Proceedings of the 2012 Winter Simulation Conference C. Laroque, J. Himmelspach, R. Pasupathy, O. Rose, and A. M. Uhrmacher, eds.

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1 Proceedings of the 22 Winter Simulation Conference C. Laroque, J. Himmelspach, R. Pasupathy, O. Rose, and A. M. Uhrmacher, eds. ON ERROR RATES IN RARE EVENT SIMULATION WITH HEAVY TAILS Søren Asmussen Aarhus University Department of Mathematics Ny Munkegade DK-8 Aarhus C, DENMARK Dominik Kortschak Université Lyon Laboratoire SAF, Institut de Science Financière et d Assurances 5 Avenue Tony Garnier F-697 Lyon, FRANCE ABSTRACT For estimating PS n > by simulation where S k = Y + +Y k with Y,...,Y n are non-negative and heavytailed with distribution F, Asmussen and Kroese 26 suggested the estimator nf M n S n where M k = may,...,y k. The estimator has shown to perform ecellently in practice and has also nice theoretical properties. In particular, Hartinger and Kortschak 29 showed that the relative error goes to as. We identify here the eact rate of decay and propose some related estimators with even faster rates. INTRODUCTION This paper is concerned with the efficient simulation of z = z = PS n >, where Y,...,Y n are i.i.d. with a common subeponential distribution F, S n = Y + +Y n and is large so that z is small for background on subeponential distributions, see, e.g., Embrechts, Klüppelberg, and Mikosch 997, Asmussen and Albrecher 2, X., or Foss, Korshunov, and Zachary 2. Recall from the outset the standard fact or definition of subeponentiality! that z nf as n where F = F is the tail. In this paper we will consider a subclass of subeponential distributions the regularly varying distributions. Regularly varying distributions have a tail that approimately behaves like a power, i.e. F = α L where α > and L is slowly varying lim Lu/L = for all u >. If the density of the distribution is also regularly varying then it fulfills f α α L as c.f. Bingham, Goldie, and Teugels 989, Proposition.5.. This problem has a long history. As is traditional in the literature, we denote by a simulation estimator a r.v. Z = Z that can be generated by simulation and is unbiased, EZ = z. The usual performance measure is the relative error e = VarZ /2 /z. The relative error is bounded if limsup e <, and the estimator Z is logarithmically efficient if limsup z ε e < for all ε >. Efficient estimators have long been known with light tails see e.g. Asmussen and Glynn 27, VI.2, Bucklew 99, Heidelberger 995, Juneja and Shahabuddin 26 for surveys, and are typically based on ideas from large deviations theory implemented via eponential change of measure. The heavy tailed case is more recent. In Asmussen, Binswanger, and Højgaard 2, some of the difficulties in a literal translation of the light-tailed ideas are eplained. However, Asmussen and Binswanger 997 gave the first logarithmically efficient estimator for PS > using a conditional Monte Carlo idea. The idea was further improved in Asmussen and Kroese 26, which as of today stands as a model of an efficient and at the same time easily implementable algorithm, and is also at the core of this paper. The idea is to combine an echangeability argument with the conditional Monte Carlo idea. More precisely assuming /2/$3. 22 IEEE 49

2 eistence of densities to eclude multiple maima one has z = nps >, M n = Y n. where M k = ma Y,...,Y k. An unbiased simulation estimator of z based on simulated values Y,...,Y n is therefore the conditional epectation Z AK = nf M n S n of this epression given Y,...,Y n, where S n = Y + +Y n.this estimator, baptized the Asmussen- Kroese estimator by the simulation community, is shown in Asmussen and Kroese 26 to have bounded relative error and in Hartinger and Kortschak 29 to have vanishing relative error e, though the argument for this is rather implicit and no quantitative rates are given. The contribution of this note is two-fold: to compute the eact error rate of Z AK in the regularly varying case; and to produce another estimator with a better rate in some cases. Both aspects combine with ideas of higher order subeponential methodology cf. Remark. For subeponential distributions with a lighter tail than regular variation like the Weibull, a corresponding theory is developed in Asmussen and Kortschak 22 and summarized in part in Section 5. For the regularly varying case, our main result is the following: Theorem Assume f = αl/ α+ where L is slowly varying. If α > 2 or, more generally, E[Y 2 ] < then VarZ AK n 2 Var[S n ] f 2 = n 2 n Var[Y ] f 2. If α = 2 and E[Y 2 ] = then If α < 2 then VarZ AK n 2 n k α F 3 where k α = = α VarZ AK 2n 2 n f 2 yfy dy. 2 α α 2 2α + α /2 [ y y α ] 2 y α dy. y α 2 y α dy Remark A main idea of higher order subeponential methodology is the Taylor epansion which easily leads to the refinement F S n = F + f S n + PS n > = nf + n f ES n + at least in the regularly varying case, cf. Omey and Willekens 987, Baltrūnas and Omey 998 and Barbe and McCormick 29. Technically, the Taylor epansion is only useful for moderate S n, and large values have to be shown to be negligible by a separate argument; this also is the case in the present paper. One may note that is only useful for heavy-tailed distributions where typically F >> f >> f >> for light-tailed distributions like the eponential typically F, f, f,... have the same magnitude. 42

3 Remark 2 The rates for VarZ AK in Theorem have to be compared with the bounded relative error rate L 2 / 2α. For α > 2, one sees an improvement to L 2 / 2α+2, for α < 2 to L 3 / 3α. In Section 3, we ehibit an estimator improving this rate for α > 2 and in Section 4 one for < α < 2. The feature of vanishing relative error is quite unusual. The few further eamples we know of are Blanchet and Glynn 28 and Dupuis, Leder, and Wang 27 in the setting of dynamic importance sampling, though it should be remarked that the algorithms there are much more complicated than those of this paper and that the rate results are not very eplicit. Remark 3 In applications to ruin theory and the M/G/ queue, the number n of terms in S n is an independent r.v. With some effort, our theory can be refined to this case, but we will not give the details here. 2 PROOF OF THEOREM Proof of Theorem. We will use the notation S n 2 = S n M n and A n, = { M n /2n }. Then the the Asmussen-Kroese estimator can be written as Z AK = nx + X 2 where X = IA c n,f M n S n, X 2 = IA n, F M n S n. Recall that the density of M n is n f yfy n 2 and hence the tail is n Fy. We have E[X k ] = E [ FM n S n k ; S n 2 >, A c ] n, + E [ FM n S n k ; S n 2, A c n,]. The first term is a O F k+ F, since we can bound F by O F and the event S n 2 >, A n, has probability O FF since it occurs only if at least one in the i.i.d. sample Y,...,Y n that is not the maimum eceeds /n 2 and another eceeds /2. For the second term, note that here M n S n M n +. Hence by repeated use of the regular variation property we get the upper bound E [ F M n M n k ; A c n, ] = /2 + /2n n /2 /2 α F y k n f yfy n 2 dy Fy k n f yfy n 2 dy /2n y α y kα dy + k + 2αk+ F k+ where we used /2 /2n F /2 y k f yfy n 2 dy F y k f ydy = /2n / /2 /2n F k f αf k+ /2 F z k f zdz / /2 /2n /2n / z αk z α dz z αk z α dz. 42

