model Poits And Risk Measurement Based Models

Transcription

1 Model poits ad Tail-VaR i life isurace Michel Deuit Istitute of Statistics, Biostatistics ad Actuarial Sciece Uiversité Catholique de Louvai (UCL) Louvai-la-Neuve, Belgium Julie Trufi Departmet of Mathematics Uiversité Libre de Bruxelles (ULB) Bruxelles, Belgium

2 Abstract Ofte, actuaries replace a group of heterogeeous life isurace cotracts (differet age at policy issue, cotract duratio, sum isured, etc.) with a represetative oe i order to speed the computatios. The preset paper aims to homogeize a group of policies by cotrollig the impact o Tail-VaR ad related risk measures. Key words ad phrases: risk measures, life isurace, model poits, supermodular order, covex order.

3 1 Itroductio ad motivatio Life isurace models are becomig more ad more sophisticated uder Solvecy 2 regulatio. Europea isurace compaies are required to base their cash-flow projectio o a policy-by-policy approach o the oe had, ad to demostrate the compliace of their iteral model by carryig out additioal testig o the other had (see EIOPA, 2010). I particular, oe of the validatio tools recommeded by the regulator is sesitivity testig, which cosists i estimatig the impact o the model outcomes of various chages i the uderlyig risk factors. Next to the baselie rus, isurers are the ivited to coduct sesitivity aalyses. Usually, all those studies eed to be performed withi tight deadlies. However, the use of Mote-Carlo simulatios based o a policy-by-policy approach ofte leads to large ruig times (up to several days for the etire portfolio with the curretly available computig power). Savig time whe ruig the models thus appears to be a issue of major importace i life isurace. A way to address this problem is to rely o groupig methods. Uder certai coditios, the regulator permits the projectio of future cash-flows based o suitable model poits. We refer the reader to EIOPA (2010) for extesive details. The basic idea is to aggregate policies ito homogeeous groups ad to replace the group of cotracts with a represetative isurace policy i order to speed the simulatio process. I this paper, we aim to homogeize a group of policies by cotrollig the impact o Tail-Value-at-Risk (Tail-VaR). Related problems have already bee cosidered i the actuarial literature. For istace, Frostig (2001) compared a heterogeeous portfolio composed of idividual risks that are idepedet but ot idetically distributed with two homogeeous portfolios i which the risks are idepedet ad idetically distributed. The first homogeeous portfolio cosidered by Frostig (2001) is made of risks that are mixtures with equal weights of the risks i the heterogeeous portfolio ad leads to a upper boud for the Tail-VaR of the heterogeeous portfolio. The secod oe cosists of risks that are the average of the risks i the heterogeeous portfolio ad turs out to be a lower boud. Here, we use a simpler approach to obtai the upper boud derived i Frostig (2001) usig a geeral compariso result obtaied by Deuit ad Müller (2002). Also, relyig o this upper boud, we show how to build coservative model poits with respect to Tail-VaR i a life isurace cotext. Fially, we improve the lower boud obtaied i Frostig (2001) ad we discuss various approximatios. The remaider of this paper is orgaized as follows. Sectio 2 recalls useful defiitios ad makes the problem uder ivestigatio more formal i terms of stochastic domiace rules ad risk measures. Sectio 3 makes the coectio with radom samplig ad mixture models. The result of Deuit ad Müller (2002) is recalled ad applied to derive stochastic iequalities amog differet samplig strategies. Sectio 4 applies these results to the derivatio of model poits. Several iequalities are derived to illustrate the impact of differet aggregatio procedures o actuarial idices i accordace with the stop-loss order (such as stop-loss premiums, Tail-VaRs, spectral risk measures with cocave distortios, etc.). 1

