix+∑s≠i,jnϕ21−pse−αswβγsx≥ψ1ϕ11+ϕ11−e−αj∗wβγjx+∑s≠i,jnϕ11−pse−αswβγsx+ψ1ϕ11+ϕ11−e−αj∗wβγjx+∑s≠i,jnϕ11−pse−αswβγsx≥ψ1ϕ11+ϕ11−e−αjwβγjx+∑s≠i,jnϕ11−pse−αswβγsx.
Observe that(58)FYn:n∗x−FYn:nx=p1ψ2∑i=1nϕ21−e−αi∗wγixβ−ψ1∑i=1nϕ11−e−αiwγixβ+∑k=1n−1∑λ∈Αki,j1,0pλψ2ϕ21+ϕ2Fx;αi∗+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ11+ϕ1Fx;αi+∑s≠i,jnϕ21−pse−αs∗wβγsx+∑k=1n−1∑λ∈Αki,j0,1pλψ2ϕ21+ϕ2Fx;αj∗+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ11+ϕ1Fx;αj+∑s≠i,jnϕ11−pse−αswβγsx+∑k=1n−1∑λ∈Αki,j0,0pλψ2ϕ21+ϕ21+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ11+ϕ11+∑s≠i,jnϕ11−pse−αs∗wβγsx+∑k=1n−1∑λ∈Αki,j1,1pλψ2ϕ2Fx;αj∗+ϕ2Fx;αi∗+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ1Fx;αj+ϕ1Fx;αi+∑s≠i,jnϕ11−pse−αs∗wβγsx≥∑k=1n−1∑λ∈Αki,j1,0pλψ2ϕ21+ϕ2Fx;αi∗+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ11+ϕ1Fx;αi+∑s≠i,jnϕ21−pse−αs∗wβγsx+∑k=1n−1∑λ∈Αki,j0,1pλψ2ϕ21+ϕ2Fx;αj∗+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ11+ϕ1Fx;αj+∑s≠i,jnϕ11−pse−αswβγsx≥∑k=1n−1∑λ∈Αki,j0,1pτi,jλψ2ϕ21+ϕ2Fx;αi∗+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ11+ϕ1Fx;αi+∑s≠i,jnϕ21−pse−αs∗wβγsx+∑k=1n−1∑λ∈Αki,j0,1pτi,jλψ2ϕ21+ϕ2Fx;αj∗+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ11+ϕ1Fx;αj+∑s≠i,jnϕ11−pse−αswβγsx=∑k=1n−1∑λ∈Αki,j0,1pτi,jλψ2ϕ21+ϕ2Fx;αi∗+∑s≠i,jnϕ21−pse−αs∗wβγsx+ψ2ϕ21+ϕ2Fx;αj∗+∑s≠i,jnϕ21−pse−αs∗wβγsx−ψ1ϕ11+ϕ1Fx;αi+∑s≠i,jnϕ21−pse−αs∗wβγsx+ψ1ϕ11+ϕ1Fx;αj+∑s≠i,jnϕ11−pse−αswβγsx≥0,where the first inequality is from (53)–(55). The second inequality is due to Lemma 6 and the last inequality is based on (57). Hence, the desired result follows.Remark 5.
In the light of Theorem5, we know that more heterogeneity among the distribution functions of claim severities leads to a larger value of the largest claim amount in terms of the usual stochastic ordering when occurrences probabilities are positively dependent via SAI and claim severities are also dependent via a general copula that is concave and Schur-concave.
Next, we give a numerical example to illustrate the result of Theorem5.Example 3.
SupposeX1,X2,X3X1∗,X2∗,X3∗ are three non-negative random variables, let Fx=1−e−γx, and generators ψ1x=η1x+1−1/η1, η1>0, ψ2x=e1/η21−ex, 0<η2≤1. Set α∗=0.6,0.5,0.2≺¯w0.5,0.45,0.35=α, β∈0.8,1.6, η1=1.6, η2=0.8, γ=0.12,0.06,0.04, p0,0,0=0.16, p1,0,0=0.05, p0,1,0=0.17, p0,0,1=0.2, p1,1,0=0.02, p1,0,1=0.1, p0,1,1=0.18, and p1,1,1=0.15. It is easy to verify that the conditions of Theorem 5 are all satisfied. The survival function of Y3:3 can be expressed as(59)F¯Y3:3x=1−p0,0,0+p1,1,1ψ1ϕ11−e−α11eγ1x−1β+ϕ11−e−α12eγ2x−1β+ϕ11−e−α13eγ3x−1β+p1,0,0ψ1ϕ11−e−α11eγ1x−1β+2ϕ11+p0,1,0ψ1ϕ11−e−α12eγ2x−1β+ϕ11+ϕ11+p0,0,1ψ1ϕ11+ϕ11+ϕ11−e−α13eγ3x−1β+p1,1,0ψ1ϕ11−e−α11eγ1x−1β+ϕ11−e−α12eγ2x−1β+ϕ11+p1,0,1ψ1ϕ11−e−α11eγ1x−1β+ϕ11+ϕ11−e−α13eγ3x−1β+p0,1,1ψ1ϕ11+ϕ11−e−α12eγ2x−1β+ϕ11−e−α13eγ3x−1β.
The survival functions ofY3:3 and Y3:3∗, denoted as F¯Y3:3x and F¯Y3:3∗x are plotted in Figure 3, respectively, for all x=−lnu and u∈0,1, from which we see that survival function F¯e−Y3:3x is always larger than that of F¯e−Y3:3∗x. Thus, the effectiveness of Theorem 5 is validated.
The proof of the following Theorem6 is similar to that of Theorem 5, and thus omitted here.Figure 3
Plot of the difference functionsF¯e−Y3:3x−F¯e−Y3:3∗x, for all x=−lnu and u∈0,1.Theorem 6.
SupposeI is SAI, Xi∼W−Gαi,β,γi and the associated Archimedean copula is with ψ1ϕ1=ψ1−1 and Xi∗∼W−Gαi,β,γi∗, and the associated Archimedean copula with generator ψ2ϕ2=ψ2−1, for all i=1,2,…,n and γi,γi∗,αi∈ℝ+, αi,∈D+. Assume the following conditions hold:(i)
γ1∗,γ2∗,…,γn∗≺¯mγ1,γ2,…,γn. for γ,γ∗∈D+(ii)
ϕ2°ψ1 is super additive
Then, we haveYn:n≥stYn:n∗.Remark 6.
Theorem6 means more heterogeneity among the distribution functions of claim severities leads to a larger value of the largest claim amount among scale parameter γ in terms of the usual stochastic ordering when occurrences probabilities are positively dependent via SAI and claim severities are also dependent via a Archimedean copula.


