Moment-Based Approximation with Mixed Erlang Distributions

Transcription

1 Moment-Based Approximation with Mixed Erlang Distributions by Hélène Cossette, David Landriault, Etienne Marceau, Khouzeima Moutanabbir Abstract Moment-based approximations have been extensively analyzed over the years (see, e.g., Osogami and Harchol-Balter (2006) and references therein). A number of speci c phase-type (and non phasetype) distributions have been considered to tackle the moment-matching problem (see, for instance, Johnson and Taa e (989)). Motivated by the development of more exible moment-based approximation methods, we develop and examine the use of nite mixture of Erlangs with a common rate parameter for the moment-matching problem. This is primarily motivated by Tijms (994) who shows that this class of distributions can approximate any continuous positive distribution to an arbitrary level of accuracy, as well as the tractability of this class of distributions for various problems of interest in quantitative risk management. We consider separately situations where the rate parameter is either known or unknown. For the former case, a direct connection with a discrete moment-matching problem is established. A parallel to the s-convex stochastic order (e.g., Denuit et al. (998)) is also drawn. Numerical examples are considered throughout. Keywords : Risk Theory; Mixed Erlang distributions; Moment-matching; Distribution Fitting; Phase-type approximation. Introduction Mixed Erlang distributions are known to yield analytic solutions to many risk management problems of interest. This is primarily due to the tractable features of this distributional class. Among others, the class of mixed Erlang distributions is closed under various operations such as convolutions and Esscher transformations (e.g., Willmot and Woo (2007) and Willmot and Lin (20)). As such, risk aggregation and ruin problems can more easily be tackled under mixed Erlang assumptions (e.g., Cheung and Woo (204), Cossette et al. (202), and Landriault and Willmot (2009)). Also, Tijms (994) showed that the class of mixed Erlang distributions is dense in the set of all continuous and positive distributions. Therefore, we consider a moment-based approximation method which capitalizes on the aforementioned properties of the mixed Erlang distribution. More precisely, we propose to approximate a distribution with known moments by a moment-matching mixed Erlang distribution. Moment-based approximations have been extensively developed in various research areas including performance evaluation, queueing theory, and risk theory, to name a few. Osogami and Harchol-Balter (2006) identi es the following four criteria to evaluate moment-matching algorithms: () the number of moments matched; (2) the computational e ciency of the algorithm; (3)

2 the generality of the solution; (4) the minimality of the number of parameters (phases). It also seems desirable for the approximation to be in itself a distribution. This is not mentioned in Osogami and Harchol-Balter (2006) for the obvious reason that they consider phase-type distributions as their momentbased approximation class. There exists an extensive literature on the approximation of distributions by a speci c subset of phase-type distributions using moment-based techniques. For instance, Whitt (982) proposed a mixture of two exponential distributions or a generalized Erlang distribution as a moment-based approximation when either the coe cient of variation (CV) is greater than or less than, respectively. Also, both Altiok (985) and Vanden Bosch et al. (2000) proposed an alternative to the moment-based approximation of Whitt (982) when CV> using a Coxian distribution. Alternatively, Johnson and Taa e (989) considered a mixture of Erlangs with a common shape (order) parameter as their moment-based approximation. Most predominantly, there exists a substantial body of literature on the three-moment approximation within the phase-type class of distributions (e.g., Telek and Heindl (2002), Bobbio et al. (2005), and references therein). Matching the rst three moments is often viewed as e ective to provide a reasonable approximation to the underlying system (e.g., Osogami and Harchol-Balter (2006) and references therein). However, as illustrated in this paper and many others, three moments does not always su ce, triggering the development of more exible moment-based approximations. Among others, we mention the work of Johnson and Taa e (989) on mixed Erlang distributions of common order. Also, Dufresne (2007) proposes two approximation techniques based on Jacobi polynomial expansions and the logbeta distribution to t combinations of exponential distributions. This paper is obviously complementary to the aforementioned ones by considering the family of nite mixture of Erlangs with common rate parameter to approximate a distribution on R +, as theoretically justi ed in the continuous case by Tijms (994, Theorem 3.9.). The reader is also referred to Lee and Lin (200) where tting of the same class of distributions is considered using the EM algorithm (which relies on the knowledge of the approximated distribution rather than only its moments). It is worth pointing out that other non-phase type approximation methods have been widely used in actuarial science. A good survey paper on this topic is Chaubey et al. (998). One of these approximation classes are re nements to the Normal approximation such as the Normal Power and the Cornish Fisher approximations (e.g., Ramsay (99), Daykin et al. (994), and Lee and Lin (992)). These approximations are based on the rst few moments. However, the resulting approximation is often not a proper distribution. Other moment-based distributional approximations are the translated gamma distribution (e.g., Seal (977)), translated inverse gaussian distribution (e.g., Chaubey et al. (998)) and the generalized Pareto distribution (e.g., Venter (983)). It should be noted that all these approximation methods are designed to t a speci c number of moments, and thus lack the exibility to match an arbitrary number of moments. The rest of the paper is constructed as follows. In Section 2, a brief review on admissible moments, mixed Erlang distributions and the approximation method of Johnson and Taa e (989) is provided. Section 3 is devoted to our class of nite mixture of Erlangs with common rate parameter. Theoretical and practical considerations related to the approximation method are drawn. Various examples are considered to examine the quality of the resulting approximation. In Section 4, we consider applications of our moment-based approximations of Section 3 when the underlying distribution is of mixed Erlang form with known rate parameter. A parallel is drawn with a discrete moment-matching problem and certain stochastic orderings, notably the s-convex stochastic order (e.g. Denuit et al. (998)). An application of Cossette et al. (2002) will later be examined in more details. 2

