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This article studies the estimation of the precision matrix of a high-dimensional Gaussian network. We investigate the graphical selector operator with shrinkage, GSOS for short, to maximize a penalized likelihood function where the elastic net-type penalty is considered as a combination of a norm-one penalty and a targeted Frobenius norm penalty. Numerical illustrations demonstrate that our proposed methodology is a competitive candidate for high-dimensional precision matrix estimation compared to some existing alternatives. We demonstrate the relevance and efficiency of GSOS using a foreign exchange markets dataset and estimate dependency networks for 32 different currencies from 2018 to 2021. [ABSTRACT FROM AUTHOR]

Wind energy production depends not only on wind speed but also on wind direction. Thus, predicting and estimating the wind direction for sites accurately will enhance measuring the wind energy potential. The uncertain nature of wind direction can be presented through probability distributions and Bayesian analysis can improve the modeling of the wind direction using the contribution of the prior knowledge to update the empirical shreds of evidence. This must align with the nature of the empirical evidence as to whether the data are skew or multimodal or not. So far mixtures of von Mises within the directional statistics domain, are used for modeling wind direction to capture the multimodality nature present in the data. In this paper, due to the skewed and multimodal patterns of wind direction on different sites of the locations understudy, a mixture of multimodal skewed von Mises is proposed for wind direction. Furthermore, a Bayesian analysis is presented to take into account the uncertainty inherent in the proposed wind direction model. A simulation study is conducted to evaluate the performance of the proposed Bayesian model. This proposed model is fitted to datasets of wind direction of Marion island and two wind farms in South Africa and show the superiority of the approach. The posterior predictive distribution is applied to forecast the wind direction on a wind farm. It is concluded that the proposed model offers an accurate prediction by means of credible intervals. The mean wind direction of Marion island in 2017 obtained from 1079 observations was 5.0242 (in radian) while using our proposed method the predicted mean wind direction and its corresponding 95 % credible interval based on 100 generated samples from the posterior predictive distribution are obtained 5.0171 and (4.7442, 5.2900). Therefore, our results open a new approach for accurate prediction of wind direction implementing a Bayesian approach via mixture of skew circular distributions. [ABSTRACT FROM AUTHOR]

This paper introduces a new family of matrix variate distributions based on the mean-mixture of normal (MMN) models. The properties of the new matrix variate family, namely stochastic representation, moments and characteristic function, linear and quadratic forms as well as marginal and conditional distributions are investigated. Three special cases including the restricted skew-normal, exponentiated MMN and the mixed-Weibull MMN matrix variate distributions are presented and studied. Based on the specific presentation of the proposed model, an EM-type algorithm can be directly implemented for obtaining maximum likelihood estimate of the parameters. The usefulness and practical utility of the proposed methodology are illustrated through two conducted simulation studies and through the Landsat satellite dataset analysis. [ABSTRACT FROM AUTHOR]

This research is inspired from monitoring the process covariance structure of q attributes where samples are independent, having been collected from a multivariate normal distribution with known mean vector and unknown covariance matrix. The focus is on two matrix random variables, constructed from different Wishart ratios, that describe the process for the two consecutive time periods before and immediately after the change in the covariance structure took place. The product moments of these constructed random variables are highlighted and set the scene for a proposed measure to enable the practitioner to calculate the run-length probability to detect a shift immediately after a change in the covariance matrix occurs. Our results open a new approach and provides insight for detecting the change in the parameter structure as soon as possible once the underlying process, described by a multivariate normal process, encounters a permanent/sustained upward or downward shift. [ABSTRACT FROM AUTHOR]

The normal distribution and its perturbation have left an immense mark on the statistical literature. Several generalized forms exist to model different skewness, kurtosis, and body shapes. Although they provide better fitting capabilities, these generalizations do not have parameters and formulae with a clear meaning to the practitioner on how the distribution is being modeled. We propose a neat integration approach generalization which intuitively gives direct control of the body and tail shape, the body-tail generalized normal (BTGN). The BTGN provides the basis for a flexible distribution, emphasizing parameter interpretation, estimation properties, and tractability. Basic statistical measures are derived, such as the density function, cumulative density function, moments, moment generating function. Regarding estimation, the equations for maximum likelihood estimation and maximum product spacing estimation are provided. Finally, real-life situations data, such as log-returns, time series, and finite mixture modeling, are modeled using the BTGN. Our results show that it is possible to have more desirable traits in a flexible distribution while still providing a superior fit to industry-standard distributions, such as the generalized hyperbolic, generalized normal, tail-inflated normal, and t distributions. [ABSTRACT FROM AUTHOR]

The Dirichlet distribution is a well-known candidate in modeling compositional data sets. However, in the presence of outliers, the Dirichlet distribution fails to model such data sets, making other model extensions necessary. In this paper, the Kummer–Dirichlet distribution and the gamma distribution are coupled, using the beta-generating technique. This development results in the proposal of the Kummer–Dirichlet gamma distribution, which presents greater flexibility in modeling compositional data sets. Some general properties, such as the probability density functions and the moments are presented for this new candidate. The method of maximum likelihood is applied in the estimation of the parameters. The usefulness of this model is demonstrated through the application of synthetic and real data sets, where outliers are present. [ABSTRACT FROM AUTHOR]