4 2 Stochastic domiace, risk measures ad the problem of iterest 2.1 Stochastic domiace rules Before settig up the scee, we recall the defiitio of the stochastic domiace ad of the covex order. We refer the iterested reader, e.g., to Müller ad Stoya (2002), Deuit et al. (2005) or Shaked ad Shathikumar (2007) for more details. Give two radom variables X 1 ad X 2 with respective distributio fuctios F X1 ad F X2, X 1 precedes X 2 i the usual stochastic order, deoted as X 1 ST X 2, if or equivaletly if F X1 (x) F X2 (x) for all x, F X1 (x) F X2 (x) for all x, where F X1 = 1 F X1 ad F X2 = 1 F X2 are the excess, or survival fuctios correspodig to F X1 ad F X2, respectively. The latter is also equivalet to the iequality E[h(X 1 )] E[h(X 2 )] for ay o-decreasig fuctio h such that the expectatios exist. The usual stochastic order compares the sizes of the risks ad traslates i mathematical terms the cocept of beig smaller tha. The covex order focuses o the variabilities ad eables the actuary to compare two risks with idetical meas. For two radom variables X 1 ad X 2 such that E[X 1 ] = E[X 2 ], X 1 precedes X 2 i the covex order, deoted as X 1 CX X 2, whe x F X1 (u) du The iequality i (2.1) ca be equivaletly writte as x F X2 (u) du for all x. (2.1) E[(X 1 x) + ] E[(X 2 x) + ] for all x. (2.2) From (2.2) it follows that X 1 CX X 2 if ad oly if E[h(X 1 )] E[h(X 2 )] for all covex fuctios h, provided the expectatios exist. 2.2 Correspodig risk measures The stochastic order relatios ST ad CX ca be defied by meas of risk measures. Recall that the Value-at-Risk (VaR) of X 1 at probability level p is just the pth quatile of X 1, that is, VaR[X 1 ; p] = F 1 X 1 (p) = if{x R F X1 (x) p}. The, it is easily deduced that X 1 ST X 2 VaR[X 1 ; p] VaR[X 2 ; p] for all probability levels p. So, ST -iequalities ca easily be iterpreted as iequalities betwee VaRs, or more geerally betwee weighted averages of VaRs (the so-called spectral risk measures, the weights beig defied from distortio fuctios). 2

5 Besides VaRs, Tail-VaRs also play a importat role i risk maagemet, measurig the risk i the right tail. Specifically, the Tail-VaR of X 1 at probability level p is a average of the VaRs from that level o, i.e. TVaR[X 1 ; p] = 1 1 p 1 p F 1 X 1 (π)dπ. The, it ca be show that give two risks with equal meas, X 1 CX X 2 TVaR[X 1 ; p] TVaR[X 2 ; p] for all probability levels p. Thus, CX -iequalities ca be iterpreted as iequalities betwee TVaRs, or more geerally betwee spectral risk measures with appropriate distortio fuctios. 2.3 Lik to the problem of iterest Cosider idepedet risks X 1,..., X causig a aggregate loss amout X 1 +X X. Let F i be the distributio fuctio of X i ad let us assume that the X i s are raked i icreasig magitude, i.e. the stochastic iequalities X 1 ST X 2 ST... ST X hold true. Our aim is to build two sets of idepedet ad idetically distributed radom variables, heceforth deoted as X 1 +,..., X + ad X1,..., X, such that X 1 + X X CX X 1 + X X CX X X X +. (2.3) 3 Methodological results The derivatio of the upper boud i (2.3) turs out to be related to repeated samplig schemes from a give populatio. Mixtures of distributios are ivolved ad this is why we cosider i this sectio coditioally idepedet radom variables, give a mixig radom vector. Specifically, cosider the radom vector (Z 1,..., Z ) with coditioal distributio depedig o a mixig vector parameter (Θ 1,..., Θ ) as follows: P1 Compoet Z i depeds oly o Θ i, i.e. the idetity Pr[Z i t Θ 1 = θ 1,..., Θ = θ ] = Pr[Z i t Θ i = θ i ] = F i (t θ i ) holds for every i {1,..., }, where F i ( θ i ) is the coditioal distributio fuctio of the ith compoet, give Θ i = θ i. P2 The compoets Z 1,..., Z are coditioally idepedet, i.e. the idetity Pr[Z 1 t 1,..., Z t ] =... F i (t i θ i )dθ 1... dθ θ 1 holds for every t 1,..., t. 3 θ

