Dokument-ID Dokumenttyp Verfasser/Autoren Herausgeber Haupttitel Abstract Auflage Verlagsort Verlag Erscheinungsjahr Seitenzahl Schriftenreihe Titel Schriftenreihe Bandzahl ISBN Quelle der Hochschulschrift Konferenzname Bemerkung Quelle:Titel Quelle:Jahrgang Quelle:Heftnummer Quelle:Erste Seite Quelle:Letzte Seite URN DOI Zugriffsart Link Abteilungen OPUS4-10755 Habilitation Gaigall, Daniel, gaigall@fh-aachen.de On selected problems in multivariate analysis 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. 2023 17 Seiten Gottfried Wilhelm Leibniz Universität Hannover 10.15488/14304 weltweit https://doi.org/10.15488/14304 Fachbereich Medizintechnik und Technomathematik OPUS4-10541 Wissenschaftlicher Artikel Gaigall, Daniel, gaigall@fh-aachen.de; Gerstenberg, Julian, Cramér-von-Mises tests for the distribution of the excess over a confidence level 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. Taylor & Francis 2023 Journal of Nonparametric Statistics 10.1080/10485252.2023.2173958 bezahl https://doi.org/10.1080/10485252.2023.2173958 Fachbereich Medizintechnik und Technomathematik OPUS4-10588 Wissenschaftlicher Artikel Gaigall, Daniel, Gaigall@fh-aachen.de AitSahlia, Farid Allocating and forecasting changes in risk 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. London Infopro Digital Risk 2023 23 Journal of risk 25 3 1 24 10.21314/JOR.2022.048 bezahl https://doi.org/10.21314/JOR.2022.048https://doi.org/10.1080/02331888.2023.2193748 Fachbereich Medizintechnik und Technomathematik OPUS4-10373 Wissenschaftlicher Artikel Baringhaus, Ludwig, ; Gaigall, Daniel, gaigall@fh-aachen.de 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. Amsterdam Elsevier 2022 Journal of Multivariate Analysis 195 Article 105154 10.1016/j.jmva.2022.105154 bezahl https://doi.org/10.1016/j.jmva.2022.105154 Fachbereich Medizintechnik und Technomathematik OPUS4-10415 Wissenschaftlicher Artikel Baringhaus, Ludwig, ; Gaigall, Daniel, gaigall@fh-aachen.de; Thiele, Jan Philipp, Statistical inference for L²-distances to uniformity 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. Berlin Springer 2018 33 Computational Statistics 2018 33 1863 1896 10.1007/s00180-018-0820-0 campus https://doi.org/10.1007/s00180-018-0820-0 Fachbereich Medizintechnik und Technomathematik OPUS4-10416 Wissenschaftlicher Artikel Baringhaus, Ludwig, ; Gaigall, Daniel, gaigall@fh-aachen.de Efficiency comparison of the Wilcoxon tests in paired and independent survey samples The efficiency concepts of Bahadur and Pitman are used to compare the Wilcoxon tests in paired and independent survey samples. A comparison through the length of corresponding confidence intervals is also done. Simple conditions characterizing the dominance of a procedure are derived. Statistical tests for checking these conditions are suggested and discussed. Berlin Springer 2018 39 Metrika 2018 81 891 930 10.1007/s00184-018-0661-4 campus https://doi.org/10.1007/s00184-018-0661-4 OPUS4-10417 Wissenschaftlicher Artikel Baringhaus, Ludwig, ; Gaigall, Daniel, gaigall@fh-aachen.de On Hotelling's T² test in a special paired sample case 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. London Taylor & Francis 2017 10 Communications in Statistics - Theory and Methods 48 2 257 267 10.1080/03610926.2017.1408828 bezahl https://doi.org/10.1080/03610926.2017.1408828 Fachbereich Medizintechnik und Technomathematik OPUS4-10419 Wissenschaftlicher Artikel Baringhaus, Ludwig, ; Gaigall, Daniel, gaigall@fh-aachen.de Hotelling's T² tests in paired and independent survey samples: An efficiency comparison 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. Amsterdam Elsevier 2017 21 Journal of Multivariate Analysis 2017 154 177 198 10.1016/j.jmva.2016.11.004 weltweit https://doi.org/10.1016/j.jmva.2016.11.004 Fachbereich Medizintechnik und Technomathematik OPUS4-10420 Wissenschaftlicher Artikel Baringhaus, Ludwig, ; Gaigall, Daniel, gaigall@fh-aachen.de On an independence test approach to the goodness-of-fit problem 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 Cramér-von Mises in the special cases where F is the family of gamma distributions or the family of inverse Gaussian distributions. Amsterdam Elsevier 2015 15 Journal of Multivariate Analysis 2015 140 193 208 10.1016/j.jmva.2015.05.013 weltweit https://doi.org/10.1016/j.jmva.2015.05.013 Fachbereich Medizintechnik und Technomathematik OPUS4-10423 Dissertation Gaigall, Daniel, gaigall@fh-aachen.de Vergleich von