On the convex transform and right-spread orders of smallest claim amounts

Accepted Manuscript On the convex transform and right-spread orders of smallest claim amounts Ghobad Barmalzan, Amir T. Payandeh Najafabadi PII: DOI: Reference:

S0167-6687(15)00102-X http://dx.doi.org/10.1016/j.insmatheco.2015.07.001 INSUMA 2107

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Insurance: Mathematics and Economics

Received date: November 2014 Revised date: May 2015 Accepted date: 3 July 2015 Please cite this article as: Barmalzan, G., Payandeh Najafabadi, A.T., On the convex transform and right-spread orders of smallest claim amounts. Insurance: Mathematics and Economics (2015), http://dx.doi.org/10.1016/j.insmatheco.2015.07.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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On the convex transform and right-spread orders of smallest claim amounts Ghobad Barmalzan and Amir T. Payandeh Najafabadi Department of Mathematical Sciences, Shahid Beheshti University, G.C. Evin, Tehran, Iran May 20, 2015 Abstract Suppose Xλ1 , · · · , Xλn is a set of Weibull random variables with shape parameter α > 0, scale parameter λi > 0 for i = 1, · · · , n and Ip1 , · · · , Ipn are independent Bernoulli random variables, independent of the Xλi ’s, with E(Ipi ) = pi , i = 1, · · · , n. Let Yi = Xλi Ipi , for i = 1, · · · , n. In particular, in actuarial science, it corresponds to the claim amount in a portfolio of risks. In this paper, under certain conditions, we discuss stochastic comparison between the smallest claim amounts in the sense of the right-spread order. Moreover, while comparing these two smallest claim amounts, we show that the right-spread order and the increasing convex orders are equivalent. Finally, we obtain the results concerning the convex transform order between the smallest claim amounts and find a lower and upper bound for the coefficient of variation. The results established here extend some well-known results in the literature.

Keywords: Smallest Claim Amount; Convex Transform Order; Right-Spread Order; Increasing Convex Order; Wiebull Distribution; Coefficient of Variations.

1

Introduction

It is quite important for an actuary to be able to express preferences between random future gains or losses. For this purpose, stochastic ordering results become very useful. Stochastic orders have been used in various areas including management science, financial economics, insurance, actuarial science, operations research, reliability theory, queuing theory and survival analysis. Interested readers may refer to M¨ uller and Stoyan (2002) and Shaked and Shanthikumar (2007) for comprehensive discussions on univariate and multivariate stochastic orders. Annual premium is the amount paid the policyholder on an annual basis to cover the cost of the insurance policy being purchased. Indeed, it is the primary cost to the policyholder of transferring the risk to the insurer which depend on the type of insurance (life, health, auto, etc). It is of interest to note that smallest claim amounts can have a critical role in insurance for determining annual premium. This reason can be one of the motivations our work. The Weibull distribution is one of the commonly used distributions in reliability, life testing and actuarial science. A random variable X is said to have the Weibull distribution with shape parameter α > 0 and scale parameter λ > 0 (denoted by X ∼ W (α, λ)) if its probability density function is given by α

f (x; α, λ) = α λα xα−1 e−(λ x) ,

x > 0, α > 0, λ > 0.

It is well-known that the hazard rate of Weibull distribution is decreasing for α < 1, constant for α = 1, and increasing when α > 1. One may refer to Johnson et al. (1994) and Murthy et al. (2004) for comprehensive discussions on various properties and applications of the Weibull distribution. 1