6 Ucoditioally, however, there may be depedece amog the radom variables Z 1,..., Z iduced by the depedece structure of (Θ 1,..., Θ ). Deuit ad Müller (2002) ivestigated how the distributio of (Θ 1,..., Θ ) affects the distributio of (Z 1,..., Z ), especially how the depedece structure of (Z 1,..., Z ) depeds o the oe of (Θ 1,..., Θ ). We recall their result i the ext property. To this ed, we eed the supermodular order. Let S = (S 1,..., S m ) ad T = (T 1,..., T m ) be two radom vectors where, for each i, S i ad T i have the same margial distributios. The, S is less tha T uder supermodular order, deoted S SM T, if E [φ(s)] E [φ(t )] for all supermodular fuctios φ, give that the expectatios exist. Recall that a fuctio φ : R m R is supermodular if φ(x 1,..., x i + ε,..., x j + δ,..., x m ) φ(x 1,..., x i + ε,..., x j,..., x m ) φ(x 1,..., x i,..., x j + δ,..., x m ) φ(x 1,..., x i,..., x j,..., x m ) holds for all x = (x 1,..., x m ) R m, 1 i j m ad all ε, δ > 0. See Marshall ad Olki (1979) for examples of supermodular fuctios. The supermodular order is used to compare radom vectors S ad T with differet levels of depedece. Property 3.1 (Theorem 4.1 i Deuit ad Müller, 2002). Cosider two radom vectors (Z 1,..., Z ) ad ( Z 1,..., Z ) satisfyig P1-P2 for two mixig vectors (Θ 1,..., Θ ) ad ( Θ 1,..., Θ ), respectively. Assume that Z i (resp. Zi ) is stochastically icreasig i Θ i (resp. Θ i ) for all i = 1, 2,...,, that is, the fuctio θ F i (t θ) = Pr[Z i t Θ i = θ] = Pr[ Z i t Θ i = θ] is decreasig for all t ad every i = 1, 2,...,. The, the followig implicatio holds: (Θ 1,..., Θ ) SM ( Θ 1,..., Θ ) (Z 1,..., Z ) SM ( Z 1,..., Z ). This property allows us to derive the ext geeral result, where we radomly sample from the compoets of a set of radom vectors. Propositio 3.2. Cosider positive itegers m ad such that m. Cosider m idepedet -dimesioal radom vectors (Y 11,..., Y 1 ), (Y 21,..., Y 2 ),..., (Y m1,..., Y m ) such that Y ij ST Y ij+1 holds for every i {1,..., m} ad j {1,..., 1}. Let N = (N 1,..., N m ) ad M = (M 1,..., M m ) be two radom vectors valued i {1,..., } m such that N SM M. The, (Y 1N1,..., Y mnm ) SM (Y 1M1,..., Y mmm ). (3.1) Proof. The radom vectors appearig i (3.1) obey mixture models, with mixig parameters N ad M. Clearly, their ith compoet depeds oly o N i ad M i, respectively. Give N, the compoets of (Y 1N1,..., Y mnm ) are idepedet, ad give M, the compoets of (Y 1M1,..., Y mmm ) are also idepedet. Moreover, the coditio Y ij ST Y ij+1 esures that the compoets of every radom vector are stochastically icreasig i their idex. The result the follows from Property

7 Corollary 3.3. Assume that for each i {1,..., m} the radom variables Y ij are partial sums j Y ij = k=1 of idepedet ad idetically distributed o-egative radom variables Z i1, Z i2,... so that Y ij Y ij+1 almost surely holds ad Y ij ST Y ij+1 is valid. The, Propositio 3.2 tells us that if N SM M, we have ( N1 ) ( N m M1 ) M m Z ik,..., Z ik SM Z ik,..., Z ik, k=1 k=1 which is the stochastic iequality stated uder item (iv) of Propositio 2 i Deuit, Geest ad Marceau (2002). 4 Coservative model poits 4.1 Covex upper boud o heterogeeous sums We are ow i positio to establish the upper boud described i (2.3). Propositio 4.1. Cosider the problem stated i Sectio 2.3. Defie idepedet ad idetically distributed radom variables X + i with commo distributio fuctio The, F + (x) = 1 Z ik k=1 F i (x), x R. X X CX X X +. k=1 Proof. Let (X (i) 1,..., X (i) ), i = 1,...,, be idepedet copies of (X 1,..., X ). that N correspods to a radom sample without replacemet from {1,..., }, i.e. Pr[N 1 = i 1,..., N m = i m ] = 1 for i ( 1)...( m+1) 1,..., i m {1,..., } 0 otherwise. Assume Also, let M be a radom sample with replacemet from {1,..., }, i.e. 1 for i m 1,..., i m {1,..., } Pr[M 1 = i 1,..., M m = i m ] = 0 otherwise. We kow from Karli (1974) that N SM M (see also Example 9.A.27 i Shaked ad Shathikumar (2007)). With m =, Propositio 3.2 the gives ( ) ( ) X (1) N 1,..., X () N SM X (1) M 1,..., X () M 5