3 2 Background 2. Admissible moments Karlin and Studden (966) provide the necessary and su cient conditions for a set of (raw) moments m = ( ; :::; m ) to be from a probability distribution de ned on R +. To state this result, de ne the matrices P k and Q k (k ) as 0 0 k 2 k+ 2 k+ P k = C. A ; Q 2 3 k+2 k = C. A. k k+ 2k k+ k+2 2k+ As stated in Courtois and Denuit (2007), there exists a non-negative random variable (rv) with distribution function (df) F and rst m moments m if and only if the following two conditions are satis ed: det P k > 0, for k = ; :::; b(m )=2c ; det Q k > 0, for k = ; :::; bm=2c ; where bxc holds for the integer part of x. In what follows, we silently assume the moment set m is from a probability distribution on R Mixed Erlang Distribution We now review some known properties of mixed Erlang distributions with common rate parameter. A more elaborate review of this class of distributions can be found in Willmot and Woo (2007), Lee and Lin (200), and Willmot and Lin (20). Let W be a mixed Erlang rv with common rate parameter > 0 and df F W (x) = X k2a l k H(x; k; ); () where A l = f; 2; :::; lg, and f k g l k= is the probability mass function (pmf) of a discrete rv K with support A l for a given l 2 f; 2; :::g [ fg. The Erlang df H is de ned as Xk H(x; k; ) H(x; k; ) = e x i=0 (x) i, x 0. (2) i! where the parameters k and of the Erlang df are known as the shape and rate parameters, respectively. An alternative and useful representation of the mixed Erlang rv W is as a compound rv, i.e. W = P K k= C k where fc k g k are iid exponential rv s with mean =, independent of K. Remark As in e.g., Willmot and Woo (2007), we consider the class of mixed Erlang dfs () rather than the more general class of combinations of Erlangs where some k s are possibly negative. For the latter 3

4 class, additional constraints on f k g l k= exist to ensure that the right-hand side of () is a non-decreasing function in x. This presents additional challenges in the subsequent moment-matching application, challenges which do not arise in the mixed Erlang case. It is well known that the j-th moment of W is given by E W j = j P n k= Qj k i=0 o. (k + i) Of particular importance in actuarial science and quantitative risk management (see, e.g., McNeil et al. (2005) and references therein) are the VaR and TVaR risk measures. For the mixed Erlang rv W, there is in general no closed form expression for VaR (W ) = inf fx 2 R : F W (x) g where 0 <, but its value can be obtained using a routine numerical procedure. As for its TVaR, Lee and Lin (200) showed that TVaR (W ) Z VaR u (W ) du = X k= k k H (VaR (W ) ; k + ; ) : (3) Another quantity of interest is the stop-loss premium de ned as W (b) = E (W b) + with (x)+ = max fx; 0g. For the mixed Erlang df (), we have W (b) = X k= k k H (b; k + ; ) bh (b; k; ) ; b 0. (see also Willmot and Woo (2007, Eq. 3.6) for the higher-order stop-loss moments). Tijms (994) showed that this class of distributions can approximate any continuous positive distribution with an arbitrary level of accuracy. For completeness, the theoretical foundation of this result is given next. Theorem 2 (Tijms (994, Theorem 3.9.)) Let F be the df of a positive rv. For any given h > 0, de ne F h (x) = X (F (kh) F ((k ) h)) H k= Then, lim h!0 F h (x) = F (x) for any continuity point x of F. x; k;. (4) h Note that F h in (4) is a mixed Erlang df of the form () with k = (F (kh) F ((k ) h)) (k = ; 2; :::) and rate parameter = =h. Several approximation methods motivated by Tijms theorem were proposed over the years (see Section for more details). In general, these moment-based approximations propose to work with a speci c subclass of all nite and in nite mixed Erlang distributions. Among them, we recall the method of Johnson and Taa e (989) who will be used later for comparative purposes. 2.3 Method of Johnson and Taa e (989) Johnson and Taa e (989) investigated the use of mixtures of Erlang distributions with common shape parameter for moment-matching purposes. More precisely, mixtures of n (or fewer) Erlangs with common shape parameters are used to match the rst (2n ) moments (whenever the set of moments is within the feasible set). For the three-moment matching problem, Johnson and Taa e (989) generalized the 4

