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Univariate continuous distributions are one of the fundamental components on which statistical modelling, ancient and modern, frequentist and Bayesian, multi-dimensional and complex, is based. In this article, I review and compare some of the main general techniques for providing families of typically unimodal distributions on ℝ with one or two, or possibly even three, shape parameters, controlling skewness and/or tailweight, in addition to their all-important location and scale parameters. One important and useful family is comprised of the 'skew-symmetric' distributions brought to prominence by Azzalini. As these are covered in considerable detail elsewhere in the literature, I focus more on their complements and competitors. Principal among these are distributions formed by transforming random variables, by what I call 'transformation of scale'—including two-piece distributions—and by probability integral transformation of nonuniform random variables. I also treat briefly the issues of multi-variate extension, of distributions on subsets of ℝ and of distributions on the circle. The review and comparison is not comprehensive, necessarily being selective and therefore somewhat personal.

The sinh-arcsinh transform is used to obtain a flexible four-parameter model that provides a natural framework with which to perform inference robust to wide-ranging departures from the logistic distribution. Its basic properties are established and its distribution and quantile functions, and properties related to them, shown to be highly tractable. Two important subfamilies are also explored. Maximum likelihood estimation is discussed, and reparametrisations designed to reduce the asymptotic correlations between the maximum likelihood estimates provided. A likelihood-ratio test for logisticness, which outperforms standard empirical distribution function based tests, follows naturally. The application of the proposed model and inferential methods is illustrated in an analysis of carbon fibre strength data. Multivariate extensions of the model are explored.

This paper presents two approaches for qualitative, quantitative and comparative concepts of skewness to be defined with respect to the spatial median for multivariate distributions. They extend the known quantile-based notions defined for real distributions. The main tool for such extensions consists of a family of central parts that provide suitable generalizations of the real interquantile intervals.

This paper deals with a survey of different types of tests, parametric, nonparametric, robustified and adaptive ones, and with an application to the two-sided c-sample location problem. Some concepts of robustness are discussed, such as breakdown point, influence function, gross-error sensitivity and especially α- and β-robustness. A robustness study on level α in the case of heteroscedasticity and nonnormal distributions is carried out via Monte Carlo methods and also a power comparison of all the tests considered. It turns out that robustified versions of the F-test and Welch-test where the original observations are replaced by its ranks behave well over a broad class of distributions, symmetric ones with different tail weight and asymmetric ones, but, on the whole, an adaptive test is to prefer. [PUBLICATION ABSTRACT]

SummaryFor the general two-sample problem we propose an adaptive test which is based on tests of Kolmogorov-Smirnov- and Cramèr- von Mises type. These tests are modifications of the Kolmogorov-Smirnov- and Cramèr- von Mises tests by using various weight functions in order to obtain higher power than its classical counterparts for short-tailed and right-skewed distributions. In practice, however, we generally have no information about the underlying distribution of the data. Thus, an adaptive test should be applied which takes into account the given data. The proposed adaptive test is based on Hogg’s concept, i.e., first, to classify the unknown distribution function with respect to two measures, one for skewness and one for tailweight, and second, to use an appropriate test of Kolmogorov-Smimov- and Cramèr- von Mises type for this classified type of distribution. We compare the distribution-free adaptive test with the tests of Kolmogorov-Smirnov- and Cramèr-von Mises type as well as with the Lepage test and a modification of its in the case of location and scale alternatives including the same shape and different shapes of the distributions of the X- and Y- variables. The power comparison of the tests is carried out via Monte Carlo simulation assuming short-, medium- and long-tailed distributions as well as distributions skewed to the right. It turns out that, on the whole, the adaptive test is the best one for the broad class of distributions considered.

An increase in kurtosis is achieved through the location- and scale-free movement of probability mass from the \"shoulders\" of a distribution into its centre and tails. We introduce a coherent structure of ordering and measures, requiring no symmetry assumption, that represent different formalizations of this movement. For this purpose spread functions and spread-spread plots are defined. The orderings impose growth patterns on the spread-spread plot of the distributions involved, and the weakest involve both a specific scale-matching technique and placement of \"shoulders\". The role of existing kurtosis orderings and measures in this general context is identified and examples discussed throughout. /// On peut augmenter le coefficient d'aplatissement (kurtosis) par un mouvement de la masse de probabilité, ne dépendant pas de la position et de l'échelle, des \"épaules\" d'une distribution vers le centre et les ailes de celle-ci. On introduit une structure cohérente de relations d'ordre et de mesures, ne supposant pas la symétrie, qui représentent différentes formalisations de ce mouvement. À cette fin on définit des fonctions d'étendue et des graphiques étendue-étendue. Ces relations d'ordre imposent des formes de croissance sur les graphiques étendue-étendue des distributions impliquées. Les plus faibles de ces relations demandent une technique spécifique pour apparier l'échelle et la position des \"épaules\". Les mesures connues du coefficient d'aplatissement et les façons de les ordonner sont revues dans ce contexte général et des exemples sont présentés.

In areas such as financial and insurance risk and communication network design the heaviness of the tail of the underlying distribution is crucial for the calculations. However, although it seems straightforward theoretically to distinguish between (say) exponential tails and power tails, this requires unexpectedly large samples in practice. Here we will use quantiles to compare the tails of distributions which are standardised to unit interquartile range to allow for possible infinite variance. We present some chi-squared tests of goodness-of-fit focussed on the tails. We also provide methods of quick comparison of distributions using counts over high thresholds and using extreme values.