Compound distributions

Abstract

Let N be a counting random variable with probability function q n = Pr{N = n}, n = 0, 1, 2,. . .. Also, let {X n , n = 1, 2,. . .} be a sequence of independent and identically distributed positive random variables (also independent of N) with common distribution function P (x). The distribution of the random sum S = X 1 + X 2 + · · · + X N , (1) with the convention that S = 0, if N = 0, is called a compound distribution. The distribution of N is often referred to as the primary distribution and the distribution of X n is referred to as the secondary distribution. Compound distributions arise from many applied probability models and from insurance risk models in particular. For instance, a compound distribution may be used to model the aggregate claims from an insurance portfolio for a given period of time. In this context, N represents the number of claims (claim frequency) from the portfolio, {X n , n = 1, 2,. . .} represents the consecutive individual claim amounts (claim severities), and the random sum S represents the aggregate claims amount. Hence, we also refer to the primary distribution and the secondary distribution as the claim frequency distribution and the claim severity distribution, respectively. Two compound distributions are of special importance in actuarial applications. The compound Pois-son distribution is often a popular choice for aggregate claims modeling because of its desirable properties. The computational advantage of the compound Poisson distribution enables us to easily evaluate the aggregate claims distribution when there are several underlying independent insurance portfolios and/or limits, and deductibles are applied to individual claim amounts. The compound negative binomial distribution may be used for modeling the aggregate claims from a nonhomogeneous insurance portfolio. In this context, the number of claims follows a (con-ditional) Poisson distribution with a mean that varies by individual and has a gamma distribution. It also has applications to insurances with the possibility of multiple claims arising from a single event or accident such as automobile insurance and medical insurance. Furthermore, the compound geometric distribution, as a special case of the compound negative binomial distribution, plays a vital role in analysis of ruin probabilities and related problems in risk theory. Detailed discussions of the compound risk models and their actuarial applications can be found in [10, 14]. For applications in ruin problems see [2]. Basic properties of the distribution of S can be obtained easily. By conditioning on the number of claims, one can express the distribution function F S (x) of S as F S (x) = ∞ n=0 q n P * n (x), x ≥ 0, (2) or 1 − F S (x) = ∞ n=1 q n [1 − P * n (x)], x ≥ 0, (3) where P * n (x) is the distribution function of the n-fold convolution of P (x) with itself, that is P * 0 (x) = 1, and P * n (x) = Pr n i=1 X i ≤ x for n ≥ 1. Similarly, the mean, variance, and the Laplace transform are obtainable as follows. E(S) = E(N)E(X n), Var(S) = E(N)Var(X n) + Var(N)[E(X n)] 2 , (4) and˜f and˜ and˜f S (z) = P N (˜ p X (z)) , (5) where P N (z) = E{z N } is the probability generating function of N, and˜fand˜ and˜f S (z) = E{e −zS } and˜pand˜ and˜p X (z) = E{e −zX n } are the Laplace transforms of S and X n , assuming that they all exist. The asymptotic behavior of a compound distribution naturally depends on the asymptotic behavior of its frequency and severity distributions. As the distribution of N in actuarial applications is often light tailed, the right tail of the compound distribution tends to have the same heaviness (light, medium, or heavy) as that of the severity distribution P (x). Chapter 10 of [14] gives a short introduction on the definition of light, medium, and heavy tailed distributions and asymptotic results of compound distributions based on these classifications. The derivation of these results can be found in [6] (for heavy-tailed distributions), [7] (for light-tailed distributions), [18] (for medium-tailed distributions) and references therein.