entropies. Maximizing this entropy without any constraints produces an extremely U-shaped (=bipolar) distribution. Maximizing the cumulative entropy under the constraints of fixed mean and variance tries to transform a distribution in the direction of a bipolar distribution, as far as it is allowed by the constraints. A bipolar distribution represents so-called contradictory information, which is in contrast to minimum or no information. In the literature, to date, only a few maximum entropy distributions for cumulative entropies have been derived. In this paper, we extended the results to well known flexible distributions (like the generalized logistic distribution) and derived some special distributions (like the skewed logistic, the skewed Tukey 𝜆

and the extended Burr XII distribution). The generalized maximum entropy principle was applied to the generalized Tukey 𝜆

distribution and the Fechner family of skewed distributions. Finally, cumulative entropies were estimated such that the data was drawn from a maximum entropy distribution. This estimator will be applied to the daily S&P500 returns and time durations between mine explosion}, author = {Klein, Ingo and Doll, Monika}, doi = {10.3390/e22010091}, faupublication = {yes}, journal = {Entropy}, month = {Jan}, pages = {91}, peerreviewed = {Yes}, title = {({Generalized}) {Maximum} {Cumulative} {Direct}, {Residual}, and {Paired} {Φ} {Entropy} {Approach}}, volume = {22}, year = {2020} } @article{faucris.231459027, author = {Klein, Ingo and Doll, Monika}, doi = {10.3390}, faupublication = {yes}, journal = {Entropy}, month = {Jan}, peerreviewed = {Yes}, title = {({Generalized}) {Maximum} {Cumulative} {Direct}, {Residual}, and {Paired} {Φ} {Entropy} {Approach}}, year = {2020} } @incollection{faucris.117690364, address = {Boston}, author = {Klein, Ingo and Fischer, Matthias}, booktitle = {Contributions to modern econometrics}, doi = {10.1007/978-1-4757-3602-1}, editor = {Klein I., Mittnik S.}, faupublication = {yes}, isbn = {978-4419-5331-5}, pages = {119-134}, peerreviewed = {No}, publisher = {Kluwer Academic Publishers}, title = {gh-transformed of symmetrical distributions}, year = {2002} } @article{faucris.117683764, author = {Klein, Ingo and Fischer, Matthias}, doi = {10.1007/s101820400158}, faupublication = {yes}, journal = {Allgemeines Statistisches Archiv}, keywords = {Kurtosis - variable transformation - normal transformation - tail elongation}, note = {WiSo-Import:2015-03-26:970}, pages = {35-50}, peerreviewed = {Yes}, title = {{Kurtosis} modelling by means of the {J}-transformation}, volume = {88}, year = {2004} } @incollection{faucris.114213484, address = {Nürnberg}, author = {Klein, Ingo}, booktitle = {Versicherungswissenschaft - im Wandel wirtschaftsrechtlicher und rechtsökonomischer Analysen mit Finanzmarkt- und Freiberufsrecht. FS für Harald Herrmann}, editor = {Holzer-Thieser A., Roth S.}, faupublication = {yes}, isbn = {978-3-934674-00-4}, note = {WiSo-Import:2015-04-20:3075}, pages = {323-331}, peerreviewed = {No}, publisher = {IF Verlag}, title = {{Möglichkeiten} der empirischen {Erfassung} von {Automietpreisen}}, year = {2011} } @misc{faucris.113763584, author = {Klein, Ingo and Grottke, Michael}, faupublication = {yes}, title = {{On} {J}.{M}. {Keynes}' "{The} {Principal} {Averages} and the {Laws} of {Error} which {Lead} to {Them}" - {Refinement} and {Generalisation}}, url = {https://www.iwf.rw.fau.de/research/iwf-discussion-paper-series/}, year = {2008} } @article{faucris.113303344, author = {Klein, Ingo and Fischer, Matthias}, doi = {10.1007/s10182-006-0241-1}, faupublication = {yes}, journal = {Advances in Statistical Analysis}, note = {WiSo-Import:2015-03-26:974}, pages = {395-401}, peerreviewed = {Yes}, title = {{Power} {Kurtosis} {Transformations}: {De} finitions, {Properties} and {Ordering}}, volume = {90}, year = {2006} } @article{faucris.275134712, abstract = {Sample size analysis is a key part of the planning phase of any research. So far, however, hardly any literature focuses on sample size analysis methods for two-sample linear rank tests, although these methods have optimal properties for different distributions. This article provides a new sample size analysis method for linear rank tests for location shift alternatives based on score-generating functions. Results show a slightly anti-conservative behavior, no severe risk of an occurring circular argument at small to moderate variances of the population’s distribution, and good performance compared to alternate sample size analysis methods for the most well-known linear rank test, the Wilcoxon-Mann-Whitney tes}, author = {Klein, Ingo and Doll, Monika}, doi = {10.1080/03610926.2022.2068029}, faupublication = {yes}, journal = {Communications in Statistics-Theory and Methods}, peerreviewed = {Yes}, title = {{Sample} size analysis for two-sample linear rank tests}, url = {https://www.tandfonline.com/doi/full/10.1080/03610926.2022.2068029}, year = {2022} } @article{faucris.117684644, abstract = {There are several possibilities for introducing skewness into a symmetric distribution. One of these procedures applies two different parameters of scale - with possibly different weights - to the positive and the negative part of a symmetric density. Within this work we show that this technique incorporates a well-defined parameter of skewness, i.e., that the generated distributions are skewed to the right (left) if the parameter of skewness takes values less (greater) than one. Second, we prove that the skewness parameter is compatible with the skewness ordering of van Zwet (1964), which is the strongest ordering in the hierarchy of orderings discussed by Oja (1981). Hence the generated (skewed) distributions can be ordered by the skewness paramete}, author = {Klein, Ingo and Fischer, Matthias}, doi = {10.1080/03610920600692730}, faupublication = {yes}, journal = {Communications in Statistics-Theory and Methods}, keywords = {Score function; Skewness; Skewness ordering; Skewness to the right}, note = {WiSo-Import:2015-03-26:991}, pages = {1159-1171}, peerreviewed = {Yes}, title = {{Skewness} by {Splitting} the scale parameter}, volume = {35}, year = {2006} } @article{faucris.284498152, abstract = {Negation of a discrete probability distribution was introduced by Yager. To date, several papers have been published discussing generalizations, properties, and applications of negation. The recent work by Wu et al. gives an excellent overview of the literature and the motivation to deal with negation. Our paper focuses on some technical aspects of negation transformations. First, we prove that independent negations must be affine-linear. This fact was established by Batyrshin et al. as an open problem. Secondly, we show that repeated application of independent negations leads to a progressive loss of information (called monotonicity). In contrast to the literature, we try to obtain results not only for special but also for the general class of (Formula presented.) -entropies. In this general framework, we can show that results need to be proven only for Yager negation and can be transferred to the entire class of independent (=affine-linear) negations. For general (Formula presented.) -entropies with strictly concave generator function (Formula presented.), we can show that the information loss increases separately for sequences of odd and even numbers of repetitions. By using a Lagrangian approach, this result can be extended, in the neighbourhood of the uniform distribution, to all numbers of repetition. For Gini, Shannon, Havrda–Charvát (Tsallis), Rényi and Sharma–Mittal entropy, we prove that the information loss has a global minimum of 0. For dependent negations, it is not easy to obtain analytical results. Therefore, we simulate the entropy distribution and show how different repeated negations affect Gini and Shannon entropy. The simulation approach has the advantage that the entire simplex of discrete probability vectors can be considered at once, rather than just arbitrarily selected probability vectors.}, author = {Klein, Ingo}, doi = {10.3390/math10203893}, faupublication = {yes}, journal = {Mathematics}, keywords = {(h,ϕ)-entropy; Dirichlet distribution; Gini entropy; Havrda–Charvát (Tsallis) entropy; Monte Carlo simulation; negation; Rényi entropy; Shannon entropy; Sharma–Mittal entropy; ϕ-entropy}, note = {CRIS-Team Scopus Importer:2022-11-04}, peerreviewed = {Yes}, title = {{Some} {Technical} {Remarks} on {Negations} of {Discrete} {Probability} {Distributions} and {Their} {Information} {Loss}}, volume = {10}, year = {2022} } @article{faucris.111632664, author = {Klein, Ingo and Köck, Christian and Tinkl, Fabian}, faupublication = {yes}, journal = {Ekonomia Menedżerska}, keywords = {multivariate GARCH models; Copulas; univariate GARCH models}, note = {WiSo-Import:2015-03-26:997}, pages = {43-62}, peerreviewed = {Yes}, title = {{Spatial}-serial dependency in multivariate {GARCH} models and dynamic copulas: {A} simulation study}, volume = {7}, year = {2010} } @article{faucris.116972724, author = {Klein, Ingo and Stroebel, Armin and Bergner, Matthias and Reulbach, Udo and Biermann, Teresa and Groemer, Teja W. and Kornhuber, Johannes}, faupublication = {yes}, journal = {Journal of Circadian Rhythms}, note = {WiSo-Import:2015-03-26:998}, pages = {1-10}, peerreviewed = {Yes}, title = {{Statistical} methods for detecting and comparing periodic data and their application to the circadian rhythm of bodily harm}, url = {http://www.jcircadianrhythms.com/articles/10.1186/1740-3391-8-10/}, volume = {8}, year = {2010} } @incollection{faucris.114242304, address = {Bamberg}, author = {Klein, Ingo}, booktitle = {Arbeiten aus der Statistik - Festschrift für F. Vogel}, editor = {Dobbener R, Kolb U, Wan Hussin D}, faupublication = {yes}, note = {WiSo-Import:2015-04-20:973}, pages = {1-18}, peerreviewed = {No}, title = {{Streuungsmessung} ordinalskalierter {Merkmale} mittels {Rangordnungsstatistiken}}, year = {2005} } @incollection{faucris.110280324, address = {Boston}, author = {Klein, Ingo and Fischer, Matthias}, booktitle = {Contributions to Modern Econometrics}, editor = {Klein Ingo, Mittnik Stefan}, faupublication = {yes}, isbn = {1-4020-7334-8}, pages = {119-134}, peerreviewed = {No}, publisher = {Kluwer Academic Publishers}, title = {{Symmetrical} gh-transformed distributions}, year = {2002} } @incollection{faucris.117689924, address = {Stuttgart}, author = {Klein, Ingo and Fischer, Matthias}, booktitle = {Finanzintermediation: Theoretische, wirtschaftspolitische und praktische Aspekte aktueller Entwicklungen im Bank- und Börsenwesen. Festschrift für Wolfgang Gehrke zum 60. Geburtstag}, editor = {M. Bank, B. Schiller}, faupublication = {yes}, isbn = {3791022571}, note = {WiSo-Import:2015-04-20:971}, pages = {69-102}, peerreviewed = {No}, publisher = {Schäffer-Pöschel Verlag}, title = {{Tailabhängigkeit} und {Asymmetrie} in multivariaten {Finanzmarktdaten}}, year = {2004} } @article{faucris.237896453, abstract = {Skewness is a well-established statistical concept for continuous and, to a lesser extent, for discrete quantitative statistical variables. However, for ordered categorical variables, limited literature concerning skewness exists, although this type of variables is common for behavioral, educational, and social sciences. Suitable measures of skewness for ordered categorical variables have to be invariant with respect to the group of strictly increasing, continuous transformations. Therefore, they have to depend on the corresponding maximal-invariants. Based on these maximal-invariants, we propose a new class of skewness functionals, show that members of this class preserve a suitable ordering of skewness and derive the asymptotic distribution of the corresponding skewness statistic. Finally, we show the good power behavior of the corresponding skewness tests and illustrate these tests by applying real data example}, author = {Klein, Ingo and Doll, Monika}, doi = {10.1080/02664763.2020.1757045}, faupublication = {yes}, journal = {Journal of Applied Statistics}, keywords = {Ordered categorical variables, skewness analysis, skewness ordering, maximal invariants}, peerreviewed = {Yes}, title = {{Tests} on asymmetry for ordered categorical variables}, year = {2020} } @article{faucris.117286884, abstract = {Using the Gaussian distribution as probabilistic model for (leptokurtic) financial data is widespread, especially in practice. However, departure from normality seems to be more the rule than the exception. The H-distributions, introduced by Tukey (1960, 1977), are generated by a single transformation (H-transformation) of the standard normal distribution or, more generally, of a symmetric "parent" distribution Z and allow for leptokurtosis represented by the (elongation) parameter H > 0. Alternatively, the J-distributions of Fischer and Klein (2004) or the K-distributions of Haynes et al. (1997) may be applied. In order to additionally introduce skewness, some have these distribution families have been generalized subsequently. Within this work we "complete" the class of so-called Tukey-type distributions by introducing KQ- and JQ-distributions, on the one side, and KK-, JJ-, and [image omitted]-distributions, on the other side. Moreover, we investigate the goodness-of-fit of such Tukey-type distributions for different parent distributions Z in the context of financial return data. In particular, the interplay between Z and different transformations is focussed. Finally, our results are compared to those of popular multi-parametric distribution model}, author = {Fischer, Matthias and Klein, Ingo and Horn, Armin}, doi = {10.1080/03610920500476382}, faupublication = {yes}, journal = {Communications in Statistics-Theory and Methods}, keywords = {Financial return data; Kurtosis; Normal transformation; Skewness; Variable transformation}, note = {WiSo-Import:2015-03-26:993}, pages = {23-35}, peerreviewed = {Yes}, title = {{Tukey}-type {Distributions} in the context of financial data}, volume = {36}, year = {2007} } @misc{faucris.110648384, author = {Klein, Ingo}, faupublication = {yes}, title = {{Van} {Zwet} {Ordering} and the {Ferreira}-{Steel} {Family} of {Skewed} {Distributions}}, url = {https://www.iwf.rw.fau.de/research/iwf-discussion-paper-series/}, year = {2011} } @misc{faucris.115983824, author = {Klein, Ingo}, faupublication = {yes}, title = {{Van} {Zwet} {Ordering} for {Fechner} {Asymmetry}}, url = {https://www.iwf.rw.fau.de/research/iwf-discussion-paper-series/}, year = {2011} } @article{faucris.112015684, author = {Klein, Ingo and Fischer, Matthias and Pleier, Thomas}, doi = {10.3233/MAS-180436}, faupublication = {yes}, journal = {Model Assisted Statistics and Applications}, pages = {253-270}, peerreviewed = {Yes}, title = {{Weighted} {Power} {Mean} {Copulas}: {Theory} and {Application}}, volume = {13}, year = {2018} }