8 which implies X (1) N X () N CX X (1) M X () M. It is easily see that the radom variables X (j) M j are idepedet with commo distributio ( ) fuctio F +. Now, the radom vector X (1) N 1,..., X () N is distributed as a radom permutatio of (X 1,..., X ) so that Pr[X X x] = Pr[X (1) N 1 holds for all x, which eds the proof X () N x] I coclusio, for a group of heterogeeous cotracts with respective losses X 1,..., X, replacig the losses by homogeeous X + 1,..., X + with commo distributio fuctio F + obtaied by averagig F 1,..., F turs out to be a coservative strategy as this iflates the Tail-VaR associated to the aggregate loss. Remark 4.2. Propositio 4.1 ca be foud i Frostig (2001). We offer here a alterative derivatio of this result. Notice that Propositio 4.1 may also have iterestig cosequeces i data aalysis, as show ext. Cosider idepedet observatios X i with distributio fuctio F i, i = 1,...,. Assume that the risk aalyst decides to treat X 1,..., X as a homogeous sample, i.e. to cosider that X 1,..., X are idepedet with commo distributio fuctio F +. Cosiderig the sample mea X = 1 estimatig the pure premium, this approach is coservative as it yields a upper boud i the covex sese o X. The variace of X is thus iflated as well as the width of large-sample cofidece itervals for the pure premium, for istace. 4.2 Life isurace applicatios Propositio 4.1 fids direct applicatios i life isurace, as show i the followig. To this ed, we cosider a portfolio made of life isurace cotracts, ad we deote by T i the remaiig lifetime of policyholder i, i = 1,...,. Also, we deote by b i g(t i ) the radom loss associated to policy i. Before goig o, let us give some examples of specific life isurace cotracts. Example 4.3. Let g(t i ) = T i k=1 v(k), where v(t) is the preset value of a uit paymet made at time t ad the fuctio T i deotes the greatest iteger smaller tha or equal to T i. Heceforth, we assume that the preset value fuctio v is decreasig, as it is usually the case. The radom loss b i g(t i ) the correspods to the preset value of a life auity cotract payig b i at the ed of each year, as log as policyholder i survives. Here, g is o-decreasig. Example 4.4. Cosider a whole life isurace cotract with paymet b i at time of death of policyholder i. The preset value of such a cotract is give by b i v(t i ) so that g(t i ) = v(t i ). Here, g is o-icreasig. 6 X i

9 Example 4.5. Let g(t i ) = v(t i )I[T i < m] + v(m)i[t i m], where I[A] is the idicator variable of the evet A, equal to 1 if A is realized ad to 0 otherwise. The radom loss b i g(t i ) for policy i the correspods to the preset value of a edowmet isurace, i.e. the combiatio of a temporary life isurace with duratio m ad beefit b i ad a pure edowmet with beefit b i payable upo survival i m years. Here, g is o-icreasig Homogeeous lifetimes Let us first cosider a group of policies subject to the same mortality but differig i the amouts of beefits. Corollary 4.6. Assume that T 1,..., T are idepedet ad idetically distributed. The, we have b g(t i ) CX b i g(t i ) CX B i g(t i ) where b = 1 b i ad where the idepedet radom variables B 1,..., B are idepedet of T 1,..., T with commo distributio b 1 with probability 1 b 2 with probability 1 B i =. b with probability 1. Proof. The lower boud is a direct cosequece of the results derived i Deuit ad Vermadele (1998); see also the proof of Propositio 5.1 below. For the upper boud, defie X i = b i g(t i ), i = 1,...,, so that F + (x) = 1 Pr[b i g(t i ) x] = Pr[B 1 g(t 1 ) x]. Propositio 4.1 the shows that the aouced upper boud is ideed valid. Notice that o particular assumptio is made about the fuctio g defiig the radom loss appearig i Corollary 4.6 (except measurability). Therefore, whe the lifetimes are homogeeous but the amouts of beefits vary betwee cotracts, it is coservative to replace the determiistic beefits b i with a stochastic oe B i radomly draw from {b 1,..., b }. O the cotrary, replacig the beefits b i with their average value b decreases the Tail-VaR of the aggregate loss Heterogeeous lifetimes Let us ow cosider that the policyholders lifetimes are stochastically ordered but the beefits are the same for all cotracts. 7

