Comparison of Approximations for Compound Poisson Processes Choirat, Christine University of Navarra, Department of Economics Edificio de Bibliotecas (Entrada Este) 31080 Pamplona, Spain E-mail: cchoirat@unav.es Seri, Raffaello University of Insubria, Department of Economics Via Monte Generoso 71 21100 Varese, Italy E-mail: raffaello.seri@uninsubria.it Introduction The aim of this paper 1 is to provide a comparison of the error in several approximation methods for the cumulative aggregate claim distribution customarily used in the collective model of insurance theory (see e.g. Cram´ er, 1955, or Beard et al., 1990). In this theory it is usually supposed that a portfolio is at risk for a time period of length t. The claims take place according to a Poisson process of intensity µ, so that the number of claims in [0,t] is a Poisson random variable N with parameter λ = µt. Each single claim is a random variable X i for i =1,...,N with a common distribution. We consider the random sum S N = ∑ N i=1 X i , i.e. a compound Poisson process representing the aggregate claim or total claim amount process in [0,t]. We denote as µ i the i-th noncentral moment of X i . Then the aggregate claim process has moments: ES N = µ 1 λ, V (S N )= µ 2 λ. We will write γ 1 := µ 3 µ 3/2 2 λ 1/2 (skewness index), γ 2 := µ 4 µ 2 2 λ (kurtosis index) and γ 3 := µ 5 µ 5/2 2 λ 3/2 . Then let S ⋆ N := S N -ES N √ V(S N ) . We evaluate the accuracy of nine approximations available in the literature to: F (x) := P {S ⋆ N ≤ x} , as the Poisson intensity diverges to infinity, i.e. as λ →∞. We consider the difference between the true distribution and the approximating one and we propose to use expansions of this difference related to Edgeworth series to measure the accuracy of the approximation. In order to do so, we will need the Hermite polynomials He j (x) for j =0, 1,... . The first Hermite polynomials, in the formulation customarily used in Statistics, are given by the formulas He 0 (x) = 1, He 1 (x)= x, He 2 (x)= x 2 - 1 and He 3 (x)= x 3 - 3x. Edgeworth Expansion The following Edgeworth expansion for compound Poisson processes (see e.g. Cram´ er, 1955) will be used in the following. Theorem. Consider a compound Poisson process with intensity λ. Let µ j ’s be the noncentral moments of the random variable X, whose characteristic function is such that lim |t|→∞ sup |φ (t)| < 1. 1 The present short paper is based on a longer one by the same authors in which several more methods are considered. The original paper contains full proofs of the results as well as more complete historical accounts concerning their introduction in Insurance. Int. Statistical Inst.: Proc. 58th World Statistical Congress, 2011, Dublin (Session CPS018) p.4290