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This paper provides an introductory overview of a portion of distribution theory which is currently under intense development. The starting point of this topic has been the so-called skew-normal distribution, but the connected area is becoming increasingly broad, and its branches include now many extensions, such as the skew-elliptical families, and some forms of semiparametric formulations, extending the relevance of the field much beyond the original theme of 'skewness'. The final part of the paper illustrates connections with various areas of application, including selective sampling, models for compositional data, robust methods, some problems in econometrics, non-linear time series, especially in connection with financial data, and more.

In present study, a calculator, probDistriCal, was developed for probability distributions. In the calculator, the probability distributions as normal distribution, t distribution, F distribution, chi-square distribution, exponential distribution, power law distribution, Weibull distribution, Zhang distribution, binomial distribution, Poission distribution, and negative binomial distribution, etc., were available for use. Given a random value, the corresponding probability for a known probability distribution can be calculated. The calculator is web browser based that includes both online and offline versions and can be used on various computing devices (PCs, iPads, smartphones, etc.), operating systems (Windows, Mac, Android, Harmony, etc.) and web browsers (Chrome, Firefox, etc).

A fairly general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities. The approach is sufficiently general to encompass some recent proposals in the literature, variously related to the skew normal distribution. The special case of skew elliptical densities is examined in detail, establishing connections with existing similar work. The final part of the paper specializes further to a form of multivariate skew t-density. Likelihood inference for this distribution is examined, and it is illustrated with numerical examples.

The distribution theory literature connected to the multivariate skew-normal distribution has grown rapidly in recent years, and a number of extensions and alternative formulations have been put forward. Presently there are various coexisting proposals, similar but not identical, and with rather unclear connections. The aim of this paper is to unify these proposals under a new general formulation, clarifying at the same time their relationships. The final part sketches an extension of the argument to the skew-elliptical family.

Skew-symmetric families of distributions such as the skew-normal and skew-t represent supersets of the normal and t distributions, and they exhibit richer classes of extremal behaviour. By defining a non-stationary skew-normal process, which allows the easy handling of positive definite, non-stationary covariance functions, we derive a new family of max-stable processes – the extremal skew-t process. This process is a superset of non-stationary processes that include the stationary extremal-t processes. We provide the spectral representation and the resulting angular densities of the extremal skew-t process and illustrate its practical implementation.

In this paper, we introduce the skew-symmetric generalized normal and the skew-symmetric generalized t distributions, which are skewed extensions of symmetric special cases of generalized skew-normal and generalized skew-t distributions, respectively. We derive key distributional properties for these new distributions, including a recurrence relation and an explicit form for the cumulative distribution function (cdf) of the skew-symmetric generalized t distribution. Numerical examples including a simulation study and a real data analysis are presented to illustrate the practical applicability of these distributions.

Deep learning has gained immense attention from researchers in medicine, especially in medical imaging. The main bottleneck is the unavailability of sufficiently large medical datasets required for the good performance of deep learning models. This paper proposes a new framework consisting of one variational autoencoder (VAE), two generative adversarial networks, and one auxiliary classifier to artificially generate realistic-looking skin lesion images and improve classification performance. We first train the encoder-decoder network to obtain the latent noise vector with the image manifold’s information and let the generative adversarial network sample the input from this informative noise vector in order to generate the skin lesion images. The use of informative noise allows the GAN to avoid mode collapse and creates faster convergence. To improve the diversity in the generated images, we use another GAN with an auxiliary classifier, which samples the noise vector from a heavy-tailed student t-distribution instead of a random noise Gaussian distribution. The proposed framework was named TED-GAN, with T from the t-distribution and ED from the encoder-decoder network which is part of the solution. The proposed framework could be used in a broad range of areas in medical imaging. We used it here to generate skin lesion images and have obtained an improved classification performance on the skin lesion classification task, rising from 66% average accuracy to 92.5%. The results show that TED-GAN has a better impact on the classification task because of its diverse range of generated images due to the use of a heavy-tailed t-distribution.

In ecological systems, extremes can happen in time, such as population crashes, or in space, such as rapid range contractions. However, current methods for joint inference about temporal and spatial dynamics (e.g., spatiotemporal modeling with Gaussian random fields) may perform poorly when underlying processes include extreme events. Here we introduce a model that allows for extremes to occur simultaneously in time and space. Our model is a Bayesian predictive-process GLMM (generalized linear mixed-effects model) that uses a multivariate-t distribution to describe spatial random effects. The approach is easily implemented with our flexible R package glmmfields. First, using simulated data, we demonstrate the ability to recapture spatiotemporal extremes, and explore the consequences of fitting models that ignore such extremes. Second, we predict tree mortality from mountain pine beetle (Dendroctonus ponderosae) outbreaks in the U.S. Pacific Northwest over the last 16 yr. We show that our approach provides more accurate and precise predictions compared to traditional spatiotemporal models when extremes are present. Our R package makes these models accessible to a wide range of ecologists and scientists in other disciplines interested in fitting spatiotemporal GLMMs, with and without extremes.

Let X be a random variable with finite second moment. We investigate the inequality: P{|X-E[X]|≤Var(X)}≥P{|Z|≤1}, where Z is a standard normal random variable. We prove that this inequality holds for many familiar infinitely divisible continuous distributions including the Laplace, Gumbel, Logistic, Pareto, infinitely divisible Weibull, Log-normal, Student’s t and Inverse Gaussian distributions. Numerical results are given to show that the inequality with continuity correction also holds for some infinitely divisible discrete distributions.

We introduce the sinh-arcsinh transformation and hence, by applying it to a generating distribution with no parameters other than location and scale, usually the normal, a new family of sinh-arcsinh distributions. This four-parameter family has symmetric and skewed members and allows for tailweights that are both heavier and lighter than those of the generating distribution. The central place of the normal distribution in this family affords likelihood ratio tests of normality that are superior to the state-of-the-art in normality testing because of the range of alternatives against which they are very powerful. Likelihood ratio tests of symmetry are also available and are very successful. Three-parameter symmetric and asymmetric subfamilies of the full family are also of interest. Heavy-tailed symmetric sinh-arcsinh distributions behave like Johnson SU distributions, while their light-tailed counterparts behave like sinh-normal distributions, the sinh-arcsinh family allowing a seamless transition between the two, via the normal, controlled by a single parameter. The sinh-arcsinh family is very tractable and many properties are explored. Likelihood inference is pursued, including an attractive reparameterization. Illustrative examples are given. A multivariate version is considered. Options and extensions are discussed.