10 Corollary 4.7. Assume that T 1,..., T are idepedet ad such that T 1 ST... ST T holds true. So, if g is a mootoic fuctio, it comes b g(t i ) CX b g(t + i ) where T + 1,..., T + are idepedet with commo survival fuctio Pr[T 1 + > ξ] = 1 Pr[T i > ξ]. Corollary 4.7 directly follows from Propositio 4.1. I practice, lifetimes T 1,..., T are stochastically ordered whe they all obey the same life table ad correspod to idividuals aged x 1,..., x with x 1 >... > x, for istace. Homogeizig life tables thus appears to be a safe strategy. It is worth oticig that a mootoicity costrait has ow to be imposed o the fuctio g. This is the case i Examples 4.3, 4.4 ad 4.5. Whe the cotracts specify periodic premium paymets, the radom loss may or may ot be expressed as a mootoic fuctio of T i, i = 1,...,, as illustrated i the two followig examples. Example 4.8. Cosider a whole life isurace cotract for policyholder i with a beefit of c due at the time of death. I exchage, policyholder i pays a premium amout of p at the begiig of each year as log as he survives. The radom loss of such a cotract is the give by c v(t i ) p T i k=0 v(k), which is obviously a decreasig fuctio of T i ad thus Corollary 4.7 applies. Example 4.9. Cosider a life auity cotract payig c at the begiig of each year from year m + 1 ad for which policyholder i pays a premium amout of p at the begiig of the ext m years. The, the correspodig radom loss is c I[T i m] T i k=m v(k) p mi( T i,m 1) k=0 v(k), which is clearly ot a mootoic fuctio of T i. Overall, i case the cotracts specify periodic premium paymets, the radom loss bg(t i ) ca the be writte as c f(t i ) p h(t i ), i = 1,...,. While h is icreasig with T i, the fuctio f ca be either decreasig (as i Example 4.8) or icreasig (as i Example 4.9), i which case the radom loss is ot a mootoic fuctio of T i. I such situatio, we ca evertheless artificially decompose the cotracts i fictitious subcotracts to derive a covex upper boud, as show ext. This idea ca be foud i Klig ad Wolthuis (1992). Corollary Assume that T 1,..., T are idepedet ad such that T 1 ST... ST T, ad let T i,1 ad T i,2 be two idepedet radom variables distributed as T i. The, for icreasig fuctios f ad h, we have ( (c f(t i ) p h(t i )) CX c f(t + i,1 ) p h(t + i,2 )) where T + i,1 ad T + i,2 are idepedet radom variables distributed as T + i. 8

11 Proof. Sice (c f(t i,1 ), p h(t i,2 )) SM (c f(t i ), p h(t i )), Theorem 9.A.18 i Shaked ad Shathikumar (2007) implies that c f(t i ) p h(t i ) CX c f(t i,1 ) p h(t i,2 ). Therefore, the radom variable (c f(t i) p h(t i )) verifies the stochastic iequality (c f(t i ) p h(t i )) CX (c f(t i,1 ) p h(t i,2 )). Now, from Corollary 4.7, we kow that c f(t i,1 ) CX c f(t + i ) ad p h(t i,2 ) CX p h(t + i ). Hece, sice p h(t i,2) CX p h(t + i ) p h(t i,2) CX p h(t + i ) (see, e.g., Theorem 3.A.12 i Shaked ad Shathikumar (2007)), it comes c f(t i,1 ) which completes the proof. p h(t i,2 ) CX c f(t + i,1 ) p h(t + i,2 ), Thus, whe both fuctios f ad h are icreasig, a safe strategy for the isurer is to homogeize life tables o the oe had, ad to split each cotract i ito two idepedet subcotracts (with radom losses c f(t + + i,1 ) ad p h(t i,2 )) o the other had. 5 Model poits, lower boud Let us ow derive cadidates for X1,..., X described i (2.3). servig to defie the covex lower boud Propositio 5.1. Let U 1,..., U be idepedet radom variables uiformly distributed over the uit iterval. Defie idepedet ad idetically distributed radom variables X i as X i = 1 j=1 F 1 j (U i ), i = 1, 2,...,. The, X X CX X X. Proof. Let Z 1, Z 2,..., Z be idepedet ad idetically distributed radom variables ad ψ 1, ψ 2,..., ψ be measurable real fuctios. Defie the fuctio ψ as ψ(z) = 1 ψ j (z). j=1 9

