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In this article, a large class of univriate Birnbaum–Saunders distributions based on the scale shape mixture of skew normal distributions is introduced which generates suitable subclasses for modeling asymmetric data in a variety of settings. The moments and maximum likelihood estimation procedures are disscused via an ECM-algorithm. The observed information matrix to approximate the asymptotic covariance matrix of the parameter estimates is then derived in some subclasses. A simulation study is also performed to evaluate the finite sample properties of ML estimators and finally, a real data set is analyzed for illustrative purposes.

In this paper, we establish efficient recursive algorithms for the computation of the cumulative distribution function (cdf) of multivariate Student’s t and multivariate unified skew-t distributions. The recurrence relations are over $$\nu $$ (the degrees of freedom), and starting from the explicit results for $$\nu $$ =1 and $$\nu $$ =2, they enable the recursive evaluation of the cdf for any positive integral value of $$\nu $$ . Using these, we obtain results for the computation of orthant probabilities of multivariate Student’s t distribution. We then demonstrate the usefulness of the established results in some problems involving order statistics and reliability systems. Finally, we use two real data sets to illustrate the methods established here.

A scale-shape mixtures of flexible generalized skew normal (SSMFGSN) distributions is proposed as a novel device for modeling asymmetric data. Computationally feasible EM-type algorithms derived from the selection mechanism are presented to compute maximum likelihood (ML) estimates of SSMFGSN distributions. Some characterizations and probabilistic properties of the SSMFGSN distributions are also studied. Monte Carlo simulations show that the proposed estimating procedures can provide desirable asymptotic properties of the ML estimates and demand less computational burden in comparison with other existing algorithms based on convolution representations. The usefulness of the proposed methodology is illustrated by analyzing a real dataset.

The probabilistic distribution of wind speed is critical information required in evaluating wind resources, designing wind farms, and mitigating the possible risks in wind power expansion. Various distributions have been used in the research literature to estimate wind speed distribution. In this paper, we propose a flexible family of mixture distributions, whose elements are convex linear combinations of the skew-t-normal Birnbaum-Saunders distributions and is suitable for modelling heavy-tailed data with a heterogeneous population. These distributions are then used to estimate the wind speed distribution. The performance of the proposed family has been compared with five mixture models of the already used distributions, using wind speed data collected at nine stations across Ontario, Canada. The results indicate that mixture models generally provide a better fit than unimodal distributions, according to the model selection criterion. Based on the obtained results, the proposed family provides highly flexible models at all selected stations. It functions better than the other considered distributions at seven of the stations, whereas it is ranked second in the remaining two stations. The proposed family can be utilized in a widespread manner to describe the wind speed in Canada, as well as other regions with similar features.

In this paper, we discuss stochastic orderings of lifetimes of two heterogeneous parallel and series systems with heterogeneous dependent components having generalized Birnbaum–Saunders distributions. The comparisons presented here are based on the vector majorization of parameters. The ordering results are established in some special cases for the generalized Birnbaum–Saunders distribution based on the multivariate elliptical, normal, t, logistic, and skew-normal kernels. Further, we use these results by considering Archimedean copulas to model the dependence structure among systems with generalized Birnbaum–Saunders components. These results have been used to derive some upper and lower bounds for survival functions of lifetimes of parallel and series systems.

Denoising of natural images is a basic problem in image processing. The present paper proposes a new algorithm for image denoising based on the maximum a-posteriori (MAP) estimator in undecimated dual-tree complex wavelet transform. The undecimated dual-tree complex wavelet transform (UDT-CWT), along with the directional selectivity of the dual-tree complex wavelet transform (DT-CWT), offers exact translational invariance property through removing the down-sampling of filter outputs together with the up-sampling of the complex filter pairs of DT-CWT. These properties are very important in image denoising. The performance of the MAP estimator depends strongly on the probability of noise-free wavelet coefficients. In our proposed denoising method, multivariate t-distribution is applied as the prior probability of noise-free coefficients. The t-distribution can accurately model the statistics of wavelet coefficients, which have peaky and heavy-tailed characteristics. On the other hand, the multivariate model makes it possible to take into account the dependencies of wavelet coefficients and their neighbors. Also, in our work, the necessary parameters of the multivariate distribution will be estimated in a locally-adaptive way to improve the denoising results via using the correlations among the amplitudes of neighbor coefficients. Simulation results delineate that the proposed algorithm outperforms state-of-the-art denoising algorithms in the literature.