## 4. On the Orderings of Smallest Claim Amounts

In this section, we study the stochastic comparisons of the smallest claim amounts with respect to the usual stochastic and hazard rate orders when the smallest claim amounts are independent and dependent.


### 4.1. Insurance Portfolios of Independent Risks

In this part, we discuss the stochastic comparisons between the smallest claim amounts from two insurance portfolios with W-G claims in the sense of the usual stochastic and hazard rate orders. SupposeX1,X2,…,Xn are independent, the survival function of the smallest claim amounts Y1:n among n claims can be expressed as(60)F¯Y1:n=∏i=1npie−αiwβγix,x∈ℝ+,αi,β,γi∈ℝ+.Theorem 7.
SupposeX1,X2,…,XnX1∗,X2∗,…,Xn∗ are independent nonnegative random variables with Xi∼W−Gαi,β,γiXi∗∼W−Gαi∗,β,γi. Let Ip1,Ip2,…,IpnIp1∗,Ip2∗,…,Ipn∗ be a set of independent Bernoulli random variables, independent of Xi′sXi∗′s, with EIpi=piEIpi∗=pi∗, i=1,2,…,n. If(i)
β≥1 and wγx is an increasing convex function for x∈ℝ+(ii)
∏i=1npi≤∏i−1npi∗(iii)
α1∗,α2∗,…,αn∗≺¯mα1,α2,…,αn. Then, for α,α∗,γ∈ℐ+, we have(61)Y1:n∗≥hrY1:n.Proof.
Note that(62)F¯Y1:n∗xF¯Y1:nx=∏i=1npi∗e−∑i=1nαi∗wγixβ∏i=1npie−∑i=1nαiwγixβ=∏i=1npi∗∏i=1npiF¯X1:n∗xF¯X1:nx.
According to (3.1), the hazard rate function ofX1:n can be presented as(63)rX1:nx=β∑i=1nαiγiwγixβ−1w′γix.
Let(64)ψα=∑i=1nαiγiwγixβ−1w′γix.
Then,(65)αi−αj∂ψα∂αi−∂ψα∂αj=αi−αjγiwγixβ−1w′γix−γjwγjxβ−1w′γjx.
Note thatwx is increasing and convex in x, then x≥0,β≥1 and i≤j imply γiwγixβ−1≤γjwγjxβ−1 and w′γix≤w′γjx. It follows that (65) is nonnegative, and hence, ψα is schur-convex in α. By Lemma 1, we have(66)ψα1,α2,…,αn≥ψα1∗,α2∗,…,αn∗.that is rX1:nx≥rX1:n∗x, which is equivalent to F¯X1:n∗x/F¯X1:nx is increasing in x. Note that, when ∏i=1npi∗/∏i=1npi≥1 implies(67)limx⟶0+F¯Y1:n∗xF¯Y1:nx=limx⟶0+∏i=1npi∗F¯X1:n∗x∏i=1npiF¯X1:nx≥F¯X1:n∗0F¯X1:n0,that is, F¯Y1:n∗x/F¯Y1:nx is increasing in x∈ℝ+. The desired result is proved.Remark 7.
According to Theorem7, more heterogeneity among the scale parameter α obtains the greater the hazard rate function of the smallest claim amounts.
The following Example4 illustrates the result of Theorem 7.Example 4.
AssumeX1,X2,X3X1∗,X2∗,X3∗ are independent nonnegative random variables with Xi∼W−Gαi,β,γiXi∗∼W−Gαi∗,β,γi, i=1,2,3. Taking the baseline distribution as Fx=1−e−γx, and let α∗=0.23,0.26,0.33≺¯m0.19,0.23,0.4=α, β∈1.1,3.1, p1,p2,p3=0.23,0.49,0.59, p1∗,p2∗,p3∗=0.45,0.64,0.88, γ1=0.12,γ2=0.21,γ3=0.62, Figure 4(a) plots the ratio function F¯Y1:3∗x/F¯Y1:3x, for all x=−lnu, u∈0,1, which confirms the effectiveness of Theorem 7.
On the other hand, setβ=0.65 and γ=0.11,0.12,0.13 with the remaining conditions, from 4(b) see that the difference of the hazard rate functions hY1:3x−hY1:3∗x≥0, for all x=−lnu, u∈0,1, hence, the condition β≥1 of Theorem 7 is sufficient and not necessarily condition for the need of mathematical proof.Figure 4
