@phdthesis{Gaigall2023, author = {Gaigall, Daniel}, title = {On selected problems in multivariate analysis}, doi = {10.15488/14304}, pages = {17 Seiten}, year = {2023}, abstract = {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.}, 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{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{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} } @phdthesis{Gaigall2016, author = {Gaigall, Daniel}, title = {Vergleich von statistischen Tests im verbundenen und unabh{\"a}ngigen Stichprobenfall}, publisher = {Gottfried Wilhelm Leibniz Universit{\"a}t Hannover}, address = {Hannover}, doi = {10.15488/8678}, pages = {281 Seiten}, year = {2016}, abstract = {Es werden Effizienzbegriffe zum Vergleich von statistischen Tests basierend auf verschiedenen statistischen Experimenten eingef{\"u}hrt. 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 f{\"u}r Likelihood-Quotienten-Tests und Waldsche Tests im Rahmen eines allgemeinen multivariaten parametrischen Modells erhalten. Statistische Tests zur Pr{\"u}fung von Hypothesen {\"u}ber 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 unabh{\"a}ngiger 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 Konfidenzbereichssch{\"a}tzverfahren werden aufgezeigt. Ausf{\"u}hrlichere Untersuchungen betreffen die korrespondierenden Hotellingschen T²-Tests im multivariaten Normalverteilungsfall, die klassischen Homogenitatstests bei k × k-Kontingenztafeln und die Wilcoxon Tests in nichtparametrischen Lagealternativmodellen}, language = {de} } @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{BaringhausGaigall2018, author = {Baringhaus, Ludwig and Gaigall, Daniel}, title = {Efficiency comparison of the Wilcoxon tests in paired and independent survey samples}, series = {Metrika}, volume = {2018}, journal = {Metrika}, number = {81}, publisher = {Springer}, address = {Berlin}, issn = {1435-926X}, doi = {10.1007/s00184-018-0661-4}, pages = {891 -- 930}, year = {2018}, abstract = {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.}, language = {de} } @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} }