5 approximation of Whitt (982) and Altiok (985) by enlarging the set of feasible moments 3 when CV >. Their method is also valid for some combinations of 3 when CV <. Their three-moment approximation is a mixture of two Erlangs with common shape parameter r (see Theorem 3 of Johnson and Taa e (989)), i.e. F (x) = ph (x; r; ) + ( p) H (x; r; 2 ), (5) where p = r 2 = 2, and f=i g 2 i= are the solutions of As2 + Bs + C = 0 with r + A = r (r + 2) 2 B = r + r 3 r r C = 3 r + r r (r + 2) + r + r r 2 + (n + 2) 2 r r The choice of the shape parameter r is discussed in Johnson and Taa e (989, Proposition 4).! 3 Moment-based approximation with mixed Erlang distribution In this section, we propose to use a di erent subclass of mixed Erlang distributions to examine momentbased approximation techniques. 3. Description of the approach For a given l 2 N +, let ME( m ; A l ) be the set of all nite mixture of Erlangs with df () and rst m moments m. From Section 2.2, this consists in the identi cation of all solutions to the following problem: lx Q j i=0 (k + i) k j = j ; j = ; :::; m, (6) k= under the constraints that > 0 and f k g l k= is a probability measure on A l. Remark 3 For a rv W with df () and rst m moments m, we indi erently write W 2 ME( m ; A l ) or F W 2 ME( m ; A l ). This will also apply to the other distributional classes. Also, let ME res ( m ; A l ) be the (restricted) subset of ME( m ; A l ) with at most m non-zero mixing probabilities f k g l k=. Given that MEres ( m ; A l ) has a nite number of solutions, we propose to use ME res ( m ; A l ) as our approximation class. It is clear that ME res ( m ; A l ) ME res ( m ; A l 0) for l l 0. Note that for a continuous positive distribution with moments m, we know from Theorem 2 that there exists a l large enough such that ME( m ; A l ) is not empty. Even though no formal conclusion can 5

6 be reached for the restricted class ME res ( m ; A l ), all our numerical studies have shown that this set has a large number of distributions (see, for instance, the examples of subsections 3.2. and 3.2.2) for a given m when l is chosen large enough. Distributions in the ME res ( m ; A l ) class are identi ed as follows: for a given set fi k g m k= A l with i < i 2 < ::: < i m l, (6) can be rewritten in matrix form as G m m = M ; where m = ( i ; i2 ; :::; im ) T, M = ( ; 2 2 ; :::; m m ) T, and 0 i i 2 i m i (i + ) i 2 (i 2 + ) i m (i m + ) G m =. B mq mq mq A (i + i) (i 2 + i) (i m + i) i=0 i=0 It follows that m = Gm M under the constraint that T me =, where e is a column vector of s. Note that T me = Gm T M e is a polynomial of degree (at most) m in. Thus, we only consider the real and positive solutions (in ) of Gm T M e =, and complete their mixed Erlang representation with the identi cation of the mixing weights m = Gm M. The procedure is systematically repeated for all l m possible sets of m distinct elements in Al. i=0 l Remark 4 Given that the above procedure is repeated m -times, the computational e ciency of the proposed methodology is mostly driven by this number, and hence the parameters m and l should be chosen accordingly. For a given number of moments m, we observe that: (a) larger values of l result in a more time-consuming numerical procedure; (b) however, l should be chosen large enough for the approximation class ME res ( m ; A l ) to have a reasonable number of members (to legitimally produce a "good" approximation). From our numerical studies, we observe that the selection of l (for a given m) can be problem-speci c, and thus this tradeo in the choice of l should be handled with care. However, as a rule of thumb when m is relatively small (i.e. m 6) which is traditionally in moment-matching exercises, a value of l between 50 and 00 leads to reasonable mixed Erlang approximations. We refer the reader to the numerical illustrations and subsequent remarks in Section 3.2 for a more detailed discussion on this topic. Among all ME res ( m ; A l ) distributions, we propose to use as a criterion of the quality of these approximations the Kolmogorov-Smirnov (KS) distance de ned for two rv s S and W (with respective dfs F S and F W ) as d KS (S; W ) = sup x0 jf S (x) F W (x)j. The KS distance is commonly used in the context of continuous distributions (e.g., Denuit et al. (2005)). Therefore, the chosen mixed Erlang approximation within ME res ( m ; A l ) is the one minimizing the KS distance with the true df F S. We denote by F Wm;l this approximation, i.e. d KS (S; W m;l ) = inf sup jf S (x) F W (x)j, F W 2ME res ( m ;A l ) where W m;l is a rv with df F Wm;l. This requires the calculation of the KS distance for each mixed Erlang distribution in ME res ( m ; A l ) to identify its minimizer W m;l. In general, an explicit expression for this x0 6

7 KS distance does not exist, and hence we propose to numerically nd this value by evaluating the distance between the two dfs over all multiples (up to a given high value) of a small discretization span. Note that other distances such as the stop-loss distance (e.g., Gerber (979)) could have been used to select our approximation distribution. Alternatively, one could have relied on another criterion to identify this approximation distribution (for instance, select the distribution in ME res ( m ; A l ) with the closest subsequent moment to the true distribution). 3.2 Numerical examples We consider a few simple examples to illustrate the quality of the approximation. For comparative purposes, other approximation methods will also be discussed. Some concluding remarks on the mixed Erlang approximation method are later made based on the numerical experiment conducted next Lognormal distribution: Dufresne (2007, Example 5.4) Let S = exp (Z) where Z is a normal rv with mean 0 and variance The rst 5 moments of S are: 5 = (:33; :6487; 3:0802; 7:389; 22:7599). We consider the class of mixed Erlang distributions ME res ( m ; A 70 ) (m = 3; 4; 5) which have a total of 398, and distributions, respectively. The resulting mixed Erlang approximations F Wm;70 (m = 3; 4; 5) are F W3;70 (x) = 0:8209H(x; 6; 6:329) + 0:727H(x; 2; 6:329) + 0:0064H(x; 26; 6:329); F W4;70 (x) = 0:6350H(x; 7; 8:3334)+0:2950H(x; 2; 8:3334)+0:0672H(x; 20; 8:3334)+0:0029H(x; 40; 8:3334); and F W5;70 (x) = 0:6273H(x; 7; 8:3608) + 0:3063H(x; 2; 8:3608) + 0:0609H(x; 20; 8:3608) + 0:0055H(x; 34; 8:3608) + 0:000H(x; 69; 8:3608); for x 0 with respective KS distances of d KS (S; W 3;70 ) = 0:0040, d KS (S; W 4;70 ) = 0:008 and d KS (S; W 5;70 ) = 0:00. Note that the quality of the mixed Erlang approximation (as measured by the KS distance) increases with the number of moments matched. For comparative purposes, the three-moment approximation (5) of Johnson and Taa e (989) is given by F WJT (x) = 0:0087H (x; 4; :2804) + 0:993H (x; 4; 3:5855), In Figure, we compare the density functions of W m;70 (m = 3; 4; 5) ; W JT and S. All three mixed Erlang approximations provide an overall good t to the exact distribution. To further examine the tail t, speci c values of VaR and TVaR for the exact and approximated distributions are provided in Tables and 2, respectively. 7

