Convex comparisons for random sums in random environments and applications
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“S0269964808000235” — 2008/5/17 — 13:37 — page 389 — #1 i i i i Probability in the Engineering and Informational Sciences,22, 2008, 389–413. Printed in the U.S.A. doi:10.1017/S0269964808000235 CONVEX COMPARISONS FOR RANDOM SUMS IN RANDOM ENVIRONMENTS AND APPLICATIONS JOSÉ MARÍA FERNÁNDEZ-PONCE Departamento Estadística e Investigación Operativa Facultad de Matemáticas Universidad de Sevilla 41012 Sevilla, Spain E-mail: [email protected] EVA MARÍA ORTEGA Centro de Investigación Operativa Escuela Politécnica Superior de Orihuela Universidad Miguel Hernández 03312 Orihuela (Alicante), Spain E-mail: [email protected] FRANCO PELLEREY Dipartimento di Matematica Politecnico diTorino c.so Duca Degli Abruzzi 24 10129Torino, Italy E-mail: [email protected] Recently, Belzunce, Ortega, Pellerey, and Ruiz [3] have obtained stochastic comparisons in increasing componentwise convex order sense for vectors of random sums when the summands and number of summands depend on a common random environment, which prove how the dependence among the random environmental parameters influences the variability of vectors of random sums. The main results presented here generalize the results in Belzunce et al. [3] by considering vectors of parameters instead of a couple of parameters and the increasing directionally convex order. Results on stochastic directional convexity of families of random sums under appropriate conditions on the families of summands and number of summands are obtained, which lead to the convex comparisons between random sums mentioned earlier. Different applications in actuarial science, reliability, and population growth are also provided to illustrate the main results. © 2008 Cambridge University Press 0269-9648/08 $25.00 389
“S0269964808000235” — 2008/5/17 — 13:37 — page 390 — #2 i i i i 390 J. M. Fernández-Ponce, E. M. Ortega, and F. Pellerey 1. INTRODUCTION Much research has been devoted to study conditions for the increasing convex order (also known as variability order, second stochastic dominance, or stop-loss order) of random sums (see Shaked and Shanthikumar [39], Pellerey [28] and [29], Denuit, Genest, and Marceau [7] or Kulik [17], among others). These results have found a wide field of applications in actuarial science, reliability, epidemics, economics, or queueing, where the random sums have been used to describe total claim amounts over a fixed time, accumulated wear of systems during time in cumulative damage shock models, number of individuals in a population that grows by means of a branching process, number of infected individuals in epidemic models, and so forth. Dependencies between summands and number of summands are common in applicative problems and several models for such dependence have been studied in the last few years. In real problems, the random variables in the sum usually depend on some economical, physical, or geographical random environment. Recently, the impact of dependencies among the random environments on variability comparisons of multivariate vectors of random sums has been studied in Belzunce, Ortega, Pellerey and Ruiz [3] and Frostig and Denuit [12]. In addition, stochastic comparisons of random sums involving Bernoulli random variables have become of growing in interest and have been applied in insurance, engineering, and medicine (see Lefèvre and Utev [18], Hu and Wu [14], Frostig [11], or Hu and Ruan [13]). In the literature, there are different multivariate extensions of the convex order from several extensions of convexity: in particular, the multivariate convex order, the componentwise convex order, and the directionally convex order (see the monograph by Shaked and Shanthikumar [39]). The directional convexity takes into account the order structure on the space, which the usual notion of convexity does not. The directionally convex order was introduced by Shaked and Shanthikumar [38] and has been proved to be useful in problems involving dependence in several contexts of applied probability (see, e.g., Meester and Shanthikumar [23,24]), Bäuerle and Rolski [2], Li and Xu [19], or Rüschendorf [35]). This order is strictly weaker than the supermodular order, which compares only dependence structure of vectors with fixed equal marginals. The directionally convex order tells about the dependence and variability of the marginals, which are not necessarily equal. Belzunce et al. [3] have studied variability comparisons by means of the increasing componentwise convex order for two vectors of random sums. In that paper, the summands and the number of summands are dependent by means of a couple of random parameters, which represent some environmental conditions. They have considered random sums defined by Zi(θ1,θ2)= Ni(θ1) k=1 Xk,i(θ2)(1.1) for i=1, 2, ...,m, where (θ1,θ2)∈T⊆R2and Xi(θ2)={Xk,i(θ2),k∈N}, i=1, ...,m, is a sequence of nonnegative random variables, (N1(θ1),...,Nm(θ1))
“S0269964808000235” — 2008/5/17 — 13:37 — page 391 — #3 i i i i CONVEX COMPARISONS FOR RANDOM SUMS 391 is a vector of integer-valued random variables, and X1(θ2),...,Xm(θ2)and N1(θ1),...,Nm(θ1)are mutually independent. In this article, we extend the above setting by considering dependence by means of a multivariate random vector of parameters. A main motivation for introducing multivariate random environments is clear from a practical point of view. For example, severity and number of claims in insurance for nature catastrophes such as hurricanes or earthquakes depend on geography as well as some other physical factors; in motor third-party liability insurance, there are several factors influencing the driving abilities (see Denuit, Dhaene, Goovaerts, and Kaas [6] for other examples). Formally, let T⊆Rn1and L⊆Rn2be two sublattices in Rn1and Rn2, respectively, and let θ=(θ1,...,θn1)∈Tand λ=(λ1,...,λn2)∈L. Consider the sums defined by Zi(θ,λ)= Ni(θ) k=1 Xk,i(λ)(1.2) for i=1, 2, ...,m, where X1,1(λ),X2,1(λ),...,X1,m(λ),X2,m(λ),... and N1(θ),..., Nm(θ)are mutually independent. Now, let (,)=(1,...,n1,1,...,n2)be a random vector taking on values in T×L. We are interested in stochastic comparisons of vectors of random sums given by Z(,)=(Z1(,),...,Zm(,)).