(a) Plot of the ratio functionF¯Y1:3∗x/F¯Y1:3x, for all x=−lnu, u∈0,1. (b) Plot of difference of the hazard rate functions hY1:3x−hY1:3∗x, for all x=−lnu, u∈0,1.
(a)(b)Theorem 8.
SupposeX1,X2,…,XnX1∗,X2∗,…,Xn∗ are independent non-negative random variables with Xi∼W−Gαi,β,γiXi∗∼W−Gαi∗,β,γi Let Ip1,Ip2,…,IpnIp1∗,Ip2∗,…,Ipn∗ be a set of independent Bernoulli random variables, independent of Xi′sXi∗′s, with EIpi=piEIpi∗=pi∗,i=1,2,…,n. If β≥2, and(i)
w⋅ is increasing convex and twice differentiable, and w″x is increasing.(ii)
∏i=1npi≤∏i=1npi∗.(iii)
γ1∗,γ2∗,…,γn∗≺¯mγ1,γ2,…,γn. Then, for α,γ,γ∗∈D+, we have(68)Y1:n≤hrY1:n∗.Proof.
In view point of Theorem7, the hazard rate function of X1:n is(69)rX1:nx=β∑i=1nαiγiwβ−1γixw′γix.
For convenience, let(70)ψγ=∑i=1nαiγiwβ−1γixw′γix,we have, for i=1,2,…,n(71)∂ψγ∂γi=αiwβ−2γixwγixw′γix+β−1γixw′γix2+γiwγixw″γixx.For β≥2 and i≤j, we obtain that(72)∂ψγ∂γi−∂ψγ∂γj=ψ1γ+ψ2γ,where(73)ψ1γ=wγixw′γix+β−1γixw′γix2+γiwγixw″γixx×αiwβ−2γix−αjwγjxβ−2,(74)ψ2γ=wγixw′γix−wγjxw′γjx+β−1γixw′γix2−β−1γjxw′γjx2+γiwγixw″γixx−γjwγjxw″γjxx]wβ−2γjx.
From the increasing convexity ofw⋅ and the increasing property of w″x, It holds that ψ1γ≥0 and ψ2γ≥0. Thus, for any i,j, we have(75)γi−γj∂ψγ∂γi−∂ψγ∂γj≥0,which shows that ψγ is schur-convex in γ. According to condition iii, we further have(76)ψγ1,γ2,…,γn≥ψγ1∗,γ2∗…,γn∗.
Similar to the proof Theorem7 we can prove that Y1:n≤hrY1:n∗. The proof is finished.Remark 8.
As per Theorem8, more heterogeneity among the scale parameter γ claim severities obtains the greater the hazard rate function of the smallest claim amounts. As one anonymous reviewer suggested, Theorem 3.10(iii) in Torrado and Navarro [19] holds for dependent insurance portfolios, they give the necessary and sufficient conditions for the hazard rate order between smallest claim amounts from portfolios with arbitrary distribution functions. Our Theorem 7 and Theorem 8 are the independent case and for a specific distribution function (Weibull-G distributions), it is evident that Theorem 3.10(iii) of Torrado and Navarro [19] is broader than our Theorem 7 and 8.
Next, Example5 is provided to illustrate the result of Theorem 8.Example 5.
LetFx=1−1+x/γ−μ, μ>2, γ>0, then wx=1+x/γμ. Set γ=4.65,4.55,4.5≽m4.95,4.75,4=γ∗, β=2.06, μ∈2.1,4.1, α=0.12,0.06,0.01, p1,p2,p3=0.3,0.4,0.5, p1∗,p2∗,p3∗=0.55,038,0.66. One can verify that the conditions of Theorem 8 are all satisfied. Figure 5(a) plots the difference hY1:3x−hY1:3∗x of two hazard rate function, for all x=−lnu, u∈0,1, which check the result of Theorem 8.
On the other hand, setβ=0.001, μ=2.5, α=0.13,0.12,0.11, p1,p2,p3=0.2,0.3,0.4, p1∗,p2∗,p3∗=0.75,0.68,0.86 with the remaining conditions, from 5(b) see that the difference of the hazard rate functions hY1:3x−hY1:3∗x is crossing at line y=0, x∈0,10, hence, the condition β≥2 of Theorem 8 is an necessarily condition and not is removed.Figure 5
Plot of the difference functionhY1:3x−hY1:3∗x, for all x=−lnu, u∈0,1.
(a)(b)


