@article{GaigallGerstenberg2023,
  author    = {Gaigall, Daniel and Gerstenberg, Julian},
  title     = {Cram{\´e}r-von-Mises tests for the distribution of the excess over a confidence level},
  series = {Journal of Nonparametric Statistics},
  journal   = {Journal of Nonparametric Statistics},
  publisher = {Taylor \& Francis},
  issn      = {1048-5252 (Print)},
  doi       = {10.1080/10485252.2023.2173958},
  year      = {2023},
  abstract  = {The Cram{\´e}r-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 Cram{\´e}r-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.},
  language  = {en}
}
@article{Gaigall2023,
  author    = {Gaigall, Daniel},
  title     = {Allocating and forecasting changes in risk},
  series = {Journal of risk},
  volume    = {25},
  journal   = {Journal of risk},
  number    = {3},
  editor    = {AitSahlia, Farid},
  publisher = {Infopro Digital Risk},
  address   = {London},
  issn      = {1755-2842},
  doi       = {10.21314/JOR.2022.048},
  pages     = {1 -- 24},
  year      = {2023},
  abstract  = {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.},
  language  = {en}
}
@article{Gaigall2023,
  author    = {Gaigall, Daniel},
  title     = {On the applicability of several tests to models with not identically distributed random effects},
  series = {Statistics : A Journal of Theoretical and Applied Statistics},
  volume    = {57},
  journal   = {Statistics : A Journal of Theoretical and Applied Statistics},
  publisher = {Taylor \& Francis},
  address   = {London},
  isbn      = {0323-3944},
  issn      = {1029-4910},
  doi       = {10.1080/02331888.2023.2193748},
  pages     = {14 Seiten},
  year      = {2023},
  abstract  = {We consider Kolmogorov-Smirnov and Cram{\´e}r-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.},
  language  = {en}
}
@article{BaringhausGaigall2022,
  author    = {Baringhaus, Ludwig and Gaigall, Daniel},
  title     = {A goodness-of-fit test for the compound Poisson exponential model},
  series = {Journal of Multivariate Analysis},
  volume    = {195},
  journal   = {Journal of Multivariate Analysis},
  number    = {Article 105154},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0047-259X},
  doi       = {10.1016/j.jmva.2022.105154},
  year      = {2022},
  abstract  = {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.},
  language  = {en}
}
@article{BaringhausGaigall2017,
  author    = {Baringhaus, Ludwig and Gaigall, Daniel},
  title     = {On Hotelling's T² test in a special paired sample case},
  series = {Communications in Statistics - Theory and Methods},
  volume    = {48},
  journal   = {Communications in Statistics - Theory and Methods},
  number    = {2},
  publisher = {Taylor \& Francis},
  address   = {London},
  issn      = {1532-415X},
  doi       = {10.1080/03610926.2017.1408828},
  pages     = {257 -- 267},
  year      = {2017},
  abstract  = {In a special paired sample case, Hotelling's T² test based on the differences of the paired random vectors is the likelihood ratio test for testing the hypothesis that the paired random vectors have the same mean; with respect to a special group of affine linear transformations it is the uniformly most powerful invariant test for the general alternative of a difference in mean. We present an elementary straightforward proof of this result. The likelihood ratio test for testing the hypothesis that the covariance structure is of the assumed special form is derived and discussed. Applications to real data are given.},
  language  = {en}
}
@article{BaringhausGaigall2017,
  author    = {Baringhaus, Ludwig and Gaigall, Daniel},
  title     = {Hotelling's T² tests in paired and independent survey samples: An efficiency comparison},
  series = {Journal of Multivariate Analysis},
  volume    = {2017},
  journal   = {Journal of Multivariate Analysis},
  number    = {154},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0047-259X},
  doi       = {10.1016/j.jmva.2016.11.004},
  pages     = {177 -- 198},
  year      = {2017},
  abstract  = {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.},
  language  = {en}
}
@article{BaringhausGaigall2015,
  author    = {Baringhaus, Ludwig and Gaigall, Daniel},
  title     = {On an independence test approach to the goodness-of-fit problem},
  series = {Journal of Multivariate Analysis},
  volume    = {2015},
  journal   = {Journal of Multivariate Analysis},
  number    = {140},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0047-259X},
  doi       = {10.1016/j.jmva.2015.05.013},
  pages     = {193 -- 208},
  year      = {2015},