(1.3) Here, the random sum Zi(,)= Ni() k=1 Xk,i()(1.4) can be considered as a mixture of {Zi(θ,λ)|(θ,λ)∈T×L}, with respect to a vector (,)of random parameters describing the environmental conditions. Another generalization that we will consider in the article gives rise when some of the parameters of the random sum appear both in the summands and the number of summands. The presence of duplicates of parameters is useful in some applicative contexts (see, e.g., Section 4.3). Formally, let D⊆Rnbe a sublattice in Rnand let δ=(δ1,...,δn)∈D. Consider the sums defined by Zi(δ)= Ni(δ) j=1 Xj,i(δ)(1.5) for i=1, 2, ...,m, where Xj,i(δ)≥0 a.s. and X1,1(δ),X2,1(δ),...,X1,m(δ),X2,m(δ),... and N1(δ),...,Nm(δ)are mutually independent. Note that (1.5) includes, as a particular case, the case when the Xj,i(δ)or the Ni(δ)are actually parametrized only by a subset of the parameters δ1,...,δn.
“S0269964808000235” — 2008/5/17 — 13:37 — page 392 — #4 i i i i 392 J. M. Fernández-Ponce, E. M. Ortega, and F. Pellerey Assuming that =(1,...,n) is a random vector taking on values in D, it is interesting to study the stochastic properties of the vector of random sums Z()=(Z1(),...,Zm()),(1.6) where Zi()is a mixture of {Zi(δ)|δ∈D}with respect to the vector of random parameters. In this article we obtain results on stochastic directional convexity (see Shaked and Shanthikumar [38]) of families of random sums, under appropriate conditions on the families of summands and number of summands. From these results, we study how the dependence among multivariate random environments influences the variability of random sums and the dependence and variability of vectors of random sums by means of the increasing directionally convex order, which are the main purposes of this article; that is, we provide sufficient conditions to model, to compare, and to bound the variability as well as the strength of dependence between two vectors of random sums parameterized on multivariate random environments. In this way, this article completes the study started in Belzunce et al. [3]. The article proceeds as follows. In Section 2 we provide notation and tools on stochastic comparisons and multivariate stochastic convexity that will be used in the article. In Section 3 we state and prove the main results mentioned earlier concerning stochastic comparisons and stochastic directional convexity of families of random sums. Finally, applications for some models in insurance, reliability, and populations growth, defined by means of random sums, are dealt with in Section 4. 2. UTILITY NOTIONS AND PRELIMINARIES In this section we focus on providing notation and mathematical tools for the results in the article. In particular, we will recall the definitions of some stochastic orders as well as multivariate notions of stochastic convexity for a family of parameterized random variables. For that, we will consider different notions of convexity in the multivariate setting. Some conventions and notations that are used throughout the article were given previously. Let ≤denote the coordinatewise ordering (i.e., for any x,y∈Rn, then x≤yif xi≤yifor i=1, 2, ...,n) and [x,y]≤zas shorthand for x≤zand y≤z. The operators +,∨, and ∧denote respectively the componentwise sum, maximum, and minimum. The notation =st stands for equality in law and a.s. is shorthand for almost surely. For any family of parameterized random variables {Xθ|θ∈T}, with T⊆R, such that every θis a value from a random variable , whose distribution is concentrated on T, we denote by X() the mixture of the family {Xθ|θ∈T} with mixing distribution . For any random variable (or vector) Xand an event A,
“S0269964808000235” — 2008/5/17 — 13:37 — page 393 — #5 i i i i CONVEX COMPARISONS FOR RANDOM SUMS 393 [X|A]denotes a random variable whose distribution is the conditional distribution of Xgiven A. Also, according to most of the reliability literature, throughout this article we write “increasing” instead of “non-decreasing” and “decreasing” instead of “non-increasing.” 2.1. Univariate Stochastic Orderings Some of the main results in this article deal with the increasing convex order of random sums. Let us recall the definition of this ordering, also known as variability order, second stochastic dominance or stop-loss order, jointly with the stochastic order. For a comprehensive discussion on these stochastic orders, we refer to Shaked and Shanthikumar [39] and Müller and Stoyan [26]. DEFINITION 2.1: Let X and Y be two nonnegative random variables, with survival functions FXand FY, respectively, then X is said to be smaller than Y in the stochastic (increasing convex) order (denoted by X ≤st(icx)Y) if E[φ(X)]≤E[φ(Y)] for all increasing (increasing convex) functions φfor which the expectations exist. Equivalently, X ≤st Y if for all t ≥0it holds that FX(t)≤FY(t). A characterization of the stochastic ordering that will play a crucial role in this article is recalled now (see Theorem 1.A.1 in Shaked and Shanthikumar [39]). Given two random variables Xand Y,X≤st Yif and only if there exist two random variables Xand Y, defined on the same probability space, such that X=st X,Y=st Y, and X≤ Y, a.s. The increasing convex order has been applied in several contexts, such as reliability and actuarial science. It allows one to compare the stop-loss transforms of two insurance policies for a kind of reinsurance contract (see Müller and Stoyan [26] for applications in risk theory). 2.2. Multivariate Notions of Convexity Next, we recall the concepts of convex, directionally convex, and supermodular functions. For a complete discussion on convex functions, we refer to the monograph by Rockafellar [31]. For a definition and properties of directionally convex functions, see Shaked and Shanthikumar [38] or Meester and Shanthikumar [23]. For a discussion and background on supermodular functions (that are also called superadditive functions in the literature) we refer to Marshall and Olkin [22].