### 4.2. Insurance Portfolios of Dependent Risks

As described by Barmalzan et al. [11]; the independent assumption among portfolio claim amounts is not usually suitable in practice and we shall need to explore portfolio risks with statistical dependent claims. In this subsection, we establish usual stochastic order of the smallest claim amounts with dependent heterogeneous Weibull-G insurance portfolios under the Archimedean survival copula. To establish the stochastic comparisons of the smallest claim amounts, we first give two Lemmas.Lemma 7.
SupposeXi∼W−Gαi,β,γiXi∗∼W−Gαi∗,β,γii=1,2,…,n, and they have the associated Archimedean surival copula with the generator ψ1ψ2 denoting the pseudo-inverse of ψi by ϕii=1,2. If ϕ2°ψ1 is super additive, and ψ1 or ψ2 is log-convex. Then for any α1,α2,…,αn≽mα1∗,α2∗,…,αn∗ implies(77)X1:n≤stX1:n∗.Proof.
The distribution function of Weibull-G distribution is increasing inα, and the survival function of Weibull-G is log-concave in α, which satisfies conditions in Proposition 3.16 of Torrado [30]. Hence, according to Proposition 3.16 of Torrado [30]; we can easily obtain Lemma 7.
The following result can be proven by applying a similar method to that of Lemma7, and thus the detailed proof is omitted for brevity.Lemma 8.
SupposeXi∼W−Gαi,β,γiXi∗∼W−Gαi,β,γi∗i=1,2,…,n, and they have the associated Archimedean surival copula with the generator ψ1ψ2. Denote the pseudo-inverse of ψi by ϕii=1,2. If ϕ2°ψ1 is super additive, and ψ1 or ψ2 is log-convex. Then for any α,γ,γ∗∈D+, β≥1, γ1∗,γ2∗,…,γn∗≺¯mγ1,γ2,…,γn implies(78)X1:n≤stX1:n∗.Proof.
The distribution function of Weibull-G is increasing inγ, and the survival function of Weibull-G is log-concave in γ, for β≥1, which checks conditions in Proposition 3.16 of Torrado [30]. Thus, according to Proposition 3.16 of Torrado [30]; we can easily establish Lemma 8.Remark 9.
Observe that Lemmas7 and 8 can be easily proved from Theorem 3.10 (iii) of Torrado and Navarro [19]. Thus, Lemma 7 and 8 are particular cases of Theorem 3.10(iii) in Torrado and Navarro [19].
Next, we give some sufficient conditions for the usual stochastic order between the smallest claim amounts from dependent heterogeneous portfolios risks.Theorem 9.
Suppose -Xi∼W−Gαi,β,γiXi∗∼W−Gαi,β,γi∗i=1,2,…,n, and they have the associated Archimedean surival copula with the generator ψ1ψ2. Let, Ip1,Ip2…,IpnIp1∗,Ip2∗,…,Ipn∗ be a set of independent Bernoulli random variables, independent of Xi′sXi∗′s, with EIpi=piEIpi∗=pi∗,i=1,2,…,n. Further, suppose ϕ2°ψ1 is supper additive, ψ1 or ψ2 is log-convex. Then, for any ∏i=1npi≤∏i=1npi∗,α,α∗,γ∈D+, and α1∗,α2∗,…,αn∗≺¯mα1,α2,…,αn implies(79)Y1:n≤stY1:n∗.Proof.
From (60), we have(80)F¯Y1:nx=e−∑i=1nαiwγixβ∏i=1npi=F¯X1:nx∏i=1npi.And thus, the desired result follows immediately from assumption ∏i=1npi≤∏i=1npi∗ and Lemma 7.Remark 10.
As per Theorem9, we know that more heterogeneity among the distribution functions of claim severities leads to a smaller value of the smallest claim amount in the stochastic order under Archimedean copula dependence.
The next numerical Example6 illustrates the result of Theorem 9.Example 6.
Under the setting of Theorem9. Let F¯x=e−γixμ,γ>0,μ>1, and generators ψ1x=e1/η11−ex, 0<η1≤1, ψ2x=η2x+1−1/η2, η2>0. One can check chat ψ1x and ψ2x is log-concave and log-convex, respectively. It can be seen that ϕ2°ψ1x″=exeη2ex−1/η1η1−2η1+η2ex≥0, hence, ϕ2°ψ1x is super-additive. Set μ=1.9, β∈1.1,1.5, η1=0.85, η2=1.6, α=2.6,2.0,1.3,0.9≽m3.0,2.3,1.4,0.1=α∗, γ=0.16,0.15,0.13,0.11, p1,p2,p3=0.78,0.38,0.29,0.37, p1∗,p2∗,p3∗=0.82,0.42,0.31,0.40, as is seen in Figure 6, F¯Y1:4∗x−F¯Y1:4x≥0, for all x=−lnu and u∈0,1, that is, Y1:4≤stY1:4∗, which coincides with the result of Theorem 9.
One natural question arises, whether the requirementsϕ2°ψ1 is supper additive in Theorem 9 could be removed or not? The following example provides a negative answer.Figure 6
Plot of the difference functionF¯Y1:4∗x−F¯Y1:4x, for all x=−lnu and u∈0,1.Example 7.
Under the setup of Example6, set μ∈1.1,4.1,β=1.28, let two generators ψ1x=η1x+1−1/η1 , ψ2x=η2x+1−1/η2 , ηi>0, i=1,2. Note that logψix″=ηiηix+1−2≥0, for x≥0, thus, ψix is log-convex. Taking η1=12.6, η2=0.2, owing to ϕ2°ψ1x″=η2−η1η1x+1η2/η1−2≤0, which implies that ϕ2°ψ1x is not supper additive, and the plot of the difference functions F¯Y1:4∗x−F¯Y1:4x in Figure 7, which neither non-negative nor non-positive, hence, the supper additive of ϕ2°ψ1 of Theorem 9 cannot be dropped.Figure 7
Plot of the difference functionF¯Y1:4∗x−F¯Y1:4x.Theorem 10.
SupposeXi∼W-Gαi,β,γi[Xi∗∼W-Xi∼W−Gαi,β,γiXi∗∼W−Gαi,β,γi∗, i=1,2,…,n, and they have the associated Archimedean survival copula with the generator ψ1ψ2. Let Ip1,Ip2,…,IpnIp1∗,Ip2∗,…,Ipn∗ be a set of independent Bernoulli random variables, independent of Xi′sXi∗′s, with EIpi=piEIpi∗=pi∗, i=1,2,…,n. Further, let ϕ2°ψ1 is supper additive, ψ1 or ψ2 is log-convex. Then, for any ∏i=1npi≤∏i=1npi∗,α,γ,γ∗∈D+, and γ1∗,γ2∗,…,γn∗≺¯mγ1,γ2,…,γn, implies(81)Y1:n≤stY1:n∗.Proof.
From (60), we have(82)F¯Y1:nx=e−∑i=1nαiwγixβ∏i=1npi≤F¯X1:nx∏i=1npi∗≤F¯X1:n∗x∏i=1npi∗,And thus the desired result follows immediately from assumption ∏i=1npi≤∏i=1npi∗ and Lemma 8.Remark 11.
As per Theorem10, we know that more heterogeneity among the distribution functions of claim severities leads to a smaller value of the smallest claim amount in the sense of the usual stochastic ordering. As one anonymous reviewer suggested, all the main results in Torrado [30] are for dependent portfolios. Actually, Theorem 9 and 10 are particular cases of Theorem 3.10(ii) in Torrado [30].