8 Figure : Density function: Lognormal vs Approximations VaR (W JT ) VaR (W 3;70 ) VaR (W 4;70 ) VaR (W 5;70 ) VaR (S) Table. Values of VaR for W JT, W m;70 (m = 3; 4; 5) and S TVaR (W JT ) TVaR (W 3;70 ) TVaR (W 4;70 ) TVaR (W 5;70 ) TVaR (S) Table 2. Values of TVaR for W JT, W m;70 (m = 3; 4; 5) and S We observe that the VaR and TVaR values of the mixed Erlang approximations compare very well to their lognormal counterparts, especially for the 5-moment approximation. This is particularly true given that the lognormal distribution is known to have a heavier tail than the mixed Erlang distribution. Note that the improvement is indeed not monotone with the number of moments matched, as increasing this number does not necessarily lead to a higher quality approximation in moment-matching techniques. 8

9 3.2.2 Mixture of Two Gamma Distributions: Lee and Lin (200, Section 5, Example ) Let S be a mixture of two Gamma distributions with density f S (s) = 0:2 (3:2s)2:6 e 3:2s s (2:6) + 0:8 (:2s)6:3 e :2s, s 0, s (6:3) and rst 6 moments 6 = (4:3623; 25:7308; 76:9624; 369:8272; 754:249; 0674:454). We consider the class of mixed Erlang distributions ME res ( m ; A 70 ) for m = 3; 4; 5; 6 which are composed of 6000, 83797, and distributions, respectively. The resulting mixed Erlang approximations F Wm;70 (m = 3; 4; 5; 6) are F W3;70 (x) = 0:240H(x; 2; 2:0835) + 0:525H(x; 9; 2:0835) + 0:2645H(x; 5; 2:0835); F W4;70 (x) = 0:2266H(x; 2; 2:0469)+0:4440H(x; 9; 2:0469)+0:2963H(x; 3; 2:0469)+0:0330H(x; 9; 2:0469); and F W5;70 (x) = 0:2023H(x; 3; 3:727) + 0:209H(x; 2; 3:727) + 0:3936H(x; 9; 3:727) + 0:805H(x; 28; 3:727) + 0:045H(x; 42; 3:727); F W6;70 (x) = 0:0768H(x; 2; 3:073) + 0:325H(x; 3; 3:073) + 0:2739H(x; ; 3:073) + 0:3928H(x; 7; 3:073) + 0:88H(x; 25; 3:073) + 0:0052H(x; 37; 3:073); with respective KS distances of d KS (S; W 3;70 ) = 0:048, d KS (S; W 4;70 ) = 0:0050, d KS (S; W 5;70 ) = 0:0035 and d KS (S; W 6;70 ) = 0:0024. Lee and Lin (200) used the EM algorithm to t a mixed Erlang distribution to the same distribution which resulted in the following model: F WEM (x) = 0:2282H(x; 2; :9603) + 0:5430H(x; 9; :9603) + 0:2288H(x; 4; :9603); with KS distance d KS (S; W EM ) = 0:0094. Note that the KS distance for the 3-moment approximation is greater than the KS distance obtained using the EM estimation. We recall that the EM algorithm nds maximum likelihood estimates using the approximated distribution as an input, while our method is based on only partial information on the approximated distribution (e.g., its rst m moments). We observe that the t improves and the KS distance decreases when more moments are included in the approximation. For illustrative purposes, we also provide in Tables 3 and 4 some values of VaR and TVaR for W m;70 (m = 3; 4; 5; 6), W EM and S. VaR (W EM ) VaR (W 3;70 ) VaR (W 4;70 ) VaR (W 5;70 ) VaR (W 6;70 ) VaR (S) Table 3. Values of VaR for W m;70 (m = 3; 4; 5; 6) ; W EM, and S 9