In this paper, we derive the conditions to compare multivariate skew-normal random vectors X and Y. Then, we present an identity for E(f(Y)−f(X)), where the function f fulfills some weak regularity...

The Birnbaum–Saunders (BS) distribution, introduced in 1969, is a popular univariate fatigue life distribution which has been widely used to model right-skewed lifetime and reliability data. In this paper, a new class of generalized multivariate BS distributions is proposed based on mean-variance mixture models to accommodate strongly skewed and heavy tailed multivariate lifetime data. Some special cases of this class as well as their properties are then discussed. We present a hierarchical representation which facilitates an efficient EM-type algorithm for the computation of maximum likelihood estimates. Empirical results from a simulation study and real data analyses show that this class of distributions outperforms many existing extensions of the BS distribution in modeling lifetime data.

‎In this paper‎, ‎we introduce integral stochastic ordering of two‎ most important classes of distributions that are commonly used to fit data possessing high values of skewness and (or)‎ ‎kurtosis‎. ‎The first one is based on the selection distributions started by the univariate skew-normal distribution‎. ‎A broad‎, ‎flexible and newest class in this area is the scale and shape mixture of multivariate skew-normal distributions‎. ‎The second one is the general class of Normal Mean-Variance Mixture distributions‎. ‎We then derive necessary and sufficient conditions for comparing the random vectors from these two classes of distributions‎. ‎The integral orders considered here are the usual‎, ‎concordance‎, ‎supermodular‎, ‎convex‎, ‎increasing convex and directionally convex stochastic orders‎. ‎Moreover‎, ‎for bivariate random vectors‎, ‎in the sense of stop-loss and bivariate concordance stochastic orders‎, ‎the dependence strength of random portfolios is characterized in terms of order of correlations‎.

Multivariate skew-symmetric-normal (MSSN) distributions have been recognized as an appealing tool for modeling data with non-normal features such as asymmetry and heavy tails, rendering them suitable for applications in diverse areas. We introduce a richer class of MSSN distributions based on a scale-shape mixture of (multivariate) flexible skew-symmetric normal distributions, called the SSMFSSN distributions. This very general class of SSMFSSN distributions can capture various shapes of multimodality, skewness, and leptokurtic behavior in the data. We investigate some of its probabilistic characterizations and distributional properties which are useful for further methodological developments. An efficient EM-type algorithm designed under the selection mechanism is advocated to compute the maximum likelihood (ML) estimates of parameters. Simulation studies as well as applications to a real dataset are employed to illustrate the usefulness of the presented methods. Numerical results show the superiority of our proposed model in comparison to several existing competitors.

Denoising an image, while retaining the important features of the image, has been a fundamental problem in image processing. Dual-tree complex wavelet transform is a recently created transform that offers both near shift invariance and improved directional selectivity properties. This transform has been used in many techniques, including denoising. However, these techniques have used the real and imaginary components of the complex-valued sub-band coefficients separately. This paper proposes the use of coefficient magnitudes to provide an improvement in image denoising. Our proposed algorithm is based on the maximum a posteriori estimator, wherein the heavy-tailed scale mixtures of bivariate Rayleigh distribution are considered as the noise-free wavelet coefficient magnitudes’ prior distribution. Also, in our work, the necessary parameters of the bivariate distributions are estimated in a locally adaptive way to improve the denoising results via using the correlation between the amplitudes of neighbor coefficients. Simulation results delineate the performance of the proposed algorithm in both MSSIM and PSNR metrics.

This paper presents a new finite mixture model based on the multivariate normal mean–variance mixture of Birnbaum–Saunders (NMVBS) distribution. We develop a computationally analytical EM algorithm for model fitting. Due to the dependence of this algorithm on initial values and the number of mixing components, a learning-based EM algorithm and an extended variant are proposed. Numerical simulations show that the proposed algorithms allow for better clustering performance and classification accuracy than some competing approaches. The effectiveness and prominence of the proposed methodology are also shown through an application to an extrasolar planet dataset.