4 Omitting gives the same lower bound, so /2 E[X k ] n α y α y kα dy + /2n k + 2αk+ F k+. Net consider X 2. If M n /2n, then S n /2 and so X 2 = IA n, F S n = F IA c n,f + IA n, S n f Ξ = F X 2, + X 2,2 2 where S n Ξ. Now E[X k 2,] = F k PA c n, n 2 α n α F k+. To evaluate E[X2,2 k ], we split the epectation into S n ε and S n ε, for some /2n > ε >. When S n ε, then since f sup y> f y c.f. Bingham, Goldie, and Teugels 989, Theorem.5.3 we have f f Ξ f ε. This and monotone convergence gives lim lim E [ Sn k f Ξk ; S n ε ] ε f k E [ Sn k ; S n ε ] =. Further by Bingham, Goldie, and Teugels 989, Proposition.5.8 and.5.9a and partial integration ε E [ Sn ; k S n ε ] = k y k F Sn ydy ε k F Sn ε ESn k EY k <, k yk F Sn ydy α = k, ε k α α k α k F Sn α < k. If ε S n /2 then it holds uniformly in S n that and hence S n f Ξ F S n F lim = lim = S n / α, F F E [ S k n f Ξ k ; S n > ε, M n /2n ] F k E [ S n / α k ; Sn > ε, M n /2n ]. As above we can split the epectation into S n 2 and S n 2 >, such that we can prove that for E[ S n / α k ; Sn > ε, M n /2n ] E[ M n / α k ; ε < Mn /2n ] n n /2n ε /2n ε n f αn F /2n y/ α k f ydy ε /2n ε y α k f ydy y α k y α dy y α k y α dy. 422

5 Here k is of order y k at y =. Since ε was arbitrary, it follows that if E[Y 2 ] < E[X 2 2,2] f 2 E[S 2 n ]. If α = 2 and E[Y 2 ] =, Bingham, Goldie, and Teugels 989, Proposition.5.9a yields If α < 2, E[X2,2] 2 2 f 2 n yfydy. /2n E[X2,2] 2 αn F 3 y α 2 y α dy. Further if E[Y ] <, then E[X 2,2 ] f E[S n ]. If α = and E[Y ] =, then If α <, then Now recall the formula Using E[X 2,2 ] αn E[X 2,2 ] n f Fydy. /2n y α y α dyf 2. Var[X +Y ] = Var[X] + Var[Y ] + 2Cov[X,Y ]. E[X X 2 ] = E[X 2, X 2,2 ] =, E[X 2 ] F, E[X 2, ]E[X 2,2 ] = oe[x 2 2,2], we get n 2 Var[Z AK] = Var[X ] + Var[X 2, ] + Var[X 2,2 ] 2FE[X ]. Collecting all terms we get that for α 2 Var[X 2,2 ] dominates the other terms asymptotically and for α < 2 we get /2 n 2 VarZ AK n α y α y 2α dy + /2n 3 23α + 2n α /2 2α y α y α dy 2 2α /2n /2n + α y α 2 y α dy F 3 = n α /2 /2n + 2n α 2 2α + α = n + α /2 y α y 2α 2 y α dy α /2n y α 2 y α dy /2 α y α dy + /2n 3 23α + 2n α 2 2α y α 2 y α dy F 3 F 3 423

6 = n Asmussen and Kortschak 2 α α 2 2α + α /2 y α 2 y α dy F 3. Remark 4 In the case α = 2, E[Y 2 ] =, the above discussion leads to asking for more eplicit growth rates of Ly I = yfy dy = dy. y An obvious conjecture is that the order is Llog. But this turns out not be true for all L. For a more detailed analysis, assume that L cep{ a et/t dt} where et is differentiable and a,c >. Further we assume that I. If there eists a κ with e tt lim t et 2 κ then a partial integration argument yields I L/κ + e. For eample, if L log β we can choose e = β/log and κ = /β, so that hence I Llog/β +. Another eample is L ep { loglog γ} where γ >. Then e = γloglog γ /log, κ = and I Llog/γloglog γ. On the other hand if elog τ and τ, it can be shown that I Llog/τ +. Again for L log β we can choose e = β/log and τ = β and find as above that I Llog/γloglog γ. 3 AN IMPROVED ESTIMATOR FOR α > 2 In the case of finite second moment we get that the error is basically given by the variance of the term X 2,2 in 2. Since for large values of X 2,2 is close to S n f a natural idea is to use this approimation as a control variate which results in the estimator Z = Z AK + nes n S n f. 3 The net theorem shows that this estimator is indeed an improvement over Z AK. Theorem 2 Assume f = αα L/ α+2. If α > 4 or, more generally, E[Y 4 ] < then If α = 4 and E[Y 4 ] = then If 2 < α < 4 then VarZ n 2 n k α F 3 where k α = α VarZ 4 n2 Var[S 2 n ] f 2. VarZ n 2 n f 2 y 3 Fydy. z z α αz 2 z α dz. Proof. The proof is a variation of the proof of Theorem. Define A n, = { M n /2n }. At first note that VarZ = n 2 VarZ where Z = F S n M n S n f. We will use Z = Z I A c n, + Z I A n, = X + X