10 TVaR (W EM ) TVaR (W 3;70 ) TVaR (W 4;70 ) TVaR (W 5;70 ) TVaR (W 6;70 ) TVaR (S) Table 4. Values of TVaR for W m;70 (m = 3; 4; 5; 6) ; W EM and S Gompertz distribution The Gompertz distribution with df F S (x) = exp f B (c x ) = ln cg (x > 0) has been extensively applied in various life contingency contexts (e.g., Bowers et al. (997)). Lenart (204) provides an expression for the j-th moment, namely j = j! (ln c) j e B ln c E j B, (7) ln c where Es j (z) = R (ln x) j j! x s e zx dx is the generalized integro-exponential function (see Milgram (985)). Here, we consider an example of Melnikov and Romaniuk (2006) on the USA mortality data of the Human mortality database where the parameters B and c were estimated to B = 6: and c = :0959. Using (7), the rst 5 moments are 5 = (76:3437; 6037:202; :3; ; ). We consider the class of mixed Erlang distributions ME res ( m ; A 90 ). The resulting mixed Erlang approximations F W3;90 and F W4;90 are and F W3;90 (x) = 0:054H(x; 22; :0928) + 0:220H(x; 65; :0928) + 0:7637H(x; 90; :0928); F W4;90 (x) = 0:0347H(x; 35; :0972)+0:0834H(x; 66; :0972)+0:009H(x; 67; :0972)+0:780H(x; 90; :0972), with KS distances d KS (S; W 3;90 ) = 0:088, and d KS (S; W 4;90 ) = 0:0239. In Figure 2, we compare the t of the 3- and 4-moment approximations to the exact distribution by plotting their densities (left) and dfs (right). Overall, we observe that the t is quite reasonable. Note that the KS distance increases from the 3-moment to the 4-moment approximation. We notice that both F W3;90 and F W4;90 use the Erlang-90 df, where 90 is the largest element of A 90. As such, a mixed Erlang approximation with a smaller KS distance can likely be found in both cases by choosing a larger support A l (i.e., l > 90) Some remarks To provide insight on the quality of the moment-based mixed Erlang approximation proposed in this section, we brie y revisit the results of the above three examples. In the rst two (lognormal and mixture of two gammas), the mixed Erlang approximation is easy to implement and provides a very satisfactory t to the true distribution. As none of the resulting mixed Erlang approximations uses the 0

11 Figure 2: Density (left) and df (right): 3- and 4-moment approximations vs Gompertz distribution Erlang-70 df (as l = 70 in the rst two examples), it is unlikely that a better approximation (from the viewpoint of KS distance) can be found by increasing the value of l. Given that the best KS- t is found from a mixture of Erlang distributions with relatively small shape parameter (which corresponds to the parameter k in the Erlang df (2)), the proposed method seems particularly well suited for these two cases. As for any approximation method, limitations can also be found as evidenced by the Gompertz example. Indeed for this example, both the 3-moment and 4-moment mixed Erlang approximations use the Erlang-90 df (recall l = 90 for this example). As mentioned earlier, one can likely reduce the KS distance (if so desired) of the resulting mixed Erlang approximation by increasing the value of l (in light of the comments in Remark 4). This implies that the KS-optimal mixed Erlang approximation would likely involve Erlang distributions with large shape parameters (which have smaller variances for a given rate parameter ). In general, distributions with negative skewness and sharp density peak(s) may require the use of Erlang distributions with large shape parameters to provide a good approximation. Computational time of the proposed mixed Erlang methodology may become a non-negligible issue in these cases (especially as the number of moments matched increases). However, a slight adjustment to the proposed methodology may be considered to address this time-consuming issue. Indeed, one may replace the set A l = f; 2; :::; lg in () by a set of the form fa; 2a; :::; jag for positive integers a and j (note that the two sets coincide when a = and j = l). To illustrate this, we have re-considered the Gompertz example by replacing the set A 90 by the set f5; 0; 5; ::::; 200g. The resulting mixed Erlang approximation, denoted by the rv W m;5:200 when m moments are matched (m = 3; 4; 5), are given by F W3;5:200 (x) = 0:0447H(x; 45; :385) + 0:2573H(x; 85; :385) + 0:6980H(x; 0; :385); F W4;5:200 (x) = 0:068H(x; 40; :6422)+0:097H(x; 85; :6422)+0:3044H(x; 5; :6422)+0:588H(x; 40; :6422), and F W5;5:200 (x) = 0:0038H(x; 20; :9095) + 0:0393H(x; 75; :9095) + 0:262H(x; 0; :9095) + 0:3275H(x; 40; :9095) + 0:5032H(x; 65; :9095);

12 Figure 3: Density (left) and df (right): 5-moment approximation vs Gompertz distribution with KS distances d KS (S; W 3;5:200 ) = 0:07, d KS (S; W 4;5:200 ) = 0:0086, and d KS (S; W 5;5:200 ) = 0:0072. Note that the KS distance in the 4-moment approximation is considerably lower than for d KS (S; W 4;90 ) = 0:0239. In Figure 2, we compare the t of the 5-moment approximation W 5;5:200 to the Gompertz distribution by plotting their densities (left) and dfs (right). We can see that the t is quite acceptable and of a better quality than the two approximations displayed in Figure 2. 4 Moment-based mixed Erlang approximation with known 4. Basic de nitions We consider here a slightly di erent context than the one of Section 3. Instead of approximating a general df F S with known moments m, we rather assume that the df F S is known to be of mixed Erlang form () with given rate parameter > 0, and rst m moments m. However, the mixing weights f k g l k= are assumed unknown or di cult to obtain. Various applications in risk theory and credit risk fall into this context (e.g., Cossette et al. (2002), Lindskog and McNeil (2003), and McNeil et al. (2005); see also the application of Section 4.4). For a given rate parameter > 0, let ME( m ; ) be the set of all mixed Erlang distributions with df () (as l! ) and rst m moments m. Also, de ne ME( m ; A l ; ) to be the subset of ME( m ; ) with df () for a given l 2 N +, and let ME ext ( m ; A l ; ) be a further subset of ME( m ; A l ; ) such that at most (m + ) of the mixing weights f k g l k= are non-zero. Note that a distribution in ME( m; A l ; ) can be expressed as a convex combination of distributions in ME ext ( m ; A l ; ) (see, e.g., De Vylder (996)). For a given function, we consider two approaches to derive bounds and approximations for E [ (S)] when S 2 ME( m ; A l ; ) (in cases when the expectation exists). The rst approach is based on discrete s- convex extremal distributions while the second is based on moment bounds on discrete expected stop-loss transforms. 2

