A goodness-of-fit test for the compound Poisson exponential model

On the basis of bivariate data, assumed to be observations of independent copies of a random vector (S, N), we consider testing the hypothesis that the distribution of (S, N) belongs to the parametric class of distributions that arise with the compound Poisson exponential model. Typically, this model is used in stochastic hydrology, with N as the number of raindays, and S as total rainfall amount during a certain time period, or in actuarial science, with N as the number of losses, and S as total loss expenditure during a certain time period. The compound Poisson exponential model is characterized in the way that a specific transform associated with the distribution of (S, N) satisfies a certain differential equation. Mimicking the function part of this equation by substituting the empirical counterparts of the transform we obtain an expression the weighted integral of the square of which is used as test statistic. We deal with two variants of the latter, one of which being invariant under scale transformations of the S-part by fixed positive constants. Critical values are obtained by using a parametric bootstrap procedure. The asymptotic behavior of the tests is discussed. A simulation study demonstrates the performance of the tests in the finite sample case. The procedure is applied to rainfall data and to an actuarial dataset. A multivariate extension is also discussed.(c) 2022 Elsevier Inc. All rights reserved.

Automated summary

This study focuses on testing the hypothesis that the distribution of bivariate data (S, N) belongs to the parametric class of distributions in the compound Poisson exponential model. The compound Poisson exponential model is often used in stochastic hydrology and actuarial science, involving variables like raindays and total rainfall amount or loss frequency and total loss expenditure. A specific transform associated with the distribution of (S, N) satisfies a certain differential equation, and this study proposes a test statistic based on the empirical counterparts of the transform. Critical values are obtained using a parametric bootstrap procedure, and the asymptotic behavior of the tests is discussed. The performance of the tests is demonstrated through simulation studies on rainfall data and an actuarial dataset, and a potential multivariate extension is also discussed.