“S0269964808000235” — 2008/5/17 — 13:37 — page 394 — #6 i i i i 394 J. M. Fernández-Ponce, E. M. Ortega, and F. Pellerey DEFINITION 2.2: A real-valued function φdefined on Rnis said to be the following: (i) Convex (concave) (denoted by φ∈cx(cv))if φ(αx+(1−α)y)≤(≥)αφ(x)+(1−α)φ(y) for all x,y∈Rnand α∈[0, 1]. If in addition, φis increasing (decreasing), [i.e., for all x≤y, then φ(x)≤(≥)φ(y)], then we say that φis increasing (decreasing) and convex (denoted by φ∈icx(icv)). (ii) Increasing componentwise convex (denoted by φ∈iccx) if it is increasing and it is convex in each argument when the others are held fixed. (iii) Supermodular (denoted by φ∈sm) if φ(x∨y)+φ(x∧y)≥φ(x)+φ(y) for all x,y∈Rn. (iv) Directionally convex (concave) (denoted by φ∈dcx(dcv)) if for any xi∈Rn, i=1, 2, 3, 4, such that x1≤[x2,x3]≤x4and x1+x4=x2+x3, then φ(x1)+φ(x4)≥(≤)φ(x2)+φ(x3). If, in addition, φis increasing (decreasing), then we say that φis increasing (decreasing) and directionally convex (denoted by φ∈idcx(idcv)). A function φ:Rn−→ Rmdefined by φ(x)=(φ1(x),...,φm(x)) is directionally convex (concave) if each of the coordinate functions φi,i =1, 2, ...,m, is directionally convex (concave). Directional convexity neither implies nor is implied by usual convexity (see Shaked and Shanthikumar [38]). The composition of functions preserves increasing directional convexity (see Lemma 2.4 in Meester and Shanthikumar [23]). In particular, the composition of an icx function with an idcx function is an idcx function (see Corollary 2.5 in Meester and Shanthikumar [23]). A useful characterization of dcx functions is given now (see Proposition 2.1 in Shaked and Shanthikumar [38]). Given φ:Rn−→ R,φ∈dcx if and only if φis supermodular and coordinatewise convex. Remark 2.1: We note that φis a supermodular function if and only if φis supermodular in any couple of arguments when the others are held fixed (see Marshall and Olkin [22]). From this property and the previous characterization, observe that a function φ:Rn−→ Ris increasing and directionally convex in (θ1,...,θn)if and only if φis increasing, supermodular in any couple (θi,θl), whenever all other arguments are held fixed, and convex in any θi, whenever all other arguments are held fixed. LEMMA 2.1: Let T⊆Rnand let g :T−→ Nbe an increasing and directionally convex function. If {xj,j∈N}is any increasing sequence of real numbers, then the function ψ(θ):=g(θ) j=1xjis increasing and directionally convex.