13 Remark 5 Naturally, the set ME( m ; A l ; ) tends to ME( m ; ) as l!. As such, when S 2 ME( m ; ), bounds for risk measures on ME( m ; ) can be approximated by their counterparts in ME( m ; A l ; ) for l reasonably large. For a mixed Erlang rv S with df (), its j-th moment is known to satisfy (6). Using the identity j Y (k + i) = i=0 jx s (j; n) k n, n= where the s (j; n) s are the (signed) Stirling numbers of the rst kind (e.g., Abramowitz and Stegun (972)), (6) becomes jx j j = s (j; n) n, (8) n= for j = ; :::; m where n = P l k= kk n. In matrix form, we have M = s T m; where s = fs (j; n)g m j;n= (with s (j; n) = 0 for n > j) and m = ( ; :::; m ). Isolating m yields m = s M T. (9) n o m From Comtet (974, p.23) we know that s c = ( ) i+j c (i; j) where the c (i; j) s are the i;j= Stirling numbers of the second kind de ned as c (i; j) = (j!) P j k=0 ( )j k k j k i (e.g., Abramowitz and Stegun (972)). Thus, for a given > 0, the class ME( m ; A l ; ) can be found through the identi cation of all discrete distributions with support A l and rst m moments given by the right-hand side of (9). Equivalently, the class ME ext ( m ; A l ; ) can be identi ed by restricting the discrete distributions on A l to have at most (m + ) non-zero mass points. This argument is formalized in the next section. 4.2 Discrete s-convex extremal distributions Let D ( m ; A l ) be all discrete distributions with support A l and rst m moments m = ( ; 2 ; :::; m ). Also, denote by D ext ( m ; A l ) the subset of D ( m ; A l ) with distributions having at most (m + ) nonzero mass points. For a given > 0, each distribution in D ( m ; A l ) (and D ext ( m ; A l )) with m as de ned in (9) corresponds to a mixed Erlang distribution in ME( m ; A l ; ) (and ME ext ( m ; A l ; )). This is a one-to-one correspondence (e.g., De Vylder (996, part 2)). Remark 6 Because of this one-to-one correspondence between D( m ; A l ) and ME( m ; A l ; ), conditions under which ME( m ; A l ; ) is not empty can be found from its discrete counterpart D( m ; A l ). We refer the reader to e.g., De Vylder (996), Marceau (996) or Courtois and Denuit (2009). This allows us to make use of the theory developed in Prékopa (990), Denuit and Lefèvre (997), Denuit, Lefèvre and Mes oui (999), and Courtois et al. (2006) to derive bounds/approximations for E [ (S)] when S 2 ME( m ; A l ; ). 3

14 First, we brie y recall the de nitions of s-convex function and s-convex order introduced by Denuit et al. (998). De nition 7 Denuit, Lefèvre and Shaked (998, 2000). Let C be a subinterval of R or a subset of N, and a function on C. For a positive integer s and x 0 < x < < x s 2 C, we recursively de ne the divided di erences as [x 0; x ; :::; x k ] = [x ; x 2 ; :::; x k ] [x 0 ; x ; :::; x k ] x k x 0 = kx i= (x i ), k = ; 2; :::; s; ky (x i x j ) j=0;j6=i where [x k ] = (x k ) for k = 0; ; :::; s. The function is s-convex if [x 0; x ; :::; x s ] 0 for all x 0 < x < < x s 2 C. We mention that the de nition of s-convex function, which refers to higher-convexity, should not be confused with the one for Schur-convex function, also known as S-convex function. De nition 8 Denuit, Lefèvre and Shaked (2000). For two rv s X and Y de ned on C, X is smaller than Y in the s-convex sense, namely X C s cx Y, if E [ (X)] E [ (Y )] for all s-convex functions (provided the expectation exists). We mention that the -convex order corresponds to the usual stochastic dominance order, and the 2-convex order is the usual convex order (see Müller and Stoyan (2002), Denuit et al. (2005), and Shaked and Shanthikumar (2007) for a review on stochastic orders). Also, as stated in Theorem.6.3 of Müller and Stoyan (2002), the s-convex order can only be used to compare rv s with the same rst (s ) moments (which explains why s is chosen to be m + in what follows). Examples of s-convex functions are (x) = x s+j for j 2 N and (x) = exp (cx) for c 0. For a general treatment of the s-convex order, see, e.g., Denuit et al. (999, 2000) and section.6 of Müller and Stoyan (2002). Let K (m+) min and K (m+) max be the (m + )-extremum rv s on D( m ; A l ), i.e. those which satisfy E K (m+) min E [ (K)] E K(m+) max ; for any (m + )-convex function and K 2 D( m ; A l ). The general distribution forms of K (m+) min and K (m+) max are given in Prékopa (990) (see also Courtois et al. (2006, Section 4)), and are repeated here: m + even o support of K (m+) min nj ; j + ; :::; j m+ ; j m o support of K (m+) max n; j ; j + ; :::; j m ; j m + ; l 2 2 n m + odd o ; j ; j + ; :::; j m ; j m n o j ; j + ; :::; j m ; j m + ; l 2 2 (0) where < j < j + < j 2 < ::: < l. From (0), it is clear that the support of K (m+) most m + elements. min(max) has at 4

