Saddlepoint Approximations to Sensitivities of Tail

SADDLEPOINT APPROXIMATIONS TO SENSITIVITIES OF TAIL PROBABILITIES OF RANDOM SUMS AND COMPARISONS WITH MONTE CARLO ESTIMATORS Riccardo Gatto† and Chantal Peeters‡ Submitted: January 2013 Revised: June 2013 Abstract This article proposes computing sensitivities of upper tail probabilities of random sums by the saddlepoint approximation. The considered sensitivity is the derivative of the upper tail probability with respect to the parameter of the summation index distribution. Random sums with Poisson or Geometric distributed summation indices and Gamma or Weibull distributed summands are considered. The score method with importance sampling is considered as an alternative approximation. Numerical studies show that the saddlepoint approximation and the method of score with importance sampling are very accurate. But the saddlepoint approximation is substantially faster than the score method with importance sampling. Thus the suggested saddlepoint approximation can be conveniently used in various scientific problems. Key words and phrases Change of measure, exponential tilt, Gamma distribution, Geometric distribution, importance sampling, Poisson distribution, rare event, score Monte Carlo method, Weibull distribution. | downloaded: 23.9.2021 2010 Mathematics Subject Classification: 41A60, 65C05. Institute of Mathematical Statistics and Actuarial Science, University of Bern, Alpeneggstrasse 22, 3012 † Bern, Switzerland. Email: [email protected]. Email: [email protected]. ‡ The authors are grateful to the Editor, the Associate Editor and two anonymous Reviewers, for thoughtful comments, suggestions and corrections which improved the quality of the article. Riccardo Gatto is grateful to the Swiss National Science Foundation for financial support (grant 200021-121901). https://doi.org/10.7892/boris.60983 source: 1 1 Introduction Let (Ω, , m) be a measure space, where (Ω) is a σ-algebra, and let P ∈ be a F F ⊂ P { θ}θ Θ class of probabilities measures such that P m, θ Θ, where Θ is an open interval of R. θ ≪ ∀ ∈ Let X ,X ,... be independent random variables on (Ω, ) with distribution function (d.f.) 1 2 F F independent of θ, survivor function F¯ =1 F and density f , with respect to (w.r.t.) X X − X X the Lebesgue measure on R. Let N be an independent N-valued random variable on (Ω, ) def F with density pθ, w.r.t. the counting measure, depending on θ Θ. Here N = 0, 1,... N def ∈ { } We consider the random sum S = k=0 Xk, where X0 = 0, and the upper tail probability P ∞ k k def z(θ; x) = Pθ[S x]= pθ(0)I x 0 + I xj x fX (xj)dxjpθ(k), (1) ≥ { ≤ } Rk ≥ ( j=1 ) j=1 Xk=1 Z X Y θ Θ, where x R is large and where I A , or alternatively I , denote the indicator of ∀ ∈ ∈ { } A statement A. In this case, z(θ; x) is typically small and referred as a rare event probability. The goal is to approximate accurately the sensitivity w.r.t. θ of this rare event probability, namely to approximate ∂ ∂ z′(θ; x) def= z(θ; x)= P [S x], θ Θ, (2) ∂θ ∂θ θ ≥ ∀ ∈ where x R is large. ∈ Random sums with Poisson or Geometric distributed counts (N) and positive summands (X1,X2,...) are often used to model phenomena occurring e.g. in earth sciences, ecology, biology, insurance and finance. In insurance, the insurer total claim amount generated by a portfolio of risks over a fixed period is often represented by a Poisson random sum. R∗ Further, let Y1,Y2,... be +-valued independent individual claim amounts with d.f. FY and expectation µ R∗ . Here R∗ def= (0, ). Let K be a Poisson process with rate λ> 0 Y ∈ + + ∞ t at a given time t 0 and let Z = Kt Y , where Y def= 0, be the total claim amount ≥ t k=0 k 0 at time t 0. The compound Poisson risk process of the insurer financial reserve at time ≥ P t 0 is given by U = x+ct Z , where x 0 is the initial capital and c> 0 is the premium ≥ t − t ≥ rate. The probability of ruin is the probability that U ≥ ever falls below the null line. { t}t 0 A classical result is the re-expression of the probability of ruin as z(θ; x) given in (1), where N has the Geometric distribution with parameter θ = λµY /c < 1 and the summands have d.f. F (x) = x[1 F (y)]dy/µ , x 0. Thus the sensitivity of the probability of ruin X 0 − Y Y ∀ ≥ w.r.t. θ is given by z′(θ; x) in (2). Some standard references on risk processes are, for R example, Asmussen (2000), Bühlmann (1970), Gerber (1979) and Mikosch (2004). Finding analytical and accurate approximations to the distribution of a random sum, or to the distribution of a compound process at a given time, has been investigated by several authors. We mention some of them. Esscher (1932, 1963) proposed an approximation based on the exponential tilt (or the Esscher transform), which is the essence of both