statistischen Tests im verbundenen und unabhängigen Stichprobenfall Es werden Effizienzbegriffe zum Vergleich von statistischen Tests basierend auf verschiedenen statistischen Experimenten eingeführt. Dabei handelt es sich um die schon aus dem Vergleich von statistischen Tests in je demselben Modell bekannten asymptotischen relativen Effizienzen wie die Hodges-Lehmann-Effizienz, die Bahadur-Effizienz und die Pitman-Effizienz sowie um Kriterien basierend auf Volumina von Konfidenzbereichen. Effizienzaussagen werden unter anderem für Likelihood-Quotienten-Tests und Waldsche Tests im Rahmen eines allgemeinen multivariaten parametrischen Modells erhalten. Statistische Tests zur Prüfung von Hypothesen über die relative Wirksamkeit zweier Experimente werden vorgeschlagen. Auf der Grundlage der erhaltenen Ergebnisse erfolgt ein Vergleich der Wirksamkeit von korrespondierenden Verfahren bei verbundener Stichprobenerhebung und unabhängiger Stichprobenerhebung. Die Rolle der Kovarianzmatrix bei verbundener Stichprobenerhebung wird insbesondere unter der Annahme, dass die zugrunde liegenden Verteilungen durch k-parametrische Exponentialfamilien modellierbar sind, herausgearbeitet. Verbindungen zu Effizienzbegriffen bei Punkt- und Konfidenzbereichsschätzverfahren werden aufgezeigt. Ausführlichere Untersuchungen betreffen die korrespondierenden Hotellingschen T²-Tests im multivariaten Normalverteilungsfall, die klassischen Homogenitatstests bei k × k-Kontingenztafeln und die Wilcoxon Tests in nichtparametrischen Lagealternativmodellen Hannover Gottfried Wilhelm Leibniz Universität Hannover 2016 281 Seiten Dissertation, Gottfried Wilhelm Leibniz Universität Hannover, 2016 10.15488/8678 weltweit https://doi.org/10.15488/8678 Fachbereich Medizintechnik und Technomathematik OPUS4-10405 Konferenzveröffentlichung Gaigall, Daniel, gaigall@fh-aachen.de On Consistent Hypothesis Testing In General Hilbert Spaces Inference on the basis of high-dimensional and functional data are two topics which are discussed frequently in the current statistical literature. A possibility to include both topics in a single approach is working on a very general space for the underlying observations, such as a separable Hilbert space. We propose a general method for consistently hypothesis testing on the basis of random variables with values in separable Hilbert spaces. We avoid concerns with the curse of dimensionality due to a projection idea. We apply well-known test statistics from nonparametric inference to the projected data and integrate over all projections from a specific set and with respect to suitable probability measures. In contrast to classical methods, which are applicable for real-valued random variables or random vectors of dimensions lower than the sample size, the tests can be applied to random vectors of dimensions larger than the sample size or even to functional and high-dimensional data. In general, resampling procedures such as bootstrap or permutation are suitable to determine critical values. The idea can be extended to the case of incomplete observations. Moreover, we develop an efficient algorithm for implementing the method. Examples are given for testing goodness-of-fit in a one-sample situation in [1] or for testing marginal homogeneity on the basis of a paired sample in [2]. Here, the test statistics in use can be seen as generalizations of the well-known Cramérvon-Mises test statistics in the one-sample and two-samples case. The treatment of other testing problems is possible as well. By using the theory of U-statistics, for instance, asymptotic null distributions of the test statistics are obtained as the sample size tends to infinity. Standard continuity assumptions ensure the asymptotic exactness of the tests under the null hypothesis and that the tests detect any alternative in the limit. Simulation studies demonstrate size and power of the tests in the finite sample case, confirm the theoretical findings, and are used for the comparison with concurring procedures. A possible application of the general approach is inference for stock market returns, also in high data frequencies. In the field of empirical finance, statistical inference of stock market prices usually takes place on the basis of related log-returns as data. In the classical models for stock prices, i.e., the exponential Lévy model, Black-Scholes model, and Merton model, properties such as independence and stationarity of the increments ensure