“S0269964808000235” — 2008/5/17 — 13:37 — page 395 — #7 i i i i CONVEX COMPARISONS FOR RANDOM SUMS 395 PROOF: First, let us write the function ψas ψ(θ)=Sg(θ), where Sn=n j=1xj. Note that Snis increasing and convex when {xj,j∈N}is an increasing sequence of real numbers. Thus, the composition ψ=S◦gis increasing and directionally convex by Corollary 2.5 in Meester and Shanthikumar [23] and the assertion follows. 2.3. Multivariate Notions of the Increasing Convex Order The increasing convex order can be extended to the multivariate case in several ways. Here, we consider three of them. For a survey on these stochastic orderings, we refer to Shaked and Shanthikumar [39]. DEFINITION 2.3: Let X=(X1,...,Xn)and Y=(Y1,...,Yn)be two n-dimensional random vectors; then Xis said to be smaller than Yin the increasing convex (increasing componentwise convex, increasing directionally convex) order (denoted by X≤icx(iccx,idcx)Y)if E[φ(X)]≤E[φ(Y)] for all increasing convex [increasing componentwise convex, increasing directionally convex] real-valued functions φdefined on Rnfor which the expectations exist. Increasing (componentwise, directionally) concave orders are defined analogously. Clearly, the iccx order is stronger than the icx order; that is, if X≤iccx Y, then X≤icx Y. Also, if X≤iccx Y, then Xi≤icx Yi. Stochastic orders defined above by means of functionals take into account variability. The following dependence order is defined in terms of supermodular functions. The supermodular order strictly implies the increasing directionally convex order, although the supermodular order compares only dependence structure of vectors with fixed equal marginals and the increasing directionally convex order additionally compares the variability of the marginals, which might be different. For a further discussion on supermodular order of random vectors, see Marshall and Olkin [22], Shaked and Shanthikumar [40] and Müller and Stoyan [26]. DEFINITION 2.4: Let X=(X1,X2,...,Xn)and Y=(Y1,Y2,...,Yn)be two n-dimensional random vectors, with equal marginal distributions; then Xis said to be smaller than Yin the supermodular order (denoted by X≤sm Y)if E[φ(X)]≤E[φ(Y)], for every supermodular real-valued function φdefined on Rnfor which the expectations exist. For n=2, the supermodular order is equivalent to the well-known positive quadrant dependence order (for short, PQD) (see Joe [15]). The supermodular order has been recently used in several applied contexts (see Shaked and Shanthikumar [40],
“S0269964808000235” — 2008/5/17 — 13:37 — page 396 — #8 i i i i 396 J. M. Fernández-Ponce, E. M. Ortega, and F. Pellerey Müller [25], Bäuerle and Müller [1], Denuit et al. [7], Lillo, Pellerey, Semeraro and Shaked [20], Frostig [11], Rüschendorf [35], Lillo and Semeraro [21], Belzunce et al. [3] or Denuit and Müller [8], among others). 2.4. Multivariate Stochastic Convexity At this point, we recall some notions of multivariate stochastic convexity for a family of parameterized random variables. Shaked and Shanthikumar [36,37] introduced the notion of stochastic convexity. Multivariate stochastic directional convexity was introduced in Shaked and Shanthikumar [38] and it was also studied in Chang, Chao, Pinedo, and Shanthikumar [4] and Meester and Shanthikumar [23]. Stochastic directional convexity was generalized to a general space in Meester and Shanthikumar [24]. Below, we will consider a family of multivariate random variables X(θ)for θ∈T, where Tis a sublattice of either Rnor Nn. DEFINITION 2.5: A family {X(θ),θ∈T}of multivariate random variables is said to be the following: (i) Stochastically increasing (denoted by {X(θ),θ∈T}∈SI) if for any θi∈T, i=1, 2,θ1≤θ2, then X(θ1)≤st X(θ2). (ii) Stochastically increasing and directionally convex (denoted by {X(θ),θ∈T} ∈SI −DCX) if {X(θ),θ∈T}∈SI and E[φ(X(θ))]is increasing and directionally convex in θfor any φ∈idcx. (iii) Stochastically increasing and directionally convex in the sample path sense (denoted by {X(θ),θ∈T}∈SI −DCX(sp)) if for any four θi∈T, i=1, ...,4, such that θ1≤[θ2,θ3]≤θ4and θ1+θ4=θ2+θ3, there exist four random variables Xi,i=1, ...,4, defined on a common probability space, such that Xi=st X(θi),i=1, ...,4and [X2,X3]≤X4,a.s.(2.1) and X1+X4≥X2+X3,a.s.(2.2) (iv) Stochastically increasing and directionally linear in the sample path sense (denoted by {X(θ),θ∈T}∈SI −DL(sp)) if in (iii) the inequality (2.2)is replaced by X1+X4=X2+X3,a.s.(2.3) In the case that both the parameter and the random variables are univariate, then we will use the notation SI −CX(sp)instead of SI −DCX(sp). Note that stochastic directional convexity in the sample path sense strictly implies stochastic directional convexity (see Counterexample 3.1 in Shaked and Shanthikumar [38]).