7 Similar to the proof of Theorem we get that Asmussen and Kortschak EX k = E [ F S n M n S n f k ;A c n, ] E [ F M n M n M n f k ;A c n, ] = n 2n [ F z z z f ] k f zfz n 2 dz αn F k+ 2n If M n /2n, then S n /2 and so [ z z α αz ] k z α dz. X 2 = F FI A c n, + F Sn F S n f I A n, = F IA c n,f 2 IA n,s 2 n f Ξ = F X 2, + X 2,2. where S n Ξ. Proceeding as in the proof of Theorem we get that E[X k 2,] = F k PA c n, n 2 α n α F k+. To evaluate E[X k 2,2 ], we split the epectation into S n ε and S n ε, for some /2n > ε >. When S n ε, we have f f Ξ f ε. This and monotone convergence gives lim lim E [ Sn 2k f Ξ k ; S n ε ] ε f k E [ Sn 2k ; S n ε ] =. Further by Bingham, Goldie, and Teugels 989, Proposition.5.8 and.5.9a and partial integration E [ Sn ; 2k S n ε ] ε = 2k y 2k F Sn ydy ε 2k F Sn ε ESn 2k EY 2k <, 2k y2k F Sn ydy α = 2k, ε 2k α α 2k α 2k F Sn α < 2k. If ε S n /2 then it holds uniformly in S n that S 2 n f Ξ 2F F S n F S n f F S n / α α S n, and hence 2 E[ S 2k n f Ξ k ; S n > ε, M n /2n ] F k E [ S n / α αs n / k ; Sn > ε, M n /2n ] F k E[ M n / α αm n / k ; ε < Mn /2n ] /2n αn F k+ y α αy k y α dy. ε 425

8 Here k is of order y 2k at y =. Since ε was arbitrary, it follows that if E[Y 4 ] < E[X k 2,2] 2 k f k E[S 2k n ]. If α = 4 and E[Y 4 ] =, Bingham, Goldie, and Teugels 989, Proposition.5.9a yields If α < 4, E[X2,2] 2 n f 2 y 3 Fydy. /2n E[X2,2] 2 αn F 3 y α αy 2 y α dy. Further since E[Y 2] <, E[X 2,2] 2 f E[Sn 2 ]. Finally with the same collecting of terms as in the proof of Theorem the Theorem follows. 4 AN IMPROVED ESTIMATOR FOR < α < 2 If EY 2 = then the estimator 3 has VarZ = so this estimator is no improvement for α < 2. Nevertheless it is an interesting question if we can improve on Z AK also in this case. In this section we will consider the case of < α < 2 where we have a finite mean but an infinite second moment. We will denote with Y Y n the order statistic of Y,...,Y n, with M k = ma i k Y i and with S k = k i= Y i. At first note that PS n > = P S n >,Y n 4 2n + P S n >,Y n >. 5 2n We will use separate estimators for 4 and 5. By conditioning on the two largest elements and using symmetry 2 P S n >,Y n > = nn P Y > p n. 2n 2n where p n, = P S n >, min Y i M n 2 min Y i >. i {n,n} i {n,n} 2n Denote with Z c the crude Monte Carlo estimator of p n,, i.e. we first simulate Y n,y n conditioned to eceed /2n then the rest normal and use the estimator Z c = I S n > I miny n,y n M n 2 [note that conditioned r.v. generation is easy whenever inversion is available, cf. Asmussen and Glynn 27, p. 39]. The estimator is an unbiased estimator for 5 and as Z b = nn F/2n 2 Z c VarZ b = Var [ Z s ] nn 2 F/2n 4 2 4α n 2 n 4α+2 F