an independent and identically structure of the data. Specific trends during certain periods of the stock price processes can cause complications in this regard. In fact, our approach can compensate those effects by the treatment of the log-returns as random vectors or even as functional data. Orléans, Kanada Avestia Publishing 2022 Proceedings of the 4th International Conference on Statistics: Theory and Applications (ICSTA'22) Prague, Czech Republic - July 28- 30 Paper No. 157 10.11159/icsta22.157 bezahl http://dx.doi.org/10.11159/icsta22.157 Fachbereich Medizintechnik und Technomathematik OPUS4-10406 Wissenschaftlicher Artikel Ditzhaus, Marc, ; Gaigall, Daniel, gaigall@fh-aachen.de Testing marginal homogeneity in Hilbert spaces with applications to stock market returns 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. Springer 2022 21 Test 2022 31 749 770 10.1007/s11749-022-00802-5 weltweit https://doi.org/10.1007/s11749-022-00802-5 Fachbereich Medizintechnik und Technomathematik OPUS4-10407 Wissenschaftlicher Artikel Gaigall, Daniel, gaigall@fh-aachen.de; Gerstenberg, Julian, ; Trinh, Thi Thu Ha, Empirical process of concomitants for partly categorial data and applications in statistics 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. Den Haag, NL International Statistical Institute 2022 26 Bernoulli 28 2 803 829 10.3150/21-BEJ1367 bezahl https://doi.org/10.3150/21-BEJ1367 Fachbereich Medizintechnik und Technomathematik OPUS4-10408 Wissenschaftlicher Artikel Gaigall, Daniel, gaigall@fh-aachen.de Test for Changes in the Modeled Solvency Capital Requirement of an Internal Risk Model 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. Cambridge Cambridge Univ. Press 2021 24 ASTIN Bulletin 51 3 813 837 10.1017/asb.2021.20 campus https://doi.org/10.1017/asb.2021.20 Fachbereich Medizintechnik und Technomathematik OPUS4-10409 Wissenschaftlicher Artikel Gaigall, Daniel, gaigall@fh-aachen.de Hoeffding-Blum-Kiefer-Rosenblatt independence test statistic on partly not identically distributed data 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. London Taylor & Francis 2020 22 Communications in Statistics - Theory and Methods 51 12 4006 4028 10.1080/03610926.2020.1805767 bezahl https://doi.org/10.1080/03610926.2020.1805767 Fachbereich Medizintechnik und Technomathematik OPUS4-10410 Wissenschaftlicher Artikel Gaigall, Daniel, gaigall@fh-aachen.de Testing marginal homogeneity of a continuous bivariate distribution with possibly incomplete paired data 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. Springer 2020 28 Metrika 2020 83 437 465 10.1007/s00184-019-00742-5 weltweit https://doi.org/10.1007/s00184-019-00742-5 Fachbereich Medizintechnik und Technomathematik OPUS4-10411 Wissenschaftlicher Artikel Gaigall, Daniel, gaigall@fh-aachen.de Rothman-Woodroofe symmetry test statistic revisited 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. Amsterdam Elsevier 2020 Computational Statistics & Data Analysis 2020 142 Artikel 106837 10.1016/j.csda.2019.106837 bezahl https://doi.org/10.1016/j.csda.2019.106837 Fachbereich Medizintechnik und Technomathematik OPUS4-10412 Wissenschaftlicher Artikel Baringhaus, Ludwig, ; Gaigall, Daniel, gaigall@fh-aachen.de On an asymptotic relative efficiency concept based on expected volumes of confidence regions 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. London Taylor & Francis 2019 40 Statistics - A Journal of Theoretical and Applied Statistic 53 6 1396 1436 10.1080/02331888.2019.1683560 bezahl https://doi.org/10.1080/02331888.2019.1683560 Fachbereich Medizintechnik und Technomathematik OPUS4-10413 Wissenschaftlicher Artikel Gaigall, Daniel, gaigall@fh-aachen.de On a new approach to the multi-sample goodness-of-fit problem 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. London Taylor & Francis 2019 18 Communications in Statistics - Simulation and Computation 53 10 2971 2989 10.1080/03610918.2019.1618472 bezahl https://doi.org/10.1080/03610918.2019.1618472 Fachbereich Medizintechnik und Technomathematik OPUS4-10414 Wissenschaftlicher Artikel Ditzhaus, Marc, ; Gaigall, Daniel, gaigall@fh-aachen.de A consistent goodness-of-fit test for huge dimensional and functional data 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. Abingdon Taylor & Francis 2018 25 Journal of Nonparametric Statistics 30 4 834 859 10.1080/10485252.2018.1486402 bezahl https://doi.org/10.1080/10485252.2018.1486402 Fachbereich Medizintechnik und Technomathematik