15 Let W K = P K j= C j be a mixed Erlang rv, i.e. fc j g j are a sequence of iid exponential rv s with mean =, independent of K. The following result of Denuit, Lefèvre and Utev (999, Property 5.7) relates to the stability of the s-convex order under compounding. Lemma 9 If K A l s cx K0, then W K R+ s cx W K 0: We apply Lemma 9 to de ne the mixed Erlang rv s W K(m+) min and W K(m+) max, which are the (m + )-extremum rv s on ME( m ; A l ; ). It is immediate that, for W 2 ME( m ; A l ; ); W K(m+) min R+ (m+) cx W R+ (m+) cx W K (m+) max ; () For instance, using (), the (m + )-convex functions (x) = x m++j (j 2 N) and (x) = exp (cx) (c 0) yield h i E W m++j K (m+) min E W m++j h i E W m++j K (m+) max, and respectively. h i h i E exp cw K(m+) min E [exp (cw )] E exp cw K(m+) max, 4.3 Moment bounds on discrete expected stop-loss transforms Extrema on the (m + )-convex order yield bounds for E [ (W )] when is (m + )-convex and W 2 ME( m ; A l ; ). However, this approach is not appropriate to derive bounds on TVaR and the stop-loss premium when the number of known moments is greater than 2. It is well known that two rv s with the same mean and variance cannot be compared under the convex order. Consequently, we use an approach inspired from Courtois and Denuit (2009) (see also Hürlimann (2002)) to derive bounds on TVaR and the stop-loss premium. We consider D( m ; A l ) and determine lower and upper bounds for E (K k) + for all k 2 Al. From the lower bound, we de ne the corresponding rv K m down on A l via the df F Km infk2d(m;a down (k) = l ) E (K k) + inf K2D(m;Al ) E (K k ) + ; k = ; 2; :::; l ; k = l., (2) for k 2 A l. Similarly, K m up is de ned as in (2) by replacing inf by sup in the de nition. Given that, for K 2 D( m ; A l ), E (K m down k) + E (K k)+ E (Km up k) + ; for all k 2 A l, it implies that K m down(up) is smaller (larger) than K under the increasing convex order, namely K m down icx K icx K m up (see, e.g., Courtois and Denuit (2009)). Note that K m down and K m up do not belong to D( m ; A l ), but both have rst moment. The increasing convex order is stable under compounding, and thus W Km down icx W K icx W Km up : (3) 5

16 Then, from Denuit et al. (2005, Proposition 3.4.8), it follows that TVaR W Km down TVaR (W K ) TVaR W Km up ; (4) for 2 (0; ). Clearly, the rv s W Km down and W Km up will most likely not belong to ME( m ; A l ; ). Remark 0 Note that, when neither of the two aforementioned approaches are applicable, we propose to derive approximate bounds for E [ (S)] with S 2 ME ext ( m ; A l ; ) by calculating E [ (W )] for all W 2 ME ext ( m ; A l ; ) and choosing E [ (W )] min(max) = inf (sup) E [ (W )]. W 2ME ext ( m ;A l ;) Obviously, E [ (S)] does not necessarily lie between E [ (W )] min and E [ (W )] max. However, the interval estimate [E [ (W )] min ; E [ (W )] max ] may give an idea of the variability of all solutions on ME( m ; A l ; ). 4.4 Example: Portfolio of dependent risks We consider a portfolio of n dependent risks as described in the common mixture model of Cossette and al. (2002). Let S = X + ::: + X n be the aggregate claim amount with X i = B i I i. Conditional on a common mixture rv with pmf a j = P ( = j) for j = ; 2; :::, fi i g n i= are assumed to form a sequence of independent Bernoulli rv s with P (I i = j = j ) = (r i ) j for r i 2 (0; ). As for the B i s, they are assumed to form a sequence of iid exponential rv s of mean, independent of fi i g 20 i= and. In this context, it is clear that S is a two-point mixture of a degenerate rv at 0 and a mixed Erlang rv of the form () with l = n and =, i.e. its Laplace transform is given by E e ts p + pe e ty = X Qn a j i= j= (r i ) j + (r i ) j, t 0; + t where p = P j= a jq n i= (r i) j. We perform the moment-based approximation on the rv Y = (S js > 0) rather than S. For illustrative purposes, we assume n = 20 and a logarithmic pmf for, namely a j = (0:5) j =(j ln 2) for j : Also, the constants r i are set such that the (unconditional) mean of I i is q i = E h(r i ) i with q = ::: = q 0 = 0: and q = ::: = q 20 = 0:02. Under the above assumptions, the rst 5 moments of Y are found to be 5 = (:7999; 6:2270; 3:4785; 208:258; 693:7077). Using the approach on discrete (m + )-convex extremal distributions, the dfs F WK(m+) min and F WK(m+) max for m = 4; 5 are: F WK5 min (x) = 0:5365H (x; ; )+0:227H (x; 2; )+0:2232H (x; 3; )+0:0245H (x; 6; )+0:0030H (x; 7; ), F WK5 max (x) = 0:4860H (x; ; )+0:398H (x; 2; )+0:062H (x; 4; )+0:0537H (x; 5; )+0:0000H (x; 20; ), 6