12 The, we kow from Deuit ad Vermadele (1998) that ψ(z i ) CX ψ i (Z i ). Let us ow apply this result with Z i = U i ad ψ i = F 1 i. This gives ψ(z) = 1 F 1 j (z) j=1 ad ψ(u i ) = which eds the proof. ( 1 j=1 F 1 j (U i ) ) CX F 1 i (U i ), This lower boud costitutes a improvemet compared to the oe obtaied i Frostig (2001). Ideed, let X i,j, i, j = 1, 2,..., be idepedet radom variables with distributio fuctio F i. Frostig (2001) tells us that X X CX X X, where X j = X i,j. So, the ew lower boud derived i this paper cosists i replacig the sums of idepedet radom variables X i,j by the comootoic sums F 1 i (U j ). Now, from iequality (9.A.20) i Shaked ad Shathikumar (2007), we kow that ad hece 1 X i,j CX 1 F 1 i (U j ) X X CX X X. So, averagig F 1,..., F to produce F + provides the actuary with a covex upper boud o the aggregate loss whereas the covex lower boud is obtaied by averagig VaRs F1 1,..., F 1. It is iterestig to otice that Propositio 5.1 leads to the lower boud defied i Corollary 4.6. Ideed, for idepedet ad idetically distributed lifetimes T 1,..., T with commo distributio fuctio F, it is easy to show that Propositio 5.1 yields X i = 1 b j g ( F 1 (U i ) ) = b g ( F 1 (U i ) ). j=1 I case the lifetimes T 1,..., T are heterogeeous ad the amouts of beefits idetical for all cotracts, Propositio 5.1 tells us that { 1 ( b g F 1 T j (U i )) } CX b g(t i ) j=1 10

13 if we assume that g is mootoic (as i Corollary 4.7). Here we deote by F Tj the distributio fuctio of T j, j = 1,...,. I words, the lower boud is obtaied by replacig each policy by a portfolio of policies differig from the origial oe by the fact that the remaiig lifetimes are comootoic o the oe had, ad that the amouts of beefits are times smaller o the other had. Ackowledgemets This project started from iterestig discussios with Patrick Heio ad Pierre Miehe at Addactis Belux. We would like to thak Addactis for stimulatig research i that area. Also, we dedicate this work to the memory of our esteemed colleague Professor Jea-Marie Reihard who passed away last November. Jea-Marie taught actuarial methods based o stochastic orders for decades at the Uiversité Libre de Bruxelles, iitiatig the first author to that fasciatig topic. We are also grateful to a aoymous Referee for several useful suggestios which helped us to improve a previous versio of our mauscript. Refereces Deuit, M., Dhaee, J., Goovaerts, M.J., Kaas, R., Actuarial Theory for Depedet Risks: Measures, Orders ad Models. Wiley, New York. Deuit, M., Geest, C., Marceau, E., Criteria for the stochastic orderig of radom sums, with actuarial applicatios. Scadiavia Actuarial Joural, Deuit, M., Müller, A., Smooth geerators of itegral stochastic orders. Aals of Applied Probability 12, Deuit, M., Vermadele, C., Optimal reisurace ad stop-loss order. Isurace: Mathematics ad Ecoomics 22, EIOPA QIS5 Techical Specificatio. Europea commissio. Frostig, E., A compariso betwee homogeeous ad heterogeeous portfolios. Isurace: Mathematics ad Ecoomics 29, Karli, S., Iequalities for symmetric samplig plas. I. Aals of Statistics 2, Klig, B., Wolthuis, H., Orderig of risks i life isurace. Isurace: Mathematics ad Ecoomics 11, Marshall, A., Olki, I., Iequalities: Theory of Majorizatio ad Its Applicatios. Academic Press, New York. Müller, A., Stoya, D., Compariso Methods for Stochastic Models ad Risks. Wiley, Chichester. Shaked, M., Shathikumar, J.G., Stochastic Orders. Spriger, New York. 11