In this paper, by considering an (n+1)-dimensional random vector (X1,X2,….,Xn,Y)T from the multivariate generalized hyperbolic (GH) distribution, we derive the joint distribution of Y and the order...

In this paper, we show that the conditional distribution of a multivariate normal mean–variance mixture (MNMVM) distribution is also a MNMVM distribution. We show the usefulness of the established result by deriving the conditional distributions of multivariate generalized hyperbolic distribution.

In this paper, we establish some results for multivariate selection scale-mixtures of normal distributions with arbitrary mixing variable. First, we discuss their stochastic representation in terms of multivariate selection normal distributions. Next, the conditional distributions as well as the first two moments of multivariate selection scale-mixtures of normal distributions are obtained when the selection set is an arbitrary rectangle in the q-dimensional Euclidean space of Rq. The unified skew-scale mixture of normal (SUSMN) distributions are subsequently discussed as a special case. As a subclass of SUSMN distributions, the class of unified skew-symmetric generalized hyperbolic (SUSGH) distributions are studied in detail. Finally, we show that our results can be used to obtain moments of L-statistics and of multivariate concomitants from multivariate scale-mixtures of normal distributions.

In this paper, a new mixture family of multivariate normal distributions, formed by mixing multivariate normal distribution and skewed distribution, is constructed. Some properties of this family, such as characteristic function, moment generating function, and the first four moments are derived. The distributions of affine transformations and canonical forms of the model are also derived. An EM type algorithm is developed for the maximum likelihood estimation of model parameters. We have considered in detail, some special cases of the family, using standard gamma and standard exponential mixture distributions, denoted by MMNG and MMNE, respectively. For the proposed family of distributions, different multivariate measures of skewness are computed. In order to examine the performance of the developed estimation method, some simulation studies are carried out to show that the maximum likelihood estimates based on the EM type algorithm do provide good performance. For different choices of parameters of MMNE distribution, several multivariate measures of skewness are computed and compared. Because some measures of skewness are scalar and some are vectors, in order to evaluate them properly, we have carried out a simulation study to determine the power of tests, based on sample versions of skewness measures as test statistics to test the fit of the MMNE distribution. Finally, two real data sets are used to illustrate the usefulness of the proposed family of distributions and the associated inferential method.

This paper presents a robust extension of factor analysis model by assuming the multivariate normal mean-variance mixture of Birnbaum-Saunders distribution for the unobservable factors and errors. A computationally analytical EM-based algorithm is developed to find maximum likelihood estimates of the parameters. The asymptotic standard errors of parameter estimates are derived under an information-based paradigm. Numerical merits of the proposed methodology are illustrated using both simulated and real datasets.

Despite the widespread use and attractive properties of the normal and Student’s t distributions for modeling financial risks, it is widely believed that the normal-inverse Gaussian, skew-normal and skew-t distributions can be promising alternatives for describing the asymmetric features of asset returns. In this article, we propose a new benchmark model for quantifying the risk of financial assets based on the normal mean-variance Birnbaum-Saunders (NMVBS) distribution. The theoretical formulae of some popular risk measures based on the NMVBS distribution are exactly derived. Numerical evaluation of risks of some selected stock market returns reveals that the proposed method may outperform some existing approaches.

In this chapter, by considering a np-dimensional random vector \(\left ( {\mathbf {X}}_{1}^\top ,\ldots ,{\mathbf {X}}_{n}^\top \right ) ^\top \), \({\mathbf {X}}_{i} \in \mathbb {R}^{p}\), i = 1, …, n, having a multivariate elliptical distribution, we derive the exact distribution of multivariate order statistics induced by ordering linear combinations of the components. These induced multivariate order statistics are often referred to as the concomitant vectors corresponding to order statistics from multivariate elliptical distribution. Specifically, we derive a mixture representation for the distribution of the rth concomitant vector, and also for the joint distribution of the rth order statistic and its concomitant vector. We show that these distributions are indeed mixtures of multivariate unified skew-elliptical distributions. The important special cases of multivariate normal and multivariate student-t distributions are discussed in detail. Finally, the usefulness of the established results is illustrated with a real dataset.