saddlepoint approximation and importance sampling. Willmot (1989) provided an in- finite series representation of the density of the insurer total claim amount. Jensen (1991) proposed saddlepoint approximations for homogeneous compound Poisson processes with 2 Wiener perturbation, under force of interest and under Markov modulation (i.e. switch- ing between different compound Poisson processes). Gatto (2011) provided saddlepoint approximations to tail probabilities, values at risk and tail values at risk of the doubly compound insurer total claim amount, which is based on a primary counting birth process, for the number of catastrophic events, and on a secondary counting distribution, for the number of individual losses generated from each catastrophe of the primary process. Gatto (2012) provided saddlepoint approximations to tail probabilities and quantiles of the compound Poisson process under force of interest and under time inhomogeneity, with several types of periodic intensity functions. However, the problem of computing analytically the random sum sensitivity z′(θ; x) has not received much attention. Available methods rely on Monte Carlo simulation, see Asmussen and Glynn (2007), Chapter VII. For situations where x is large, the method of score with importance sampling can used to approximate z′(θ; x) accurately, see Asmussen and Rubinstein (1999), Asmussen and Glynn (2007), p. 239-240, and Section 3 of the present article. A simple analytic approximation to z′(θ; x) could be derived from the limiting normal approximation. Denote by S ∈ a stochastic process on (Ω, , P) such that the distri- { θ}θ Θ F bution of S under P is the distribution of S under P , θ Θ. Let θ¯ = sup Θ or inf Θ, θ θ ∀ ∈ according to the selected model, let E and var be the expectation and variance under P and assume S E[S ] d θ − θ (0, 1), as θ θ.¯ var(Sθ) −→ N → Then p def x E[S ] z(θ; x) z (θ; x) = 1 Φ − θ , (3) a var ≃ ≃ − (Sθ)! where means approximately equal. A rough approximationp to the sensitivity z′(θ; x) ≃ could be obtained by differentiating the last expression. However, it is well-known that asymptotic normal approximations often lead to inaccurate approximations to small tail probabilities, as they often entail large relative errors. Clearly, z(θ; x) 0, as x , → → ∞ θ Θ. As explained at the beginning of Section 3, ∀ ∈ log z′(θ; x) lim inf | | 1, θ Θ, (4) x→∞ log z(θ; x) ≥ ∀ ∈ which means that z(θ; x) and z′(θ; x) have similar logarithmic asymptotics, as x . So →∞ both z(θ; x) and z′(θ; x) are asymptotically vanishing quantities and so for none of them one should expect accurate normal approximations. Alternative approximations to sensitivities with small relative errors are thus required and the aims of this article are: to propose an analytical approximation obtained from the saddlepoint approximation, then to illustrate its high numerical accuracy and finally to compare it with an accurate Monte Carlo competitor, namely with the method of score with importance sampling. The remaining part of this article is organized as follows. In Section 2 we give an approximation to the sensitivity of a random sum obtained from the saddlepoint approxi- 3 mation of Lugannani and Rice (1980). Specific results for the Poisson random sum are given under Section 2.1 and for the Geometric random sum under Section 2.2. In Section 3 we summarize the score with importance sampling Monte Carlo method and then we present numerical comparisons: Poisson random sums with Gamma and Weibull summands, in Section 3.1, and Geometric random sums with Gamma summands, in Section 3.2. These numerical comparisons show that the saddlepoint approximation to sensitivities is very accurate and provide thus an efficient alternative to importance sampling simulation. A conclusion is given in Section 4. 2 Saddlepoint approximations to sensitivities The saddlepoint approximation or method of steepest descent is an important technique of asymptotic analysis, which provides approximations to some types of complex integrals, see e.g. de Bruijn (1981), Chapter 5. It was introduced in statistics by Daniels (1954), who provided the saddlepoint expansion of the density of the sample mean of n independent and identically distributed random variables. The relative error of the first term of the expansion is O(n−1), as n . Lugannani and Rice (1980) provided a saddlepoint approximation → ∞ to the survivor function, also with relative error O(n−1), as n , see also Daniels → ∞ E vS (1987). Let us define the cumulant generating function (c.g.f.) of S as Kθ(v) = log Pθ [e ], which is assumed existing for all v over a neighborhood of zero. Let us denote by φ the standard normal density.

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