Suppose Xλi denotes the total of random claims that can be made in an insurance period and Ipi denotes a Bernoulli random variable associated with Xλi defined as follows: Ipi = 1 whenever the i-th policyholder makes random claim Xλi and Ipi = 0 whenever he/she does not make a claim. In actuarial science, Yi = Xλi Ipi corresponds to the claim amount in a portfolio of risks. The problem of comparisons of the number of claims and aggregate claim amount with respect to some well-known stochastic orders is of interest on both theoretical and practical grounds. Several authors worked on this direction: Karlin and Novikoff (1963), Ma (2000), Frostig (2001), Hu and Ruan (2004), Denuit and Frostig (2006), Khaledi and Ahmadi (2008). Recently, Barmalzan et al. (2015) presented a complete version of the results of Khaledi and Ahmadi (2008) which have then been extended to the more general case. Suppose Xλ1 , · · · , Xλn is a set of independent non-negative random variables with Xλi ∼ F (λi x), i = 1, · · · , n, where F is an absolutely continuous distribution function with corresponding hazard rate function r. Further, suppose Ip1 , · · · , Ipn are independent Bernoulli random variables, independent of the Xλi ’s, with E(Ipi ) = pi , i = 1, · · · , n. Let Yi = Xλi Ipi , for i = 1, · · · , n. In particular, in actuarial science, Y1:n = ∗ = min(Y1 , · · · , Yn ) corresponds to the smallest claim amount in a portfolio of risks. We assume that Y1:n min(Y1∗ , · · · , Yn∗ ) denotes the smallest claim amount arising from Yi∗ = Xλ∗i Ip∗i , i = 1, · · · , n, in a another portfolio of risks. Barmalzan et al. (2014) under certain conditions, by using the concept of vector majorization and related orders, obtained stochastic comparisons between the smallest claim amounts in the sense of the usual stochastic and hazard rate orders. They also applied the results for three special cases of the scale model like generalized gamma, Marshall-Olkin extended exponential and Exponentiated Weibull distribution with possibly different scale parameters. It is of interest to note that our results do not restricted to actuarial sciences and can be used in various areas including reliability theory and survival analysis. For instance, suppose random variable Xλi presents life-length of the i-th component in a series system which may received a random shocked at binging. This random shocked may not be impact on the i-th component (set Ipi = 1) or does (set Ipi = 0.) Thus, Yi = Ipi Xλi admit Xλi when random shocked does not impact on the i-th component and zero when random shocked does. Throughout, we have considered the following heterogeneous portfolios: Suppose Xλ1 , · · · , Xλn is a set of independent non-negative random variables with Xλi ∼ W (α, λi ), i = 1, · · · , n. Further, suppose Ip1 , · · · , Ipn are independent Bernoulli random variables, independent of the Xλi ’s, respectively, with E(Ipi ) = pi , i = ∗ = min(Y ∗ , · · · , Y ∗ ) denotes the smallest claim 1, · · · , n. Let Yi = Xλi Ipi , for i = 1, · · · , n. We assume that Y1:n n 1 amount arising from Yi∗ = Xλ∗i Ip∗i , i = 1, · · · , n, in a another portfolio of risks. In this paper, we establish for ∏ ∏ any α > 0 and ni=1 pi = ni=1 p∗i , n ∑ i=1

λαi ≤

n ∑ ∗ (λ∗i )α =⇒ Y1:n ≤RS Y1:n , i=1

we also obtain a necessary and sufficient condition for the right-spread order as well as the increasing convex ∗ ≤ order between two smallest claim amounts. More precisely, we show that the three statements (i) Y1:n RS Y1:n ; ∑ ∑ n n ∗ ≤ α ≤ ∗ )α are equivalent. Finally, we obtain the results concerning the (ii) Y1:n Y and (iii) λ (λ icx 1:n i=1 i i=1 i convex transform and Lorenz orders between the smallest claim amounts as follows: 2

(i) for α > 1 and (ii) for α > 1 and

∏n

∗ i=1 pi

∏n

∗ i=1 pi

≤ ≤

∏n

i=1 pi ,

∏n

i=1 pi ,

∗ ≤ Y we have Y1:n c 1:n ; ∗ ; we have Y1:n ≤Lorenz Y1:n

and then find a lower and upper bound for the coefficient of variation of smallest claim amount. The rest of this paper is organized as follows. In Section 2, we introduce some definitions and notation pertinent to stochastic orders and related orders. In Section 3, some new sufficient conditions for the comparison of smallest claim amounts in the sense of the right-spread order are presented. Finally, in Section 4 we obtain the results concerning the convex transform order between the smallest claim amounts.

2

Preliminaries

In this section, we give some preliminaries including definitions and notation which will be useful in the subsequent sections. Throughout this article, we use ‘increasing’ to mean ‘non-decreasing’ and similarly ‘decreasing’ to mean ‘non-increasing’. Suppose X and Y are two non-negative random variables with density function f and g, distribution ¯ = 1 − G, right continuous inverses (quantile functions) functions F and G, survival functions F¯ = 1 − F and G F −1 and G−1 , respectively. Further, all expectations are implicitly assumed to exist wherever they are given. The first definition states some well-known concepts to compare magnitude two random variables. ¯ Definition 1 X is said to be smaller than Y in the usual stochastic order (denoted by X ≤st Y ) if F¯ (x) ≤ G(x) for all x ∈ R+ . This is equivalent to saying that E(ϕ(X)) ≤ E(ϕ(Y )) for all increasing functions ϕ : R+ → R. Interested readers may refer to M¨ uller and Stoyan (2002) and Shaked and Shanthikumar (2007) for a comprehensive discussions on various stochastic orderings and relations between them. Definition 2 (i) X is said to be smaller than Y in the convex transform order (denoted by X ≤c Y ) if G−1 F (x) is convex in x ∈ R+ . Equivalently, X ≤c Y if and only if F −1 G(x) is concave in x ∈ R+ ; (ii) X is said to be smaller than Y in the Lorenz order order (denoted by X ≤Lorenz Y ) if 1 E(X)