17 F WK6 min (x) = 0:526H (x; ; ) + 0:379H (x; 2; ) + 0:053H (x; 3; ) + 0:082H (x; 4; ) + 0:0059H (x; 7; ) + 0:0023H (x; 8; ), and F WK6 max (x) = 0:5322H (x; ; ) + 0:2264H (x; 2; ) + 0:20H (x; 3; ) + 0:0004H (x; 5; ) + 0:0299H (x; 6; ) + 0:0000H (x; 20; ) : Let X K(m+) min(max) be a rv with df F XK(m+)- min(max) (x) = p + pf WK(m+)- min(max) (x) for x 0. It follows from Section 4.2 that lower (upper) bounds for the higher-order moments E S j (j = 4; 5; :::) and the exponential premium principle de ned as ' (S) = ln E e S = ( > 0) can be found from their counterparts for X K(m+) min(max). A few numerical values are provided in Tables 5 and 6, respectively. i i j E hx j K5 min E hx j K6 min E S j i i E hx j K6 max E hx j K5 max 4 38: : : : : : :880 29:880 29:880 49: : : : : :676 Table 5. Higher-order moments of S, X K(m+) min and X K(m+) max (m = 4; 5) ' X K5 min ' X K6 min ' (S) ' X K6 max ' X K5 max 0:2 :5545 :5546 :5546 :5548 :5564 0: :3536 :3536 :3536 :3536 :3536 0:0 :237 :237 :237 :237 :237 Table 6. Exponential premiums of S, X K(m+) min and X K(m+) max (m = 4; 5) As expected, the bounds get sharper as the number of moments involved increases. As for the second approach based on moment bounds with discrete expected stop-loss transforms, Table 7 presents the values of TVaR for X Km down(up) (m = 4; 5) with df F XKm-down(up) (x) = p + pf WKm-down(up) (x) for x 0. Exact 4 moments 5 moments TVaR (S) TVaR X K4 down TVaR X K4 up TVaR X K5 down TVaR X K5 up 0:9 5:0696 4:9222 5:2062 4:9800 5:490 0:95 6:224 5:9708 6:4548 6:0594 6:3642 0:99 8:8460 8:2899 9:330 8:4655 9:767 0:995 9:9589 9:2500 0:5629 9:4679 0:3775 0:999 2:5066 :422 3:4854 :7323 3:382 Table 7. Values of TVaR for S, X Km down and X Km up (m = 4; 5) As expected, the inequality (4) is veri ed. Also, we observe that the interval estimate of TVaR (S) shrinks as the number of moments matched increases. Finally, given that neither method is applicable for the VaR risk measure, we make use of the technique 7

18 discussed in Remark 0. We nd the numerical values provided in Table 8. Exact ME ext ( 4 ; A 20 ; ) ME ext ( 5 ; A 20 ; ) VaR (S) VaR (X) min VaR (X) max VaR (X) min VaR (X) max 0:9 3:3965 3:3896 3:403 3:3897 3:400 0:95 4:5704 4:5539 4:6030 4:5584 4:5730 0:99 7:2334 7:282 7:2690 7:225 7:2533 0:995 8:3604 8:343 8:3946 8:3539 8:3925 0:999 0:9388 0:79 0:978 0:9226 0:9536 Table 8. Minimal/Maximal values of VaR within ME ext ( m ; A 20 ; ) (m = 4; 5) vs VaR (S) For this example, we observe that the exact values of VaR (S) are within the minimal and maximal values of their corresponding risk measures among all members of ME ext ( m ; A 20 ; ) for m = 4; 5. Also, the spread between the minimal and maximal values of VaR are reduced when we go from 4 to 5 moments. An identical exercise for the TVaR risk measure resulted in the same conclusions. 5 References Abramowitz, M., Stegun, I. (972). Handbook of mathematical functions. 9th Edition. Washington, DC: US Government Printing O ce. Altiok, T. (985). On the Phase-Type Approximations of General Distributions. IIE Transactions 7(2): 0-6. Bobbio, A., Horváth, A., Telek, M. (2005). Matching Three Moments with Minimal Acyclic Phase Type Distributions. Stochastic Models 2: Bowers, N.L., Gerber, H.U., Hickman, G.C., Jones, D.A., Nesbit, C.J. (997). Actuarial Mathematics. The Society of Actuaries, Schaumburg, Illinois. Chaubey, Y.P., Garrido, J., Trudeau, S. (998). On the computation of aggregate claims distributions: some new approximations. Insurance: Mathematics and Economics 23(3): Cheung, E. C. K., Woo, J. K. (204). On the discounted aggregate claim costs until ruin in dependent Sparre Andersen risk processes. Scandinavian Actuarial Journal. In press. Comtet, L. (974). Advanced Combinatorics. Dordrecht-Holland. Boston. Cossette, H., Mailhot, M., Marceau, E. (202). T-Var based capital allocation for multivariate compound distributions. Insurance: Mathematics and Economics 50(2), Cossette, H., Gaillardetz, P., Marceau, E. (2002). Common mixtures in the individual risk model. Bulletin de l Association suisse des actuaires, Courtois, C., Denuit, M. (2009). Moment Bounds on Discrete Expected Stop-Loss Transforms, with Applications. Methodology and Computing in Applied Probability : Courtois, C., Denuit, M. (2007). Bounds on convex reliability functions with known rst moments. European Journal of Operational Research 77:

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