“S0269964808000235” — 2008/5/17 — 13:37 — page 397 — #9 i i i i CONVEX COMPARISONS FOR RANDOM SUMS 397 Stochastic increasing directional convexity and stochastic increasing directional convexity in sample path sense are closed by composition with idcx functions (see, e.g., Lemma 2.15 in Meester and Shanthikumar [23]).Also, both notions of stochastic convexity are closed by conjunction of independent random variables (see Lemma 2.16 in Meester and Shanthikumar [23] or Theorem 3.3 and Theorem 4.4 in Meester and Shanthikumar [24]). Some examples of stochastic directional convexity of parameterized families of random variables can be found in the literature: See Shaked and Shanthikumar [38], Chang, Shanthikumar and Yao [5] or Meester and Shanthikumar [24]. For example, the Bernoulli distribution and the Poisson distribution are SI −DL(sp), the multinomial distribution and the gamma distribution are SI −DCX(sp)and the multivariate geometric distribution is SD −DCX(sp). Other examples can be obtained by using the above preservation properties. Also, under appropriate conditions, some applied stochastic models have stochastic directional convexity properties (see above references). 3. MAIN RESULTS In this section we provide results on stochastic directional convexity and stochastic directional convexity in the sample path sense for a family of parameterized random sums, under appropriate conditions on the parameterized families of nonnegative summands and number of summands. From them, we provide results for comparing two random sums in the increasing convex order and two vectors of random sums in the increasing directionally convex order sense when the summands and the number of summands are dependent by means of a multivariate random environment. THEOREM 3.1: Consider the family of random sums {Z(δ),δ∈D}defined by Z(δ)= N(δ) j=1 Xj(δ), where Dis a sublattice in Rn.If (i) all of the families {Xj(δ),δ∈D},j∈N, and {N(δ),δ∈D}are independent, (ii) {Xj(δ),δ∈D}∈SI −DCX(sp)for every fixed j ∈N, (iii) {N(δ),δ∈D}∈SI −DCX(sp), (iv) {Xj(δ),j∈N}∈SI for every fixed δ∈D, then {Z(δ),δ∈D}∈SI −DCX(sp). PROOF: Let δi, with i=1, ..., 4, be such that δ1≤[δ2,δ3]≤δ4and δ1+δ4= δ2+δ3. By assumptions (i), (ii), and (iii), we can build on the same probability
“S0269964808000235” — 2008/5/17 — 13:37 — page 404 — #16 i i i i 404 J. M. Fernández-Ponce, E. M. Ortega, and F. Pellerey COROLLARY 3.3: Consider the family of random sums {Z(θ,λ,δ),(θ,λ,δ)∈ T×L×D}defined by Z(θ,λ,δ)= N(θ,δ) j=1 Xj(λ,δ). If (i) all of the families {Xj(λ,δ),(λ,δ)∈L×D},j∈N, and {N(θ,δ),(θ,δ)∈ T×D}are independent, (ii) {Xj(λ,δ),(λ,δ)∈L×D}∈SI −DCX for every j ∈N, (iii) {N(θ,δ),(θ,δ)∈T×D}∈SI −DCX, (iv) {Xj(λ,δ),j∈N}∈SI for every fixed (λ,δ)∈L×D, then (,,)≤idcx (,,) implies Z(,,)≤icx Z(,,) PROOF: First, we will prove that for any two random vectors (1,1,1)and (2,2,2), (1,1,1)≤idcx (2,2,2)⇒((1,1),(1,1)) ≤idcx ((2,2),(2,2)). (3.5) For it, note that if g((θ,δ1),(λ,δ2)) is idcx, then also the function φ(θ,λ,δ)= g((θ,δ),(λ,δ)) is idcx. Therefore, if (1,1,1)≤idcx (2,2,2), then for any idcx function gwe have that E[g((1,1),(1,1))]=E[φ(1,1,1)] ≤E[φ(2,2,2)] =E[g((2,2),(2,2))], and this proves (3.5). Wewilldenote ˜ Z(θ,λ,δ1,δ2)=N(θ,δ1) j=1Xj(λ,δ2)andobservethatZ(,,)=st ˜ Z(,,,).
“S0269964808000235” — 2008/5/17 — 13:37 — page 405 — #17 i i i i CONVEX COMPARISONS FOR RANDOM SUMS 405 Now, let ube any increasing and convex function and let hbe defined as in Theorem 3.2. Then, by Theorem 3.2 and inequality (3.5) we get E[u(Z(,,))]=E[u(˜ Z(,,,))] =E[E[u(˜ Z(,,,))|(,,,)]] =E[h((,),(,))] ≤E[h((,),(,))] =E[E[u(˜ Z(,,,))|(,,,)]] =E[u(˜ Z(,,,))] =E[u(Z(,,))] (i.e., the assertion). Note that the above result can be generalized to a vector of random sum like for Corollary 3.2. In fact, the proof of the following corollary is similar to the proof of Corollary 3.2, but here we use Theorem 3.3 in Meester and Shanthikumar [24] instead of Theorem 4.4 in Meester and Shanthikumar [24]. COROLLARY 3.4: Consider m ∈Nrandom sums defined by Zi(θ,λ,δ)= Ni(θ,δ) j=1 Xj,i(λ,δ),i=1, ...,m, that are independent for any fixed value of (θ,λ,δ)∈T×L×Dand let Z(θ,λ,δ)=(Z1(θ,λ,δ),...,Zm(θ,λ,δ)). If (i) all of the families {Xj,i(λ,δ),(λ,δ)∈L×D},j∈N, and {Ni(θ,δ), (θ,δ)∈T×D},i=1, ...,m, are independent, (ii) {Xj,i(λ,δ),(λ,δ)∈L×D}∈SI −DCX for every j ∈Nand i =1, ...,m, (iii) {Ni(θ,δ),(θ,δ)∈T×D}∈SI −DCX for any i =1, ...,m, (iv) the sequence {Xj,i(λ,δ),j∈N}∈SI for every fixed (λ,δ)∈L×Dand i=1, ...,m, then (,,)≤idcx (,,) implies Z(,,)≤idcx Z(,,)