## 4.1. Insurance Portfolios of Independent Risks

In this part, we discuss the stochastic comparisons between the smallest claim amounts from two insurance portfolios with W-G claims in the sense of the usual stochastic and hazard rate orders. SupposeX1,X2,…,Xn are independent, the survival function of the smallest claim amounts Y1:n among n claims can be expressed as(60)F¯Y1:n=∏i=1npie−αiwβγix,x∈ℝ+,αi,β,γi∈ℝ+.Theorem 7.
SupposeX1,X2,…,XnX1∗,X2∗,…,Xn∗ are independent nonnegative random variables with Xi∼W−Gαi,β,γiXi∗∼W−Gαi∗,β,γi. Let Ip1,Ip2,…,IpnIp1∗,Ip2∗,…,Ipn∗ be a set of independent Bernoulli random variables, independent of Xi′sXi∗′s, with EIpi=piEIpi∗=pi∗, i=1,2,…,n. If(i)
β≥1 and wγx is an increasing convex function for x∈ℝ+(ii)
∏i=1npi≤∏i−1npi∗(iii)
α1∗,α2∗,…,αn∗≺¯mα1,α2,…,αn. Then, for α,α∗,γ∈ℐ+, we have(61)Y1:n∗≥hrY1:n.Proof.
Note that(62)F¯Y1:n∗xF¯Y1:nx=∏i=1npi∗e−∑i=1nαi∗wγixβ∏i=1npie−∑i=1nαiwγixβ=∏i=1npi∗∏i=1npiF¯X1:n∗xF¯X1:nx.
According to (3.1), the hazard rate function ofX1:n can be presented as(63)rX1:nx=β∑i=1nαiγiwγixβ−1w′γix.
Let(64)ψα=∑i=1nαiγiwγixβ−1w′γix.
Then,(65)αi−αj∂ψα∂αi−∂ψα∂αj=αi−αjγiwγixβ−1w′γix−γjwγjxβ−1w′γjx.
Note thatwx is increasing and convex in x, then x≥0,β≥1 and i≤j imply γiwγixβ−1≤γjwγjxβ−1 and w′γix≤w′γjx. It follows that (65) is nonnegative, and hence, ψα is schur-convex in α. By Lemma 1, we have(66)ψα1,α2,…,αn≥ψα1∗,α2∗,…,αn∗.that is rX1:nx≥rX1:n∗x, which is equivalent to F¯X1:n∗x/F¯X1:nx is increasing in x. Note that, when ∏i=1npi∗/∏i=1npi≥1 implies(67)limx⟶0+F¯Y1:n∗xF¯Y1:nx=limx⟶0+∏i=1npi∗F¯X1:n∗x∏i=1npiF¯X1:nx≥F¯X1:n∗0F¯X1:n0,that is, F¯Y1:n∗x/F¯Y1:nx is increasing in x∈ℝ+. The desired result is proved.Remark 7.
According to Theorem7, more heterogeneity among the scale parameter α obtains the greater the hazard rate function of the smallest claim amounts.
The following Example4 illustrates the result of Theorem 7.Example 4.
AssumeX1,X2,X3X1∗,X2∗,X3∗ are independent nonnegative random variables with Xi∼W−Gαi,β,γiXi∗∼W−Gαi∗,β,γi, i=1,2,3. Taking the baseline distribution as Fx=1−e−γx, and let α∗=0.23,0.26,0.33≺¯m0.19,0.23,0.4=α, β∈1.1,3.1, p1,p2,p3=0.23,0.49,0.59, p1∗,p2∗,p3∗=0.45,0.64,0.88, γ1=0.12,γ2=0.21,γ3=0.62, Figure 4(a) plots the ratio function F¯Y1:3∗x/F¯Y1:3x, for all x=−lnu, u∈0,1, which confirms the effectiveness of Theorem 7.
On the other hand, setβ=0.65 and γ=0.11,0.12,0.13 with the remaining conditions, from 4(b) see that the difference of the hazard rate functions hY1:3x−hY1:3∗x≥0, for all x=−lnu, u∈0,1, hence, the condition β≥1 of Theorem 7 is sufficient and not necessarily condition for the need of mathematical proof.Figure 4
(a) Plot of the ratio functionF¯Y1:3∗x/F¯Y1:3x, for all x=−lnu, u∈0,1. (b) Plot of difference of the hazard rate functions hY1:3x−hY1:3∗x, for all x=−lnu, u∈0,1.