  abstract  = {Let X₁,…,Xₙ be independent and identically distributed random variables with distribution F. Assuming that there are measurable functions f:R²→R and g:R²→R characterizing a family F of distributions on the Borel sets of R in the way that the random variables f(X₁,X₂),g(X₁,X₂) are independent, if and only if F∈F, we propose to treat the testing problem H:F∈F,K:F∉F by applying a consistent nonparametric independence test to the bivariate sample variables (f(Xᵢ,Xⱼ),g(Xᵢ,Xⱼ)),1⩽i,j⩽n,i≠j. A parametric bootstrap procedure needed to get critical values is shown to work. The consistency of the test is discussed. The power performance of the procedure is compared with that of the classical tests of Kolmogorov-Smirnov and Cram{\´e}r-von Mises in the special cases where F is the family of gamma distributions or the family of inverse Gaussian distributions.},
  language  = {en}
}
@article{DitzhausGaigall2022,
  author    = {Ditzhaus, Marc and Gaigall, Daniel},
  title     = {Testing marginal homogeneity in Hilbert spaces with applications to stock market returns},
  series = {Test},
  volume    = {2022},
  journal   = {Test},
  number    = {31},
  publisher = {Springer},
  issn      = {1863-8260},
  doi       = {10.1007/s11749-022-00802-5},
  pages     = {749 -- 770},
  year      = {2022},
  abstract  = {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 Cram{\´e}r-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.},
  language  = {en}
}
@article{GaigallGerstenbergTrinh2022,
  author    = {Gaigall, Daniel and Gerstenberg, Julian and Trinh, Thi Thu Ha},
  title     = {Empirical process of concomitants for partly categorial data and applications in statistics},
  series = {Bernoulli},
  volume    = {28},
  journal   = {Bernoulli},
  number    = {2},
  publisher = {International Statistical Institute},
  address   = {Den Haag, NL},
  issn      = {1573-9759},
  doi       = {10.3150/21-BEJ1367},
  pages     = {803 -- 829},
  year      = {2022},
  abstract  = {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.},
  language  = {en}
}
@article{Gaigall2021,
  author    = {Gaigall, Daniel},
  title     = {Test for Changes in the Modeled Solvency Capital Requirement of an Internal Risk Model},
  series = {ASTIN Bulletin},
  volume    = {51},
  journal   = {ASTIN Bulletin},
  number    = {3},
  publisher = {Cambridge Univ. Press},
  address   = {Cambridge},
  issn      = {1783-1350},
  doi       = {10.1017/asb.2021.20},
  pages     = {813 -- 837},
  year      = {2021},
  abstract  = {In the context of the Solvency II directive, the operation of an internal risk model is a possible way for risk assessment and for the determination of the solvency capital requirement of an insurance company in the European Union. A Monte Carlo procedure is customary to generate a model output. To be compliant with the directive, validation of the internal risk model is conducted on the basis of the model output. For this purpose, we suggest a new test for checking whether there is a significant change in the modeled solvency capital requirement. Asymptotic properties of the test statistic are investigated and a bootstrap approximation is justified. A simulation study investigates the performance of the test in the finite sample case and confirms the theoretical results. The internal risk model and the application of the test is illustrated in a simplified example. The method has more general usage for inference of a broad class of law-invariant and coherent risk measures on the basis of a paired sample.},
  language  = {en}
}
@article{Gaigall2020,
  author    = {Gaigall, Daniel},
  title     = {Hoeffding-Blum-Kiefer-Rosenblatt independence test statistic on partly not identically distributed data},
  series = {Communications in Statistics - Theory and Methods},
  volume    = {51},
  journal   = {Communications in Statistics - Theory and Methods},
  number    = {12},
  publisher = {Taylor \& Francis},
  address   = {London},
  issn      = {1532-415X},
  doi       = {10.1080/03610926.2020.1805767},
  pages     = {4006 -- 4028},
  year      = {2020},
  abstract  = {The established Hoeffding-Blum-Kiefer-Rosenblatt independence test statistic is investigated for partly not identically distributed data. Surprisingly, it turns out that the statistic has the well-known distribution-free limiting null distribution of the classical criterion under standard regularity conditions. An application is testing goodness-of-fit for the regression function in a non parametric random effects meta-regression model, where the consistency is obtained as well. Simulations investigate size and power of the approach for small and moderate sample sizes. A real data example based on clinical trials illustrates how the test can be used in applications.},
  language  = {en}
}
@article{Gaigall2020,
  author    = {Gaigall, Daniel},
  title     = {Testing marginal homogeneity of a continuous bivariate distribution with possibly incomplete paired data},
  series = {Metrika},
  volume    = {2020},
  journal   = {Metrika},
  number    = {83},
  publisher = {Springer},
  issn      = {1435-926X},
  doi       = {10.1007/s00184-019-00742-5},
  pages     = {437 -- 465},
  year      = {2020},