9 So we have to find an estimator for 4. By conditioning on the largest element and using symmetry we get that P S n >,X n = np S n >,M n 2n. 2n Let f be an importance sampling density of the form L/ α with α < 2α 2 and P, Ẽ the corresponding probability- and epectation operators. We will assume that f is bonded away from zero on finite intervals. We now combine the estimator which conditions on S n with importance sampling and a control variate to get the estimator Z s = F S n F LI n, + FPI n, where L = n i= f Y i/ f Y i is the likelihood ratio and I n, is the indicator of the event M n /2n. Since f sup z f z e.g. Bingham, Goldie, and Teugels 989, Theorem.5.3, we get by Taylor epansion and with the same arguments We have that Since for k =,,2 Ẽ [ F S n F LI n, ] f /2 Ẽ[S n LI n, ] f /2ES n Ẽ [ F S n F LI n, ] 2 f /2 2 Ẽ[S 2 n L 2 ]. [ Ẽ[Sn L 2 2 ] = n Ẽ Y 2 f Y 2 n f Y i 2 ] f Y 2 i=2 f Y i 2 f Y +n n 2Ẽ [Y 2 f Y 2Y 2 Ẽ [Y k f Y i 2 ] f Y i 2 = f Y 2 2 f Y 2 2 y k f y2 f y dy < n f Y i 2 ]. i=3 f Y i 2 the integrand is regularly varying with inde k 2α 2 + α + <, it follows by independence that Ẽ[S 2 n L2 ] < and hence ṼarZs = O f 2. Thus we have shown Theorem 3 Assume f = αl/ α+. If < α < 2 then the estimator Z = Z b + nz s is an unbiased estimator for PS n > with VarZ = O f 2 Remark 5 Note that for a fied number of simulations the confidence interval of fied level for the estimator Z has a length of order f. This is the same order as the error of the asymptotic approimation PS n > nf. 5 THE WEIBULL CASE We finally give a brief survey of our results for the Weibull case F = e β with < β < related distributions, say modified by a power, are easily included, but for simplicity, we refrain from this. We refer to Asmussen and Kortschak 22 for a more complete treatment. The density is f = β β e β and f = pf where p = β 2 2β + β β β 2. Theorem 4 If < β < log3/2/ log2, then the Asmussen-Kroese estimator s variance is asymptotically given by VarZ AK n 2 VarS n f 2 427