(a)(b)Theorem 8.
SupposeX1,X2,…,XnX1∗,X2∗,…,Xn∗ are independent non-negative random variables with Xi∼W−Gαi,β,γiXi∗∼W−Gαi∗,β,γi Let Ip1,Ip2,…,IpnIp1∗,Ip2∗,…,Ipn∗ be a set of independent Bernoulli random variables, independent of Xi′sXi∗′s, with EIpi=piEIpi∗=pi∗,i=1,2,…,n. If β≥2, and(i)
w⋅ is increasing convex and twice differentiable, and w″x is increasing.(ii)
∏i=1npi≤∏i=1npi∗.(iii)
γ1∗,γ2∗,…,γn∗≺¯mγ1,γ2,…,γn. Then, for α,γ,γ∗∈D+, we have(68)Y1:n≤hrY1:n∗.Proof.
In view point of Theorem7, the hazard rate function of X1:n is(69)rX1:nx=β∑i=1nαiγiwβ−1γixw′γix.
For convenience, let(70)ψγ=∑i=1nαiγiwβ−1γixw′γix,we have, for i=1,2,…,n(71)∂ψγ∂γi=αiwβ−2γixwγixw′γix+β−1γixw′γix2+γiwγixw″γixx.For β≥2 and i≤j, we obtain that(72)∂ψγ∂γi−∂ψγ∂γj=ψ1γ+ψ2γ,where(73)ψ1γ=wγixw′γix+β−1γixw′γix2+γiwγixw″γixx×αiwβ−2γix−αjwγjxβ−2,(74)ψ2γ=wγixw′γix−wγjxw′γjx+β−1γixw′γix2−β−1γjxw′γjx2+γiwγixw″γixx−γjwγjxw″γjxx]wβ−2γjx.
From the increasing convexity ofw⋅ and the increasing property of w″x, It holds that ψ1γ≥0 and ψ2γ≥0. Thus, for any i,j, we have(75)γi−γj∂ψγ∂γi−∂ψγ∂γj≥0,which shows that ψγ is schur-convex in γ. According to condition iii, we further have(76)ψγ1,γ2,…,γn≥ψγ1∗,γ2∗…,γn∗.
Similar to the proof Theorem7 we can prove that Y1:n≤hrY1:n∗. The proof is finished.Remark 8.
As per Theorem8, more heterogeneity among the scale parameter γ claim severities obtains the greater the hazard rate function of the smallest claim amounts. As one anonymous reviewer suggested, Theorem 3.10(iii) in Torrado and Navarro [19] holds for dependent insurance portfolios, they give the necessary and sufficient conditions for the hazard rate order between smallest claim amounts from portfolios with arbitrary distribution functions. Our Theorem 7 and Theorem 8 are the independent case and for a specific distribution function (Weibull-G distributions), it is evident that Theorem 3.10(iii) of Torrado and Navarro [19] is broader than our Theorem 7 and 8.
Next, Example5 is provided to illustrate the result of Theorem 8.Example 5.
LetFx=1−1+x/γ−μ, μ>2, γ>0, then wx=1+x/γμ. Set γ=4.65,4.55,4.5≽m4.95,4.75,4=γ∗, β=2.06, μ∈2.1,4.1, α=0.12,0.06,0.01, p1,p2,p3=0.3,0.4,0.5, p1∗,p2∗,p3∗=0.55,038,0.66. One can verify that the conditions of Theorem 8 are all satisfied. Figure 5(a) plots the difference hY1:3x−hY1:3∗x of two hazard rate function, for all x=−lnu, u∈0,1, which check the result of Theorem 8.
On the other hand, setβ=0.001, μ=2.5, α=0.13,0.12,0.11, p1,p2,p3=0.2,0.3,0.4, p1∗,p2∗,p3∗=0.75,0.68,0.86 with the remaining conditions, from 5(b) see that the difference of the hazard rate functions hY1:3x−hY1:3∗x is crossing at line y=0, x∈0,10, hence, the condition β≥2 of Theorem 8 is an necessarily condition and not is removed.Figure 5
Plot of the difference functionhY1:3x−hY1:3∗x, for all x=−lnu, u∈0,1.
(a)(b)