  abstract  = {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 Cr{\´a}mer-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.},
  language  = {en}
}
@article{Gaigall2020,
  author    = {Gaigall, Daniel},
  title     = {Rothman-Woodroofe symmetry test statistic revisited},
  series = {Computational Statistics \& Data Analysis},
  volume    = {2020},
  journal   = {Computational Statistics \& Data Analysis},
  number    = {142},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0167-9473},
  doi       = {10.1016/j.csda.2019.106837},
  pages     = {Artikel 106837},
  year      = {2020},
  abstract  = {The Rothman-Woodroofe symmetry test statistic is revisited on the basis of independent but not necessarily identically distributed random variables. The distribution-freeness if the underlying distributions are all symmetric and continuous is obtained. The results are applied for testing symmetry in a meta-analysis random effects model. The consistency of the procedure is discussed in this situation as well. A comparison with an alternative proposal from the literature is conducted via simulations. Real data are analyzed to demonstrate how the new approach works in practice.},
  language  = {en}
}
@article{BaringhausGaigall2019,
  author    = {Baringhaus, Ludwig and Gaigall, Daniel},
  title     = {On an asymptotic relative efficiency concept based on expected volumes of confidence regions},
  series = {Statistics - A Journal of Theoretical and Applied Statistic},
  volume    = {53},
  journal   = {Statistics - A Journal of Theoretical and Applied Statistic},
  number    = {6},
  publisher = {Taylor \& Francis},
  address   = {London},
  issn      = {1029-4910},
  doi       = {10.1080/02331888.2019.1683560},
  pages     = {1396 -- 1436},
  year      = {2019},
  abstract  = {The paper deals with an asymptotic relative efficiency concept for confidence regions of multidimensional parameters that is based on the expected volumes of the confidence regions. Under standard conditions the asymptotic relative efficiencies of confidence regions are seen to be certain powers of the ratio of the limits of the expected volumes. These limits are explicitly derived for confidence regions associated with certain plugin estimators, likelihood ratio tests and Wald tests. Under regularity conditions, the asymptotic relative efficiency of each of these procedures with respect to each one of its competitors is equal to 1. The results are applied to multivariate normal distributions and multinomial distributions in a fairly general setting.},
  language  = {en}
}
@article{Gaigall2019,
  author    = {Gaigall, Daniel},
  title     = {On a new approach to the multi-sample goodness-of-fit problem},
  series = {Communications in Statistics - Simulation and Computation},
  volume    = {53},
  journal   = {Communications in Statistics - Simulation and Computation},
  number    = {10},
  publisher = {Taylor \& Francis},
  address   = {London},
  issn      = {1532-4141},
  doi       = {10.1080/03610918.2019.1618472},
  pages     = {2971 -- 2989},
  year      = {2019},
  abstract  = {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.},
  language  = {en}
}
@article{DitzhausGaigall2018,
  author    = {Ditzhaus, Marc and Gaigall, Daniel},
  title     = {A consistent goodness-of-fit test for huge dimensional and functional data},
  series = {Journal of Nonparametric Statistics},
  volume    = {30},
  journal   = {Journal of Nonparametric Statistics},
  number    = {4},
  publisher = {Taylor \& Francis},
  address   = {Abingdon},
  issn      = {1029-0311},
  doi       = {10.1080/10485252.2018.1486402},
  pages     = {834 -- 859},
  year      = {2018},
  abstract  = {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 Cram{\´e}r-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.},
  language  = {en}
}
@article{BaringhausGaigallThiele2018,
  author    = {Baringhaus, Ludwig and Gaigall, Daniel and Thiele, Jan Philipp},
  title     = {Statistical inference for L²-distances to uniformity},
  series = {Computational Statistics},
  volume    = {2018},
  journal   = {Computational Statistics},
  number    = {33},
  publisher = {Springer},
  address   = {Berlin},
  issn      = {1613-9658},
  doi       = {10.1007/s00180-018-0820-0},
  pages     = {1863 -- 1896},
  year      = {2018},
  abstract  = {The paper deals with the asymptotic behaviour of estimators, statistical tests and confidence intervals for L²-distances to uniformity based on the empirical distribution function, the integrated empirical distribution function and the integrated empirical survival function. Approximations of power functions, confidence intervals for the L²-distances and statistical neighbourhood-of-uniformity validation tests are obtained as main applications. The finite sample behaviour of the procedures is illustrated by a simulation study.},
  language  = {en}
}