10 Note that log3/2/log2.585 is also found to be critical in Asmussen and Kroese 26 as the threshold for logarithmic efficiency to hold. As in Section 3 we can use the estimator defined in 3 to improve on Z AK. Theorem 5 Assume that < β < log3/2/log Then the estimator Z in 3 has vanishing relative error. More precisely, VarZ n2 4 VarS2 n f 2. The estimator Z in 3 has the form Z AK + αs n ES n, so it is a control variate estimator, using S n as control for Z AK. It is natural to ask whether the α = n f at least asymptotically coincides with the optimal α = CovZ AK,S n /VarS n cf. Asmussen and Glynn 27, V.2. The following lemma shows that this is the case: Lemma 6 CovZ AK,S n = nvars n f + o f. ACKNOWLEDGMENTS We thank Sandeep Juneja for discussions stimulating the present research. The second author was supported by the the MIRACCLE-GICC project and the Chaire d ecellence Generali - Actuariat responsable: gestion des risques naturels et changements climatiques. REFERENCES Asmussen, S., and H. Albrecher. 2. Ruin probabilities. Second ed. Advanced Series on Statistical Science & Applied Probability, 4. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ. Asmussen, S., and K. Binswanger Simulation of ruin probabilities for subeponential claims. ASTIN Bulletin 27 2: Asmussen, S., K. Binswanger, and B. Højgaard. 2. Rare events simulation for heavy-tailed distributions. Bernoulli 6 2: Asmussen, S., and P. W. Glynn. 27. Stochastic simulation: algorithms and analysis, Volume 57 of Stochastic Modelling and Applied Probability. New York: Springer. Asmussen, S., and D. Kortschak. 22. Error rates and improved algorithms for rare event simulation with heavy Weibull tails. Technical report, Working paper. Asmussen, S., and D. P. Kroese. 26. Improved algorithms for rare event simulation with heavy tails. Adv. in Appl. Probab. 38 2: Baltrūnas, A., and E. Omey The rate of convergence for subeponential distributions. Liet. Mat. Rink. 38 : 8. Barbe, P., and W. P. McCormick. 29. Asymptotic epansions for infinite weighted convolutions of heavy tail distributions and applications. American Mathematical Society. Bingham, N. H., C. M. Goldie, and J. L. Teugels Regular variation, Volume 27 of Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press. Blanchet, J., and P. Glynn. 28. Efficient rare-event simulation for the maimum of heavy-tailed random walks. Ann. Appl. Probab. 8 4: Bucklew, J. A. 99. Large deviation techniques in decision, simulation, and estimation. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. New York: John Wiley & Sons Inc. A Wiley-Interscience Publication. Dupuis, P., K. Leder, and H. Wang. 27. Importance sampling for sums of random variables with regularly varying tails. ACM TOMACS 7 3. Embrechts, P., C. Klüppelberg, and T. Mikosch Modelling etremal events, for insurance and finance, Volume 33 of Applications of Mathematics New York. Berlin: Springer-Verlag. Foss, S., D. Korshunov, and S. Zachary. 2. An introduction to heavy-tailed and subeponential distributions. Springer Series in Operations Research and Financial Engineering. New York: Springer. 428

11 Hartinger, J., and D. Kortschak. 29. On the efficiency of the Asmussen-Kroese-estimators and its application to stop-loss transforms. Blätter DGVFM 3 2: Heidelberger, P Fast simulation of rare events in queueing and reliability models. ACM Trans. Model. Comput. Simul. 5 : Juneja, S., and P. Shahabuddin. 26. Rare event simulation techniques: An introduction and recent advances. In Handbook on Simulation S. Henderson & B. Nelson, eds., North-Holland. Omey, E., and E. Willekens Second-order behaviour of distributions subordinate to a distribution with finite mean. Comm. Statist. Stochastic Models 3 3: AUTHOR BIOGRAPHIES SøREN ASMUSSEN is Professor of Applied Probability at Aarhus University, Denmark, with earlier long-time appointments at Copenhagen and Aalborg, Denmark, and Lund, Sweden. He has worked on branching processes, insurance risk, queueing theory, heavy tails, matri-analytic methods, and Monte Carlo simulation, and is the author of 4 books and more than 5 research papers. He was the editor of the Annals of Applied Probability 2-22 and is the editor of Journal of Applied Probability and Advances in Applied Probability since 25. He has received the Marcel F. Neuts Applied Probability Prize 999, the INFORMS Simulation Society Outstanding Publication Award 22, 28, the John von Neumann Theory Prize 2, and the gold medal For Great Contributions in Mathematics awarded by the Sobolev Institute of Mathematics under the Russian Academy of Sciences in 2. asmus@imf.au.dk. DOMINIK KORTSCHAK is a research scientist Cherchuer at the University Lyon. Before this he was a Collaborateur de recherche, at the Department of Actuarial Science DAS, Faculty of Business and Economics HEC of the University of Lausanne. While earning his PhD at the Graz University of Technology, he was also working at the at the Johann Radon Institute for Computational and Applied Mathematics RICAM of the Austrian Academy of Sciences. Further he was visiting the University of Aarhus and he was a collaborator in projects of the Johaneum research on insurance related questions of natural hazards in Austria. His research interests include the tail behavior of dependent risks, Monte Carlo Simulation and the assessment of natural catastrophe risk. kortschakdominik@gmail.com. 429