TY - THES A1 - Gaigall, Daniel T1 - On selected problems in multivariate analysis N2 - Selected problems in the field of multivariate statistical analysis are treated. Thereby, one focus is on the paired sample case. Among other things, statistical testing problems of marginal homogeneity are under consideration. In detail, properties of Hotelling‘s T² test in a special parametric situation are obtained. Moreover, the nonparametric problem of marginal homogeneity is discussed on the basis of possibly incomplete data. In the bivariate data case, properties of the Hoeffding-Blum-Kiefer-Rosenblatt independence test statistic on the basis of partly not identically distributed data are investigated. Similar testing problems are treated within the scope of the application of a result for the empirical process of the concomitants for partly categorial data. Furthermore, testing changes in the modeled solvency capital requirement of an insurance company by means of a paired sample from an internal risk model is discussed. Beyond the paired sample case, a new asymptotic relative efficiency concept based on the expected volumes of multidimensional confidence regions is introduced. Besides, a new approach for the treatment of the multi-sample goodness-of-fit problem is presented. Finally, a consistent test for the treatment of the goodness-of-fit problem is developed for the background of huge or infinite dimensional data. KW - Paired sample KW - Marginal homogeneity KW - Incomplete data KW - Asymptotic relative efficiency KW - Volumes of confidence regions Y1 - 2023 U6 - http://dx.doi.org/10.15488/14304 N1 - Gottfried Wilhelm Leibniz Universität Hannover 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 - http://dx.doi.org/10.1080/02331888.2023.2193748 SN - 1029-4910 VL - 57 PB - Taylor & Francis CY - London 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 - http://dx.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 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 - http://dx.doi.org/10.1080/10485252.2023.2173958 SN - 1048-5252 (Print) SN - 1029-0311 (Online) PB - Taylor & Francis ER - TY - JOUR A1 - Baringhaus, Ludwig A1 - Gaigall, Daniel T1 - On Hotelling’s T² test in a special paired sample case JF - Communications in Statistics - Theory and Methods N2 - 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. KW - complete block symmetry KW - Hotelling’s T² test KW - likelihood ratio test KW - uniformly most powerful invariant test Y1 - 2017 U6 - http://dx.doi.org/10.1080/03610926.2017.1408828 SN - 1532-415X VL - 48 IS - 2 SP - 257 EP - 267 PB - Taylor & Francis CY - London 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 - http://dx.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 - Baringhaus, Ludwig A1 - Gaigall, Daniel T1 - On an asymptotic relative efficiency concept based on expected volumes of confidence regions JF - Statistics - A Journal of Theoretical and Applied Statistic N2 - 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. KW - Volume of confidence regions KW - asymptotic relative efficiency KW - likelihood ratio test KW - multivariate normal distribution KW - multinomial distribution Y1 - 2019 U6 - http://dx.doi.org/10.1080/02331888.2019.1683560 SN - 1029-4910 VL - 53 IS - 6 SP - 1396 EP - 1436 PB - Taylor & Francis CY - London 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 - http://dx.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 - http://dx.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 - Baringhaus, Ludwig A1 - Gaigall, Daniel A1 - Thiele, Jan Philipp T1 - Statistical inference for L²-distances to uniformity JF - Computational Statistics N2 - 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. KW - Integrated empirical distribution (survival) function KW - Goodness-of-fit tests for uniformity KW - Numerical inversion of Laplace transforms KW - Coverage probability KW - Equivalence test Y1 - 2018 U6 - http://dx.doi.org/10.1007/s00180-018-0820-0 SN - 1613-9658 VL - 2018 IS - 33 SP - 1863 EP - 1896 PB - Springer CY - Berlin ER -