∫

0

F −1 (u)

xdF (x) ≥

1 E(Y )

∫

G−1 (u)

xdG(x)

0

f or all u ∈ (0, 1].

It is worthwhile to mention that the convex transform order is a well-known criterion to compare skewness of probability distributions. The following implications between these orderings are well-known: X ≤c Y =⇒ X ≤Lorenz Y =⇒ cv(X) ≤ cv(Y ), √ √ where cv(X) = V ar(X)/E(X) and cv(Y ) = V ar(Y )/E(Y ) denote the coefficients of variation of X and Y , respectively. For more details about the above implications, interested readers may refer to Marshall and Olkin 3

(2007, p. 69, 75). The Lorenze order is an important criterion in economics to compare income distributions and the risks associated with different prospects (see Marshall and Olkin (2007, p. 68)). The following definition introduces some well-known orders that compare the dispersion of two random variables. Definition 3

(i) X is said to be smaller than Y in the right-spread order (denoted by X ≤RS Y ) if ∫ ∞ ∫ ∞ ¯ ¯ F (x) dx ≤ G(x) dx f or all u ∈ (0, 1); F −1 (u)

G−1 (u)

(ii) X is said to be smaller than Y in the increasing convex order (denoted by X ≤icx Y ) if ∫ ∞ ∫ ∞ ¯ F¯ (x) dx ≤ G(x) dx f or all t ≥ 0. t

t

It is well-known that the right-spread order implies the increasing convex order (see Kochar and Carri´ ere (1997) and Shaked and Shanthikumar (1998)). It is easy to show that for nonnegative random variables X and Y , if E(X) = E(Y ), then the increasing convex order implies the order between the corresponding variances.

3

Right-Spread Order Between Smallest Claim Amounts

In this section, we obtain a sufficient condition under which the right-spread order (which is also called the excess wealth order) between two smallest claim amounts holds. The following useful Lemma of Kochar and Xu (2010), which presents an equivalent characterization of the right-spread order in one parameter family, will be needed to prove our main results. Lemma 1 Suppose {Fb |b ∈ R} is a class of distribution functions such that Fb is supported on some interval (x0 , x1 ) ⊆ R+ and has a density fb which does not vanish on any sub-interval of (x0 , x1 ). Then, (i) Fb ≤RS Fb∗ , b, b∗ ∈ R, b ≤ b∗ , if and only if ∫∞ Wb = x F¯b (t) dt with respect to b;

(ii) Fb∗ ≤RS Fb , b, b∗ ∈ R, b ≤ b∗ , if and only if ∫∞ Wb = x F¯b (t) dt with respect to b.

′

Wb (x) F¯b (x)

′

Wb (x) F¯b (x)

′

is increasing in x, where Wb (x) is the derivative of

′

is decreasing in x, where Wb (x) is the derivative of

Theorem 1 Suppose Xλ1 , · · · , Xλn are independent non-negative random variables with Xλi ∼ W (α, λi ), i = 1, · · · , n, where λi > 0. Further, suppose Ip1 , · · · , Ipn are independent Bernoulli random variables, independent ∏ ∏ of the Xλi ’s, with E(Ipi ) = pi , i = 1, · · · , n. Then for any α > 0 and ni=1 pi = ni=1 p∗i , n ∑ i=1

λαi ≤

n ∑ ∗ (λ∗i )α =⇒ Y1:n ≤RS Y1:n , i=1

∗ denote the smallest claim amount arising from Y = X I ∗ where Y1:n and Y1:n i λi pi and Yi = Xλ∗i Ip⋆i , for i = 1, · · · , n.