“S0269964808000235” — 2008/5/17 — 13:37 — page 406 — #18 i i i i 406 J. M. Fernández-Ponce, E. M. Ortega, and F. Pellerey 4. APPLICATIONS In this section we provide some examples to illustrate how the main results can be applied. 4.1. Collective Risk Models in Actuarial Sciences Consider an homogeneous portfolio of nrisks over a single period of time and assume that during that period, each policyholder ican have a nonnegative claim Xiwith probability θi∈[0, 1]⊆R. Then the total claim amount S(θ1,...,θn)during that time can be represented as S(θ1,...,θn)= n i=1 Ii(θi)Xi, where Ii(θi)denotes a Bernoulli random variable with parameter θi. As it is pointed out, for example, in Frostig [10], assumption of independence among the Bernoulli random variables Ii(θi),i=1, ...,n, is not suitable to describe real contexts, since their distributions might actually depend on some common random environment. Thus, one can replace the vector of real parameters (θ1,...,θn)by a random vector =(1,...,n), with values in [0, 1]n⊆Rnand describing both the random environment for occurrences of claims and the dependence among them. Some known results in the literature deal with stochastic comparisons of random sums involving Bernoulli random variables (see Lefèvre and Utev [18], Hu and Wu [14], Frostig [10], or Hu and Ruan [13]). Here, we state conditions for the stochastic comparison, in the increasing convex sense, of two total claim amounts defined as above. PROPOSITION 4.1: Let I(θ)=(I1(θ1),...,In(θn)), where the Ii(θi)are independent Bernoulli random variables with parameters θi,i=1, ...,n. Consider N(θ1,...,θn)=n i=1Ii(θi). Then {N(θ1,...,θn),(θ1,...,θn)∈[0, 1]n⊆Rn}∈ SI −DCX(sp). PROOF: First, note that {N(θ1,...,θn),(θ1,...,θn)∈[0, 1]n⊆Rn}is clearly stochastically increasing. Now, consider a family of Bernoulli random variables {Iθ:θ∈[0, 1]}. It is easy to see that this family is SI −DL(sp) (see, e.g., Example 5.3.8 in Chang et al. [5]). Therefore, for any fixed θi,k(k=1, ...,4,i=1, ...,n)such that θi,1 ≤[θi,2,θi,3]≤ θi,4 and θi,1 +θi,4 =θi,2 +θi,3, we can build, on the same probability space, random variables Ii(θk)=st Ii(θk)for k=1, ..., 4 and i=1, ...,n, such that Ii(θi,2), Ii(θi,3)≤ Ii(θi,4), a.s. and Ii(θi,1)+ Ii(θi,4)= Ii(θi,2)+ Ii(θi,3), a.s.
“S0269964808000235” — 2008/5/17 — 13:37 — page 407 — #19 i i i i CONVEX COMPARISONS FOR RANDOM SUMS 407 Note that, by independence, we can build all of the variables Ii(θi,k), for all i=1, ...,n, on the same probability space. Now, consider the random variables Nk=n i=1 Ii(θi,k). We observe that [ N2, N3]≤ N4, a.s. and N1+ N4= N2+ N3, a.s. Then {N(θ1,...,θn),(θ1,...,θn)∈[0, 1]n⊆Rn}∈SI −DL(sp), since (θ1,1,...,θn,1)≤[(θ1,2,...,θn,2),(θ1,3,...,θn,3)]≤(θ1,4,...,θn,4), a.s. (θ1,1,...,θn,1)+(θ1,4,...,θn,4)=(θ1,2,...,θn,2)+(θ1,3,...,θn,3), a.s. and Nk=st N(θ1,k,...,θn,k), for k=1, ..., 4. The assertion follows observing that SI −DL(sp)implies SI −DCX(sp). As immediate consequence, we get the following result. COROLLARY 4.1: Let X1,...,Xnbe independent and identically distributed nonnegative random variables and let I1(θ1),...,In(θn)be independent Bernoulli random variables with parameters θ1,...,θn,respectively, and independent of Xi,i =1, ...,n. Consider the total claim amounts S(θ1,...,θn)=n i=1Ii(θi)Xi. Then (1,...,n)≤idcx ( 1,..., n) implies S(1,...,n)≤icx S( 1,..., n). PROOF: Observe that since the claims Xiare assumed to be independent, then S(θ1,...,θn)=st N(θ1,...,θn) i=1 Xi. The assertion now follows from Proposition 4.1 and Corollary 3.1. 4.2. Population Growth Models Branching processes have been considered an appropriate mathematical model for the description of populations’ growth, where individuals produce offsprings according to some stochastic laws. Several applications involve medicine, molecular and cellular biology, human evolution, physics or actuarial science (see Rolski, Schmidli, Schmidt, and Teugels [32], Ross [33], or Kimmel and Axelrod [16]). In this subsection, we provide a result dealing with stochastic comparisons between two branching