## 4.2. Insurance Portfolios of Dependent Risks

As described by Barmalzan et al. [11]; the independent assumption among portfolio claim amounts is not usually suitable in practice and we shall need to explore portfolio risks with statistical dependent claims. In this subsection, we establish usual stochastic order of the smallest claim amounts with dependent heterogeneous Weibull-G insurance portfolios under the Archimedean survival copula. To establish the stochastic comparisons of the smallest claim amounts, we first give two Lemmas.Lemma 7.
SupposeXi∼W−Gαi,β,γiXi∗∼W−Gαi∗,β,γii=1,2,…,n, and they have the associated Archimedean surival copula with the generator ψ1ψ2 denoting the pseudo-inverse of ψi by ϕii=1,2. If ϕ2°ψ1 is super additive, and ψ1 or ψ2 is log-convex. Then for any α1,α2,…,αn≽mα1∗,α2∗,…,αn∗ implies(77)X1:n≤stX1:n∗.Proof.
The distribution function of Weibull-G distribution is increasing inα, and the survival function of Weibull-G is log-concave in α, which satisfies conditions in Proposition 3.16 of Torrado [30]. Hence, according to Proposition 3.16 of Torrado [30]; we can easily obtain Lemma 7.
The following result can be proven by applying a similar method to that of Lemma7, and thus the detailed proof is omitted for brevity.Lemma 8.
SupposeXi∼W−Gαi,β,γiXi∗∼W−Gαi,β,γi∗i=1,2,…,n, and they have the associated Archimedean surival copula with the generator ψ1ψ2. Denote the pseudo-inverse of ψi by ϕii=1,2. If ϕ2°ψ1 is super additive, and ψ1 or ψ2 is log-convex. Then for any α,γ,γ∗∈D+, β≥1, γ1∗,γ2∗,…,γn∗≺¯mγ1,γ2,…,γn implies(78)X1:n≤stX1:n∗.Proof.
The distribution function of Weibull-G is increasing inγ, and the survival function of Weibull-G is log-concave in γ, for β≥1, which checks conditions in Proposition 3.16 of Torrado [30]. Thus, according to Proposition 3.16 of Torrado [30]; we can easily establish Lemma 8.Remark 9.
Observe that Lemmas7 and 8 can be easily proved from Theorem 3.10 (iii) of Torrado and Navarro [19]. Thus, Lemma 7 and 8 are particular cases of Theorem 3.10(iii) in Torrado and Navarro [19].
Next, we give some sufficient conditions for the usual stochastic order between the smallest claim amounts from dependent heterogeneous portfolios risks.Theorem 9.
Suppose -Xi∼W−Gαi,β,γiXi∗∼W−Gαi,β,γi∗i=1,2,…,n, and they have the associated Archimedean surival copula with the generator ψ1ψ2. Let, Ip1,Ip2…,IpnIp1∗,Ip2∗,…,Ipn∗ be a set of independent Bernoulli random variables, independent of Xi′sXi∗′s, with EIpi=piEIpi∗=pi∗,i=1,2,…,n. Further, suppose ϕ2°ψ1 is supper additive, ψ1 or ψ2 is log-convex. Then, for any ∏i=1npi≤∏i=1npi∗,α,α∗,γ∈D+, and α1∗,α2∗,…,αn∗≺¯mα1,α2,…,αn implies(79)Y1:n≤stY1:n∗.Proof.
From (60), we have(80)F¯Y1:nx=e−∑i=1nαiwγixβ∏i=1npi=F¯X1:nx∏i=1npi.And thus, the desired result follows immediately from assumption ∏i=1npi≤∏i=1npi∗ and Lemma 7.Remark 10.
As per Theorem9, we know that more heterogeneity among the distribution functions of claim severities leads to a smaller value of the smallest claim amount in the stochastic order under Archimedean copula dependence.
The next numerical Example6 illustrates the result of Theorem 9.Example 6.
Under the setting of Theorem9. Let F¯x=e−γixμ,γ>0,μ>1, and generators ψ1x=e1/η11−ex, 0<η1≤1, ψ2x=η2x+1−1/η2, η2>0. One can check chat ψ1x and ψ2x is log-concave and log-convex, respectively. It can be seen that ϕ2°ψ1x″=exeη2ex−1/η1η1−2η1+η2ex≥0, hence, ϕ2°ψ1x is super-additive. Set μ=1.9, β∈1.1,1.5, η1=0.85, η2=1.6, α=2.6,2.0,1.3,0.9≽m3.0,2.3,1.4,0.1=α∗, γ=0.16,0.15,0.13,0.11, p1,p2,p3=0.78,0.38,0.29,0.37, p1∗,p2∗,p3∗=0.82,0.42,0.31,0.40, as is seen in Figure 6, F¯Y1:4∗x−F¯Y1:4x≥0, for all x=−lnu and u∈0,1, that is, Y1:4≤stY1:4∗, which coincides with the result of Theorem 9.
One natural question arises, whether the requirementsϕ2°ψ1 is supper additive in Theorem 9 could be removed or not? The following example provides a negative answer.Figure 6
Plot of the difference functionF¯Y1:4∗x−F¯Y1:4x, for all x=−lnu and u∈0,1.Example 7.
Under the setup of Example6, set μ∈1.1,4.1,β=1.28, let two generators ψ1x=η1x+1−1/η1 , ψ2x=η2x+1−1/η2 , ηi>0, i=1,2. Note that logψix″=ηiηix+1−2≥0, for x≥0, thus, ψix is log-convex. Taking η1=12.6, η2=0.2, owing to ϕ2°ψ1x″=η2−η1η1x+1η2/η1−2≤0, which implies that ϕ2°ψ1x is not supper additive, and the plot of the difference functions F¯Y1:4∗x−F¯Y1:4x in Figure 7, which neither non-negative nor non-positive, hence, the supper additive of ϕ2°ψ1 of Theorem 9 cannot be dropped.Figure 7
Plot of the difference functionF¯Y1:4∗x−F¯Y1:4x.Theorem 10.
SupposeXi∼W-Gαi,β,γi[Xi∗∼W-Xi∼W−Gαi,β,γiXi∗∼W−Gαi,β,γi∗, i=1,2,…,n, and they have the associated Archimedean survival copula with the generator ψ1ψ2. Let Ip1,Ip2,…,IpnIp1∗,Ip2∗,…,Ipn∗ be a set of independent Bernoulli random variables, independent of Xi′sXi∗′s, with EIpi=piEIpi∗=pi∗, i=1,2,…,n. Further, let ϕ2°ψ1 is supper additive, ψ1 or ψ2 is log-convex. Then, for any ∏i=1npi≤∏i=1npi∗,α,γ,γ∗∈D+, and γ1∗,γ2∗,…,γn∗≺¯mγ1,γ2,…,γn, implies(81)Y1:n≤stY1:n∗.Proof.
From (60), we have(82)F¯Y1:nx=e−∑i=1nαiwγixβ∏i=1npi≤F¯X1:nx∏i=1npi∗≤F¯X1:n∗x∏i=1npi∗,And thus the desired result follows immediately from assumption ∏i=1npi≤∏i=1npi∗ and Lemma 8.Remark 11.
As per Theorem10, we know that more heterogeneity among the distribution functions of claim severities leads to a smaller value of the smallest claim amount in the sense of the usual stochastic ordering. As one anonymous reviewer suggested, all the main results in Torrado [30] are for dependent portfolios. Actually, Theorem 9 and 10 are particular cases of Theorem 3.10(ii) in Torrado [30].