4

Proof. Let λ = (

∑n

α 1/α . i=1 λi )

The survival function of X1:n , for x ≥ 0 is as follows:

F¯λ (x) = P (Y1:n > x) = P (Xλ1 Ip1 > x, · · · , Xλn Ipn > x)

= P (Xλ1 Ip1 > x, · · · , Xλn Ipn > x|Ip1 = 1, · · · , Ipn = 1) P (Ip1 = 1, · · · , Ipn = 1) = P (Xλ1 > x, · · · , Xλn > x|Ip1 = 1, · · · , Ipn = 1) P (Ip1 = 1, · · · , Ipn = 1)

= P (Xλ1 > x, · · · , Xλn > x) P (Ip1 = 1, · · · , Ipn = 1) n ∏ = P (Xλi > x) P (Ipi = 1) = =

i=1 n ∏

(

(

pi

i=1

n ∏ i=1

pi

) )

e−(

∑n

i=1

α λα i )x

α

e−(λx) .

∗ , for x ≥ 0 is given by: Similarly, the survival function of Y1:n ) ( n ∑n ∏ ∗ α α ∗ F¯λ∗ (x) = pi e−( i=1 (λi ) )x i=1

( n ) ∑n ∏ ∗ α α pi e−( i=1 (λi ) )x = i=1

( n ) ∏ ∗ α = pi e−(λ x) , i=1

where λ∗ = (

∑n

∗ α 1/α . i=1 (λi ) )

Let Wλ (x) be defined as follows: ( n )∫ ∫ ∞ ∏ Wλ (x) = F¯λ (t) dt = pi x

α

e−(λ t) dt.

x

i=1

Taking the derivative with respect to λ, we have ( n ) ∫ ∏ ′ Wλ (x) = − pi α λα−1 i=1

∞

∞

α

tα e−(λ t) dt.

x

So, by combining these observations we obtain ′

Wλ (x) F¯λ (x)

where ϕ(x) = −

=

− α λα−1

∫∞

α −(λ t)α x t e α e−(λ t)

dt

∫ ∞ α −(λ t)α t e dt = − ∫ x∞ α−1 −(λ t)α λ x t e dt 1 = ϕ(x), say, λ ∫∞

α tα e−(λ t) dt x ∫∞ , α−1 e−(λ t)α dt x t

5

x ≥ 0.

By taking the derivative of ϕ(x) with respect to x, we have { ∫ ∞ ∫ α ′ α−1 −(λ x)α ϕ (x) = x e x tα−1 e−(λ t) dt − x

≤0

∞

α −(λ t)α

t e

x

(Since x ≤ t).

} dt

′

W (x)

Therefore, ϕ(x) is decreasing in x ≥ 0 which implies F¯ λ(x) is decreasing in x ≥ 0. Thus, the required result λ follows immediately from Part (ii) of Lemma 1. □ The following corollary is a direct consequence of Theorem 1 and this fact the right-spread order implies the order between the corresponding variances. Corollary 1 Suppose Xλ1 , · · · , Xλn are independent non-negative random variables with Xλi ∼ W (α, λi ), i = 1, · · · , n, where λi > 0. Further, suppose Ip1 , · · · , Ipn are independent Bernoulli random variables, independent ∏ ∏ of the Xλi ’s, with E(Ipi ) = pi , i = 1, · · · , n. Then for any α > 0 and ni=1 pi = ni=1 p∗i , n ∑ i=1

λαi ≤

n ∑ i=1

∗ (λ∗i )α =⇒ V ar(Y1:n ) ≤ V ar(Y1:n ),

∗ denote the smallest claim amount arising from Y = X I ∗ where Y1:n and Y1:n i λi pi and Yi = Xλ∗i Ip⋆i , for i = 1, · · · , n. □

We may question whether the result of Theorem 1 could hold if first prove the following lemma.

∑n

α i=1 λi

̸≤

∑n

∗ α i=1 (λi ) ?

For this purpose, We

Lemma 2 Suppose Xλ1 , · · · , Xλn are independent non-negative random variables with Xλi ∼ W (α, λi ), i = 1, · · · , n, where λi > 0. Further, suppose Ip1 , · · · , Ipn are independent Bernoulli random variables, independent of the Xλi ’s, with E(Ipi ) = pi , i = 1, · · · , n. Then for α > 0, the variance of Y1:n is as follows: } ( n ) ( n ){ ∏ ∏ 1 2 −1 −1 V ar(Y1:n ) = ∑n pi Γ(1 + 2α ) − pi Γ (1 + α ) . ( i=1 λαi )2/α i=1 i=1 Proof. Note that, the random variable Y1:n is a discrete-continuous type, which admits zero with probability ∏ ∏ 1 − ni=1 pi and min(Xλ1 , · · · , Xλn ) with probability ni=1 pi . By using this point, the proof of lemma is straightforward. Therefore, it is omitted here for the sake of brevity. □ ∑ ∑ From Lemma 2 and Corollary 1, we readily observe that Theorem 1 could not hold if ni=1 λαi ̸≤ ni=1 (λ∗i )α . In the following theorem, we extend the result in Theorem 1 and obtain a necessary and sufficient condition for the right-spread order as well as the increasing convex order between two smallest claim amount to hold without any restriction on the common shape parameter of the underlying Weibull random variables.