“S0269964808000235” — 2008/5/17 — 13:37 — page 408 — #20 i i i i 408 J. M. Fernández-Ponce, E. M. Ortega, and F. Pellerey processes defined on random environments, which is closely related to Theorem 2.2 in Pellerey [30]. The branching processes on random environments that we consider here are defined as follows. Let θ={θ0,θ1,...,}be a sequence of values in Tdescribing the evolutions of the environment, and define, recursively, the stochastic process Z(θ)={Zn(θ0,...,θn),n∈N}by Z0(θ0)=X1,0(θ0) and Zn(θ0,...,θn)= Zn−1(θ0,...,θn−1) j=1 Xj,n(θn),n≥1. (4.1) In order to deal with random evolutions of the environment, we consider a sequence =(0,1,...)of random variables taking on values in Tand we consider the stochastic process Z()={Zn(0,...,n),n∈N}defined by Z0(0)=X1,0(0) and Zn(0,...,n)= Zn−1(0,...,n−1) j=1 Xj,n(n),n≥1, (4.2) where, for every j,k∈N,Xj,k(k)is a nonnegative random variable such that [Xj,k(k)|k=θ]= st Xj,k(θ). First, we prove the SI −DCX(sp) property of such parameterized families of branching processes. PROPOSITION 4.2: Let θ=(θ0,θ1,...)be a sequence of values in T⊆Rand consider the stochastic process defined by (4.1). If (i) the variables {Xj,k(θk)},j∈Nand k ∈Nare all mutually independent, (ii) {Xj,k(θk),θk∈T}∈SI −CX(sp)for every fixed j ∈Nand k ∈N, (iii) {Xj,k(θk),j∈N}∈SI for every fixed θk∈Tand k ∈N, then {Zn(θ0,...,θn),(θ0,...,θn)∈Tn+1}∈SI −DCX(sp)for every n ∈N. PROOF: We will proceed by induction. First, observe that, trivially we have that {Z1(θ0),(θ0)∈T}is SI −CX(sp)and, thus, SI −DCX(sp). Now, assume that assertion is true for n−1; that is, assume that {Zn−1(θ0,...,θn−1),(θ0,...,θn−1)∈Tn}is SI −DCX(sp). Then, by Theorem 3.1 and the inductive hypothesis, it follows that Zn(θ0,...,θn)= Zn−1(θ0,...,θn−1) j=1 Xj,n(θn)(4.3) is SI −DCX(sp)in (θ0,...,θn)∈Tn+1and, thus, the assertion follows.
“S0269964808000235” — 2008/5/17 — 13:37 — page 409 — #21 i i i i CONVEX COMPARISONS FOR RANDOM SUMS 409 From the previous result, we can easily get the following comparison result for two branching processes defined on two different random environments (see Pellerey [30] for further details)). COROLLARY 4.2: Consider the stochastic processes Z(θ)={Zn(θ0,...,θn),n∈N} and Z()={Zn(0,...,n),n∈N}defined by (4.1) and (4.2), respectively. If the assumptions of Proposition 4.2 hold, then (1,...,n)≤idcx ( 1,..., n) implies Zn(1,...,n)≤icx Zn( 1,..., n). 4.3. Cumulative Damage Shock Models Shock models are of great interest in the context of reliability theory since they are commonly used to describe the lifetime or the reliability of systems or items subjected to shocks. In this context, compound Poisson processes are used to describe the wear accumulated by systems during time. Assume that a system is subjected to shocks arriving according to a Poisson process Nθhaving rate θ>0 and that the ith shock causes a nonnegative damage Xi, where the damages accumulate additively. Then the total wear accumulated up to time t≥0 by the system is given by (see Esary, Marshall, and Proschan [9]) Wθ(t)= Nθ(t) i=1 Xi,(4.4) with Wθ(t)=0 in the case Nθ(t)=0. If the system fails when the accumulated wear exceeds a fixed threshold, then some properties of the distribution of the system lifetime can be obtained from stochastic properties of the process Wθ={Wθ(t),t∈R}. In literature there are many articles dealing with stochastic comparisons among accumulated wear processes defined as in (4.4). However, almost all of them assume independence among all damages Xiand also independence between the damages and the counting process Nθ(see, e.g., Esary et al. [9], Ross and Schechner [34], or Pellerey [27]). Here, we provide a generalization of these results under conditional independence among damages and the shock arrival process. For it, assume that the system is subjected to shocks arriving according to a Poisson process Nθ. Let Xj(θ,λ) denote the damage caused by the jth shock, parameterized by the same parameter θof the process Nθand a generic environmental parameter λ that is common for all damages. Then the total wear accumulated up to time t≥0by
