TY - JOUR A1 - Baringhaus, Ludwig A1 - Gaigall, Daniel T1 - A goodness-of-fit test for the compound Poisson exponential model JF - Journal of Multivariate Analysis 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. KW - Bootstrapping KW - Collective risk model Y1 - 2022 U6 - https://doi.org/10.1016/j.jmva.2022.105154 SN - 0047-259X SN - 1095-7243 VL - 195 IS - Article 105154 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Gaigall, Daniel ED - AitSahlia, Farid T1 - Allocating and forecasting changes in risk JF - Journal of risk N2 - We consider time-dependent portfolios and discuss the allocation of changes in the risk of a portfolio to changes in the portfolio’s components. For this purpose we adopt established allocation principles. We also use our approach to obtain forecasts for changes in the risk of the portfolio’s components. To put the approach into practice we present an implementation based on the output of a simulation. Allocation is illustrated with an example portfolio in the context of Solvency II. The quality of the forecasts is investigated with an empirical study. KW - portfolio risk KW - allocation KW - forecast KW - covariance principle KW - conditional expectation principle Y1 - 2023 U6 - https://doi.org/10.21314/JOR.2022.048 SN - 1755-2842 SN - 1465-1211 VL - 25 IS - 3 SP - 1 EP - 24 PB - Infopro Digital Risk CY - London ER - TY - JOUR A1 - Gaigall, Daniel T1 - On the applicability of several tests to models with not identically distributed random effects JF - Statistics : A Journal of Theoretical and Applied Statistics N2 - We consider Kolmogorov–Smirnov and Cramér–von-Mises type tests for testing central symmetry, exchangeability, and independence. In the standard case, the tests are intended for the application to independent and identically distributed data with unknown distribution. The tests are available for multivariate data and bootstrap procedures are suitable to obtain critical values. We discuss the applicability of the tests to random effects models, where the random effects are independent but not necessarily identically distributed and with possibly unknown distributions. Theoretical results show the adequacy of the tests in this situation. The quality of the tests in models with random effects is investigated by simulations. Empirical results obtained confirm the theoretical findings. A real data example illustrates the application. KW - central symmetry test KW - exchangeability test KW - independence test KW - random effects KW - not identically distributed Y1 - 2023 SN - 0323-3944 U6 - https://doi.org/10.1080/02331888.2023.2193748 SN - 1029-4910 VL - 57 PB - Taylor & Francis CY - London ER - TY - JOUR A1 - Gaigall, Daniel A1 - Gerstenberg, Julian T1 - Cramér-von-Mises tests for the distribution of the excess over a confidence level JF - Journal of Nonparametric Statistics N2 - The Cramér-von-Mises distance is applied to the distribution of the excess over a confidence level. Asymptotics of related statistics are investigated, and it is seen that the obtained limit distributions differ from the classical ones. For that reason, quantiles of the new limit distributions are given and new bootstrap techniques for approximation purposes are introduced and justified. The results motivate new one-sample goodness-of-fit tests for the distribution of the excess over a confidence level and a new confidence interval for the related fitting error. Simulation studies investigate size and power of the tests as well as coverage probabilities of the confidence interval in the finite sample case. A practice-oriented application of the Cramér-von-Mises tests is the determination of an appropriate confidence level for the fitting approach. The adoption of the idea to the well-known problem of threshold detection in the context of peaks over threshold modelling is sketched and illustrated by data examples. KW - Cramér-von-Mises test KW - conditional excess distribution KW - confidence interval KW - goodness-of-fit test Y1 - 2023 U6 - https://doi.org/10.1080/10485252.2023.2173958 SN - 1048-5252 (Print) SN - 1029-0311 (Online) PB - Taylor & Francis ER - TY - JOUR A1 - Gaigall, Daniel T1 - Testing marginal homogeneity of a continuous bivariate distribution with possibly incomplete paired data JF - Metrika N2 - We discuss the testing problem of homogeneity of the marginal distributions of a continuous bivariate distribution based on a paired sample with possibly missing components (missing completely at random). Applying the well-known two-sample Crámer–von-Mises distance to the remaining data, we determine the limiting null distribution of our test statistic in this situation. It is seen that a new resampling approach is appropriate for the approximation of the unknown null distribution. We prove that the resulting test asymptotically reaches the significance level and is consistent. Properties of the test under local alternatives are pointed out as well. Simulations investigate the quality of the approximation and the power of the new approach in the finite sample case. As an illustration we apply the test to real data sets. KW - Marginal homogeneity test KW - Crámer–von-Mises distance KW - Paired sample KW - Incomplete data KW - Resampling test Y1 - 2019 U6 - https://doi.org/10.1007/s00184-019-00742-5 SN - 1435-926X VL - 2020 IS - 83 SP - 437 EP - 465 PB - Springer ER - TY - JOUR A1 - Gaigall, Daniel T1 - On a new approach to the multi-sample goodness-of-fit problem JF - Communications in Statistics - Simulation and Computation N2 - Suppose we have k samples X₁,₁,…,X₁,ₙ₁,…,Xₖ,₁,…,Xₖ,ₙₖ with different sample sizes ₙ₁,…,ₙₖ and unknown underlying distribution functions F₁,…,Fₖ as observations plus k families of distribution functions {G₁(⋅,ϑ);ϑ∈Θ},…,{Gₖ(⋅,ϑ);ϑ∈Θ}, each indexed by elements ϑ from the same parameter set Θ, we consider the new goodness-of-fit problem whether or not (F₁,…,Fₖ) belongs to the parametric family {(G₁(⋅,ϑ),…,Gₖ(⋅,ϑ));ϑ∈Θ}. New test statistics are presented and a parametric bootstrap procedure for the approximation of the unknown null distributions is discussed. Under regularity assumptions, it is proved that the approximation works asymptotically, and the limiting distributions of the test statistics in the null hypothesis case are determined. Simulation studies investigate the quality of the new approach for small and moderate sample sizes. Applications to real-data sets illustrate how the idea can be used for verifying model assumptions. KW - Goodness-of-fit test KW - Multi-sample problem KW - Parametric bootstrap Y1 - 2019 U6 - https://doi.org/10.1080/03610918.2019.1618472 SN - 1532-4141 VL - 53 IS - 10 SP - 2971 EP - 2989 PB - Taylor & Francis CY - London ER - TY - JOUR A1 - Ditzhaus, Marc A1 - Gaigall, Daniel T1 - A consistent goodness-of-fit test for huge dimensional and functional data JF - Journal of Nonparametric Statistics N2 - A nonparametric goodness-of-fit test for random variables with values in a separable Hilbert space is investigated. To verify the null hypothesis that the data come from a specific distribution, an integral type test based on a Cramér-von-Mises statistic is suggested. The convergence in distribution of the test statistic under the null hypothesis is proved and the test's consistency is concluded. Moreover, properties under local alternatives are discussed. Applications are given for data of huge but finite dimension and for functional data in infinite dimensional spaces. A general approach enables the treatment of incomplete data. In simulation studies the test competes with alternative proposals. KW - Cramér-von-Mises statistic KW - separable Hilbert space KW - huge dimensional data KW - functional data Y1 - 2018 U6 - https://doi.org/10.1080/10485252.2018.1486402 SN - 1029-0311 VL - 30 IS - 4 SP - 834 EP - 859 PB - Taylor & Francis CY - Abingdon ER - TY - JOUR A1 - Gaigall, Daniel A1 - Gerstenberg, Julian A1 - Trinh, Thi Thu Ha T1 - Empirical process of concomitants for partly categorial data and applications in statistics JF - Bernoulli N2 - On the basis of independent and identically distributed bivariate random vectors, where the components are categorial and continuous variables, respectively, the related concomitants, also called induced order statistic, are considered. The main theoretical result is a functional central limit theorem for the empirical process of the concomitants in a triangular array setting. A natural application is hypothesis testing. An independence test and a two-sample test are investigated in detail. The fairly general setting enables limit results under local alternatives and bootstrap samples. For the comparison with existing tests from the literature simulation studies are conducted. The empirical results obtained confirm the theoretical findings. KW - bootstrap KW - Categorial variable KW - Concomitant KW - Empirical process KW - Independence test Y1 - 2022 U6 - https://doi.org/10.3150/21-BEJ1367 SN - 1573-9759 VL - 28 IS - 2 SP - 803 EP - 829 PB - International Statistical Institute CY - Den Haag, NL ER - TY - JOUR A1 - Baringhaus, Ludwig A1 - Gaigall, Daniel T1 - Hotelling’s T² tests in paired and independent survey samples: An efficiency comparison JF - Journal of Multivariate Analysis N2 - Hotelling’s T² tests in paired and independent survey samples are compared using the traditional asymptotic efficiency concepts of Hodges–Lehmann, Bahadur and Pitman, as well as through criteria based on the volumes of corresponding confidence regions. Conditions characterizing the superiority of a procedure are given in terms of population canonical correlation type coefficients. Statistical tests for checking these conditions are developed. Test statistics based on the eigenvalues of a symmetrized sample cross-covariance matrix are suggested, as well as test statistics based on sample canonical correlation type coefficients. Y1 - 2017 U6 - https://doi.org/10.1016/j.jmva.2016.11.004 SN - 0047-259X VL - 2017 IS - 154 SP - 177 EP - 198 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Ditzhaus, Marc A1 - Gaigall, Daniel T1 - Testing marginal homogeneity in Hilbert spaces with applications to stock market returns JF - Test N2 - This paper considers a paired data framework and discusses the question of marginal homogeneity of bivariate high-dimensional or functional data. The related testing problem can be endowed into a more general setting for paired random variables taking values in a general Hilbert space. To address this problem, a Cramér–von-Mises type test statistic is applied and a bootstrap procedure is suggested to obtain critical values and finally a consistent test. The desired properties of a bootstrap test can be derived that are asymptotic exactness under the null hypothesis and consistency under alternatives. Simulations show the quality of the test in the finite sample case. A possible application is the comparison of two possibly dependent stock market returns based on functional data. The approach is demonstrated based on historical data for different stock market indices. Y1 - 2022 U6 - https://doi.org/10.1007/s11749-022-00802-5 SN - 1863-8260 VL - 2022 IS - 31 SP - 749 EP - 770 PB - Springer ER -