## 5. An Application

In this section, based on the real data of insurance, we apply the proposed methods to present some significantly insights for insurance industry. Eric and Ziegel [31] give the data of set of the third party motor insurance for 1977 for one of seven geographical zones. The data represent the total amount paid by the insurance company and the total number claims received by the insurance company. Raqab et al. [32] uses normal distribution(83)Fx=∫0x1σ2πe−1/2t−μ/σ2dt,to fit the univariate payment data set, and obtained parameter’s estimate values μ^=31.1 and σ^=24.062. Now, in order to help insurance industry to analysis the information of survival function on the smallest claim amounts under difference scale parameters, we select the previous normal distribution as the baseline distribution of the smallest amounts, and set α1=3,3,3,α2=4,3.5,3.6,α3=4.5,4,3.8,α4=5,4.5,4,p=e−0.1,e−0.1,e−0.1,e−0.1. Obviously, α1≺¯wα2≺¯wα3≺¯wα4. Based on Theorem 9, we plot the curves of survival function in Figure 8, from which it see that F¯Y1:4x;α1,p≥F¯Y1:4x;α2,p≥F¯Y1:4x;α3,p≥F¯Y1:4x;α4,p. Hence, the survival function F¯Y1:4x;α1,p can be treated as upper of other survival function, which provides useful information on the smallest claim amounts of insurance industry.Figure 8
Curves of survival functionF¯Y1:4x;αi,p,i=1,2,3,4.


## 6. Conclusion

We establish the stochastic order between the largest claim amounts and the smallest claim amounts from sets of dependent heterogeneous insurance portfolios. Sufficient conditions are provided for the usual stochastic order between the largest claim amounts when the matrix of parametershp,αhp,γ changes to another matrix in some certain mathematical sense. Meanwhile, based on the concepts of vector majorizations and related orders, we establish stochastic comparisons between the smallest claim amounts in the sense of the usual stochastic and hazard rate orders. The obtained findings present quite significantly perceptions and insights into determining the annual premium for an insurant.As discussed in Zhang [14]; Zhang et al. [33]; it is of great importance to investigate some other stochastic orderings, such as the convex transform ordering, star ordering and right spread ordering for the extreme claim amounts arising from two sets of heterogeneous portfolios risk, which remain open and merit further discussion.



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