Entropy indicates a measure of information contained in a complex system, and its estimation continues to receive ongoing focus in the case of multivariate data, particularly that on the unit simplex. Oftentimes the Dirichlet distribution is employed as choice of prior in a Bayesian framework conjugate to the popular multinomial likelihood with K distinct classes, where consideration of Shannon- and Tsallis entropy is of interest for insight detection within the data on the simplex. However, this prior choice only accounts for negatively correlated data, therefore this paper incorporates previously unconsidered mixtures of Dirichlet distributions as potential priors for the multinomial likelihood which addresses the drawback of negative correlation. The power sum functional, as the product moment of the mixture of Dirichlet distributions, is of direct interest in the multivariate case to conveniently access the Tsallis- and other generalized entropies that is incorporated within an estimation perspective of the posterior distribution using real economic data. A prior selection method is implemented to suggest a suitable prior for the consideration of the practitioner; empowering the user in future for consideration of suitable priors incorporating entropy within the estimation environment as well as having the option of certain mixture of Dirichlet distributions that may require positive correlation. [ABSTRACT FROM AUTHOR]

The joint modeling of angular and linear observations is crucial as data of this nature are prevalent in multiple disciplines, for example the joint modeling of wind direction and another climatological variable such as wind speed or air temperature, the direction an animal moves and the distance moved, or wave direction and wave height. Hence, there is a need for developing flexible distributions on the hyper-disc, which has support of the interior of the hyper-sphere, as it allows for modeling the combination of angular and linear observations. This paper addresses this need by developing flexible distributions for the disc that have the ability to capture any inherent bimodality present in the data. A new class of bivariate distributions is proposed which has support on the unit disc in two dimensions that includes, as a special case, the existing Möbius distribution on the disc. This class is obtained by expressing the density function in a general form using a measurable function termed as generator. Special cases of this generator are considered to demonstrate the flexibility. By applying a conformal mapping to the generator function a new Möbius distribution class emanates. This class of bivariate distributions on the disc is the first to account for bimodality and skewness present in the data. The flexible behavior of the proposed models in terms of bimodality and skewness is graphically demonstrated. Preliminary evidential analysis of the wind data observed at Marion Island reveals the absence of unimodality in the data. The fit of the proposed models, which account for bimodality, to the Marion Island wind data were evaluated analytically and visually. [ABSTRACT FROM AUTHOR]

Insurance loss data have peculiar features that can rarely be accounted for by simple parametric distributions. Thus, in this manuscript, we first introduce a new type of location mixture model: the mode mixture. By using convenient mode-parameterized hump-shaped distributions, we present a family of eight mode mixture of unimodal distributions. Then, we fit these models to two real insurance loss datasets, where they are evaluated in terms of goodness of fit and ability to reproduce classical risk measures. We extend the comparisons to existing models based on mode-parameterized hump-shaped distributions. Lastly, using simulated data, we further investigate the performance of the estimated risk measures of our models. [ABSTRACT FROM AUTHOR]

Proteins are found in all living organisms and constitute a large group of macromolecules with many functions. Proteins achieve their operations by adopting distinct three-dimensional structures encoded within the sequence of the constituent amino acids in one or more polypeptides. New, more flexible distributions are proposed for the MCMC sampling method for predicting protein 3D structures by applying a Möbius transformation to the bivariate von Mises distribution. In addition to this, sine-skewed versions of the proposed models are introduced to meet the increasing demand for modelling asymmetric toroidal data. Interestingly, the marginals of the new models lead to new multimodal circular distributions. We analysed three big datasets consisting of bivariate information about protein domains to illustrate the efficiency and behaviour of the proposed models. These newly proposed models outperformed mixtures of well-known models for modelling toroidal data. A simulation study was carried out to find the best method for generating samples from the proposed models. Our results shed new light on proposal distributions in the MCMC sampling method for predicting the protein structure environment. [ABSTRACT FROM AUTHOR]

Entropy is a functional of probability and is a measurement of information contained in a system; however, the practical problem of estimating entropy in applied settings remains a challenging and relevant problem. The Dirichlet prior is a popular choice in the Bayesian framework for estimation of entropy when considering a multinomial likelihood. In this work, previously unconsidered Dirichlet type priors are introduced and studied. These priors include a class of Dirichlet generators as well as a noncentral Dirichlet construction, and in both cases includes the usual Dirichlet as a special case. These considerations allow for flexible behaviour and can account for negative and positive correlation. Resultant estimators for a particular functional, the power sum, under these priors and assuming squared error loss, are derived and represented in terms of the product moments of the posterior. This representation facilitates closed-form estimators for the Tsallis entropy, and thus expedite computations of this generalised Shannon form. Select cases of these proposed priors are considered to investigate the impact and effect on the estimation of Tsallis entropy subject to different parameter scenarios. [ABSTRACT FROM AUTHOR]