Theorem 2 Suppose Xλ1 , · · · , Xλn are independent non-negative random variables with Xλi ∼ W (α, λi ), i = 1, · · · , n, where λi > 0. Further, suppose Ip1 , · · · , Ipn are independent Bernoulli random variables, independent ∏ ∏ of the Xλi ’s, with E(Ipi ) = pi , i = 1, · · · , n. Then, for any α > 0 and ni=1 pi = ni=1 p∗i , the following statements are equivalent: 6

∗ ≤ (i) Y1:n RS Y1:n ; ∗ ≤ (ii) Y1:n icx Y1:n ;

(iii)

∑n

α i=1 λi

≤

∑n

∗ α i=1 (λi ) ,

∗ denote the smallest claim amount arising from Y = X I ∗ where Y1:n and Y1:n i λi pi and Yi = Xλ∗i Ip⋆i , for i = 1, · · · , n.

Proof. For non-negative random variables, the right-spread order implies the increasing convex order (Shaked ∗ ≤ and Shanthikumar (2007), Corollary 4.A.32, p. 196), and so (i) implies (ii). Now, suppose Y1:n icx Y1:n then ∗ E(Y1:n ) ≤ E(Y1:n ). But,

and

∏ ( n pi ) E(Y1:n ) = ∑n i=1 α 1/α Γ(1 + α−1 ), ( i=1 λi )

∏ ( n p∗ ) ∗ E(Y1:n ) = ∑n i=1∗ αi 1/α Γ(1 + α−1 ), ( i=1 (λi ) )

So, (ii) implies (iii). Finally, (iii) ⇒ (i) follows readily from Theorem 1. Hence, the theorem.

4

□

Convex Transform Order for Smallest Claim Amounts

In statistical distribution theory, skewness is a criterion used for measuring the asymmetry of a distribution. Conceptually, skewness explains which side of a distribution is more stretched out than the other. For a unimodal distribution, positive skewness indicates that the tail on the right side of the probability density function is longer or fatter than the left side, and negative skewness indicates that the tail on the left side is longer or fatter than the right side. Right skewness is common when a variable is bounded on the left but unbounded on the right. van Zwet (1964) presented a well-known criterion, called the convex transformation order, to compare skewness of two probability distributions. Indeed, the convex transform order explains that one distribution is more skewed to the right than the other. He also showed that the standardized odd central moments preserve this ordering. The star order, which is weaker than the convex transform order, has been proved itself as an important concept to compare skewness between two probability distributions. Interested readers may refer to Oja (1981) and Marshall and Olkin (2007) for a comprehensive discussion about the partial orderings that compare skewness. This section concerns some certain conditions for the comparison of the smallest claim amounts, with respect to some well-known orders. Theorem 3 Suppose Xλ1 , · · · , Xλn are independent non-negative random variables with Xλi ∼ W (α, λi ), i = 1, · · · , n, where λi > 0. Further, suppose Ip1 , · · · , Ipn are independent Bernoulli random variables, independent ∏ ∏ ∗ ≤ Y of the Xλi ’s, with E(Ipi ) = pi , i = 1, · · · , n. Then, for α > 1 and ni=1 p∗i ≤ ni=1 pi , we have Y1:n c 1:n . 7

∗ as follow: Proof. The distribution functions Y1:n and Y1:n (

FY1:n (x) = 1 − P (Y1:n > x) = 1 −

and ∗ (x) = 1 − FY1:n

∗ P (Y1:n

We note that for x ≥ 0, FY−1 (x) = 1:n and hence

> x) = 1 −

(

n ∏

pi

i=1

n ∏ i=1

p∗i

)

)

e−(

e−(

∑n

i=1

∑n

α λα i )x

∗ α i=1 (λi )

,

)xα .