“S0269964808000235” — 2008/5/17 — 13:37 — page 410 — #22 i i i i 410 J. M. Fernández-Ponce, E. M. Ortega, and F. Pellerey the system is given by Wθ,λ(t)= Nθ(t) j=1 Xj(θ,λ) (4.5) (where Nθ(t) j=1Xj(θ,λ) =0 in the case Nθ(t)=0). Now, assume that the parameters are given by random environmental factors (i.e., by a random vector (,)), and consider the wear process W,(t)= N(t) j=1 Xj(,),(4.6) defined as a mixture of the families Wθ,λwith respect to the vector (,). Then by Corollary 3.1 and since Poisson random variables are SI −DL(sp), we obtain the following comparison criterion. COROLLARY 4.3: Consider the stochastic processes Wθ,λand W,defined by (4.5) and (4.6), respectively. If (i) Xj(θ,λ) are independent for all j ∈Nfor any fixed values of (θ,λ), (ii) {Xj(θ,λ),(θ,λ) ∈R+×R+}∈SI −DCX(sp)for any j ∈N, (iii) the families {Xj(θ,λ),(θ,λ) ∈R+×R+}and {Nθ,θ∈R+}are independent, (iv) {Xj(θ,λ),j∈N}∈SI for any (θ,λ) ∈R+×R+, then (,) ≤idcx (,) implies W,(t)≤icx W,(t)∀t≥0. Similar results can be stated in case the damages do not accumulate additively. For example, assume that the damage caused by the ith shock is given by a function of the previously accumulated damage and the intensity Xiof the ith shock. For that, consider a cumulative damage discrete-time process W(λ)={Wn(λ1,...,λn), n∈N,λi∈R+,i=1, ...,n}defined recursively as W1(λ1)=X1(λ1) and Wn(λ1,...,λn)=Wn−1(λ1,...,λn−1)+g(Wn−1(λ1,...,λn−1),Xn(λn)),n>1. Now, consider two processes defined as above but with parameters given by realizations of two vectors (1,...,n)and ( 1,..., n)describing different environmental conditions. Proceeding by induction and using arguments similar to those
“S0269964808000235” — 2008/5/17 — 13:37 — page 411 — #23 i i i i CONVEX COMPARISONS FOR RANDOM SUMS 411 in the previous proof and Lemma 2.4 in Meester and Shanthikumar [23], one can easily prove the following result. COROLLARY 4.4: Consider Wn(λ1,...,λn),n∈N,λi∈R+,i=1, ...,n defined as above. If (i) the families {Xi(λi),λi∈R+}, with i =1, ...,n, are independent, (ii) {Xi(λi),λi∈R+}∈SI −CX(sp), for every fixed value i =1, ...,n, (iii) {Xi(λi),i=1, ...,n}∈SI, for every fixed value λi∈R+, then (1,...,n)≤idcx ( 1,..., n)∀n∈N implies Wn(1,...,n)≤icx Wn( 1,..., n)∀n∈N whenever the function g(w,x)is increasing and directionally convex. Acknowledgment We sincerely thank Professor Moshe Shaked for useful comments and suggestions regarding the proof of Theorem 3.1. F. Pellerey is supported by the Italian PRIN-Cofin 2006 “Metodi Stocastici in Finanza” J. M. Fernández-Ponce is supported by Consejeria de Innovacion, Ciencia y Empresa of Junta deAndalucia under grant Ayuda a la Investigación (resol. 19 de septiembre de 2005). E. M. Ortega is supported by the Operations Research Centre and the Department of Statistics, Mathematics and Computer Sciences, both at the University Miguel Hernández and Ministerio de Ciencia y Tecnologia under grant BFM2003-02947. She is sincerely grateful to Dr. JoséAlouso and Prof. Laureauo F. Escudero by their time and encouragement on relation by this paper. References 1. Bäuerle, N. & Müller,A. (1998). Modeling and comparing dependencies in multivariate risk portfolios. Astin Bulletin 28: 59–76. 2. Bäuerle, N. & Rolski, T. (1998).A monotonicity result for the work-load in Markov-modulates queues. Journal of Applied Probability 35: 741–747. 3. Belzunce, F., Ortega, E.M., Pellerey, F., & Ruiz, J.M. (2006). Variability of total claim amounts under dependence between claims severity and number of events. Insurance: Mathematics and Economics 38: 460–468. 4. Chang, C.-S., Chao, X.L., Pinedo, M., & Shanthikumar, J.G. (1991). Stochastic convexity for multidimensional processes and applications. IEEE Transactions on Automated Control 36: 1347–1355. 5. Chang, C.-S., Shanthikumar, J.G., & Yao, D.D. (1994). Stochastic convexity and stochastic majorization. In D.D. Yao (ed.), Stochastic modeling and analysis of manufacturing systems. New York: Springer-Verlag. 6. Denuit, M., Dhaene, J., Goovaerts, M., & Kaas, R. (2005). Actuarial theory for dependent risks. Measures, orders and models. Chichester, UK: Wiley. 7. Denuit, M., Genest, C., & Marceau, E. (2002). Criteria for the stochastic ordering of random sums, with acturial applications. Scandinavian Actuarial Journal 1: 3–16. 8. Denuit, M. & Müller, A. (2002). Smooth generators of integral stochastic orders. Annals of Applied Probability 12: 1174–1184.
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