Comparison of Approximations for Compound Poisson Processes

Abstract

In this paper, we compare the error in several approximation methods for the cumulative aggregate claim distribution customarily used in the collective model of insurance theory. In this model, it is usually supposed that a portfolio is at risk for a time period of length t. The occurrences of the claims are governed by a Poisson process of intensity μ so that the number of claims in [0, t] is a Poisson random variable with parameter λ = μt. Each single claim is an independent replication of the random variable X, representing the claim severity. The aggregate claim or total claim amount process in [0, t] is represented by the random sum of N independent replications of X, whose cumulative distribution function (cdf) is the object of study. Due to its computational complexity, several approximation methods for this cdf have been proposed. In this paper, we consider 15 approximations put forward in the literature that only use information on the lower order moments of the involved distributions. For each approximation, 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 their accuracy as λ = μt diverges to infinity. Using these expansions, several statements concerning the quality of approximations for the distribution of the aggregate claim process can find theoretical support. Other statements can be disproved on the same grounds. Finally, we investigate numerically the accuracy of the proposed formulas

Statistical characterization of the linear attenuation coefficient in polychromatic CT scans

“…Using the compound Poisson process as a starting model, we demonstrate that the compound Poisson obtained from conventional x-ray spectra can be accurately approximated by Gamma distributions. 12,13 This reasonable assumption allows us to extend the statistical characterization to the postlog transformation of the linear attenuation coefficient, where the post-log values are known to follow a non-Poisson distribution. As a result, we show that the post-log distribution of the sinogram intensities follows a Gumbel distribution.…”

“…. We also assume that the compound Poisson distribution can be accurately described by a single Gamma distribution, 12,13 with the following PDF: where Γ(·) is the Euler’s Gamma function, a is the shape parameter, and b is the scale parameter. In what follows, we will denote a random variable, X , to follow a Gamma distribution with parameters a and b as X ∼Γ( a , b ).…”

“…(3). We also assume that the compound Poisson distribution can be accurately described by a single Gamma distribution, 12,13 with the following PDF:…”

“…This result is consistent with the literature due to the excellent approximation that a single Gamma distribution offers for the sum of independent gamma-like variables, as it is the case of the energy spectrum. 13 Quantitatively, Fig. 3(c) shows the goodness-of-fit obtained with the uniform distance.…”

“…A more suitable approach can be adopted considering the good fitting of a single Gamma distribution to lower‐bounded infinitely divisible distributions such as the compound Poisson distribution 12,13 . The Gamma approximation also fits the high SNR scenario since it converges to a Gaussian distribution as the average photon energy increases.…”