@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{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} } @inproceedings{Gaigall2022, author = {Gaigall, Daniel}, title = {On Consistent Hypothesis Testing In General Hilbert Spaces}, publisher = {Avestia Publishing}, address = {Orl{\´e}ans, Kanada}, doi = {10.11159/icsta22.157}, pages = {Paper No. 157}, year = {2022}, abstract = {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 Cram{\´e}rvon-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 L{\´e}vy 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.}, 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{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} }