TY - JOUR

T1 - A goodness-of-fit test for the compound Poisson exponential model

AU - Baringhaus, Ludwig

AU - Gaigall, Daniel

N1 - Funding information: The authors thank the Editor, the Associate Editor and the referees for constructive comments and suggestions.

PY - 2023/5

Y1 - 2023/5

N2 - 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.

AB - 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.

KW - Bootstrapping

KW - Collective risk model

KW - Compound Poisson exponential model

KW - Goodness-of-fit

UR - http://www.scopus.com/inward/record.url?scp=85145833324&partnerID=8YFLogxK

U2 - 10.1016/j.jmva.2022.105154

DO - 10.1016/j.jmva.2022.105154

M3 - Article

AN - SCOPUS:85145833324

VL - 195

JO - Journal of multivariate analysis

JF - Journal of multivariate analysis

SN - 0047-259X

M1 - 105154

ER -