( ( ))1/α 1 1−x ∏ − ∑n ln , n ( i=1 λαi ) i=1 pi

( ))1/α ( ∗ (x) 1 − FY1:n 1 ∏ ln − ∑n n ( i=1 λαi ) i=1 pi ( ∏ ( ))1/α ∑n ∗ α α ( ni=1 p∗i ) e−( i=1 (λi ) )x 1 ∏n ln = − ∑n ( i=1 λαi ) i=1 pi ( ( ∏n ( ∑n )1/α ∗) ∗ α) 1 α i=1 pi i=1 (λi ) ∏ ∑ = − ∑n ln + x . n n α ( i=1 λαi ) i=1 pi i=1 λi

∗ (x)) FY−1 (FY1:n = 1:n

∗ (x)) In order to obtain the required result, in view of Part (i) of Definition 2, it suffices to show that FY−1 (FY1:n 1:n ∗ (x)) with respect to x, respectively, are is convex in x. The partial derivatives of FY−1 (FY1:n 1:n ∗ (x)) ∂FY−1 (FY1:n sgn α−1 1:n = x ∂x

and

(

( ∏n ) ( ∑n ) )(1/α)−1 p∗i (λ∗i )α i=1 i=1 ∑n ln ∏n + xα , − ∑n α ( i=1 λαi ) i=1 pi i=1 λi 1

( ( ∏n ( ∑n )(1/α)−2 ∗) ∗ α) ∗ (x)) ∂ 2 FY−1 (FY1:n 1 sgn α−2 α 1:n i=1 pi i=1 (λi ) ∑ ∏ ∑ = (α − 1) x − ln + x n n α ∂x2 ( n λα ) i=1 pi i=1 λi ( ∏ni=1 ∗i)) ( pi 1 ln ∏i=1 , × − ∑n n α ( i=1 λi ) i=1 pi ∏ ∏ sgn where a = b means that a and b have the same sign. Thus, for any α > 1 and ni=1 p∗i ≤ ni=1 pi , we readily ∗ (x))/∂x2 is positive, which completes the proof of the theorem. observe that ∂ 2 FY−1 (FY1:n □ 1:n It is of interest to note that, unlike the magnitude and variability of orders, no restriction on the vectors of the involved Weibull scale parameters. Intuitively, the smallest claim amount from a sample with more product in Bernoulli parameters will be more skewness than that from a sample with less product in Bernoulli parameters. The following corollary, which is of independent interest in economics, is a direct consequence of Theorem 3. This theorem enables us to compare the smallest claim amounts in the sense of the Lorenz ordering. 8

Corollary 2 Suppose Xλ1 , · · · , Xλn are independent non-negative random variables with Xλi ∼ W (α, λi ), i = 1, · · · , n, where λi > 0. Further, suppose Ip1 , · · · , Ipn are independent Bernoulli random variables, independent ∏ ∏ ∗ ≤ of the Xλi ’s, with E(Ipi ) = pi , i = 1, · · · , n. Then, for α > 1 and ni=1 p∗i ≤ ni=1 pi , we have Y1:n Lorenz Y1:n .

Remark 1 Since the Lorenz order implies the order between coefficients of variation, thus we can obtain a lower and upper bound for the coefficient of variation of Y1:n . Set p˜ = min(p1 , · · · , pn ) and p˘ = max(p1 , · · · , pn ). It is ∏ ∏ clear that to show that p˜n = ni=1 p˜ ≤ ni=1 pi . So, from Corollary 2, we obtain a lower bound for the coefficient of variation of Y1:n as follows: √ Γ (1 + 2α−1 ) − (˜ pn ) Γ2 (1 + α−1 ) √ n cv(Y1:n ) ≥ , p˜ Γ(1 + α−1 ) ∏ ∏ also, we readily see that ni=1 pi ≤ ni=1 p˘ = p˘n . Therefore, by using Corollary 2, a upper bound for the coefficient of variation of Y1:n is given by: √ Γ (1 + 2α−1 ) − (˘ pn ) Γ2 (1 + α−1 ) √ n cv(Y1:n ) ≤ . p˘ Γ(1 + α−1 )

5

Conclusions and Suggestions

In this paper, under certain conditions, we discuss stochastic comparison between the smallest claim amounts in the sense of the right-spread order. Moreover, we show that the right-spread order and the increasing convex orders are equivalent. Finally, we obtain the results concerning the convex transform order between the smallest claim amounts and find a lower and upper bound for the coefficient of variation. The results established here extend some well-known results in the literature. Our results do not restricted to actuarial sciences and can be used in various areas including reliability theory and survival analysis. We may question whether the above results can be held for largest claim amounts. We are currently looking into this problem and hope to report the findings in a future paper.

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