Spatial regression models are used to analyze quantitative spatial responses based on linear and non-linear relationships with explanatory variables. Usually, the spatial correlation of responses is modeled with a Gaussian random field based on a multivariate NORMAL DISTRIBUTION. However, in practice, we encounter SKEWed responses, which are analyzed using SKEW-NORMAL DISTRIBUTIONs. CLOSED SKEW-NORMAL DISTRIBUTION is one of the extended families of SKEW-NORMAL DISTRIBUTIONs, which has similar properties to NORMAL DISTRIBUTIONs. This article presents a hierarchical Bayesian analysis based on a flexible subclass of CLOSED SKEW-NORMAL DISTRIBUTIONs. Given the time-consuming nature of Monte Carlo methods in hierarchical Bayes analysis, we have opted to use the variational Bayes approach to approximate the posterior DISTRIBUTION. This decision was made to expedite the analysis process without compromising the accuracy of our results. Then, the proposed model is implemented and analyzed based on the real earthquake data of Iran.

Gaussian random field is usually used to model Gaussian spatial data. In practice, we may encounter non-Gaussian data that are SKEWed. One solution to model SKEW spatial data is to use a SKEW random field. Recently, many SKEW random fields have been proposed to model this type of data, some of which have problems such as complexity, non-identifiability, and non-stationarity. In this article, a flexible class of CLOSED SKEW-NORMAL DISTRIBUTION is introduced to construct valid stationary random fields, and some important properties of this class such as identifiability and CLOSEDness under marginalization and conditioning are examined. The reasons for developing valid spatial models based on these SKEW random fields are also explained. Additionally, the identifiability of the spatial correlation model based on empirical variogram is investigated in a simulation study with the stationary SKEW random field as a competing model. Furthermore, spatial predictions using a likelihood approach are presented on these SKEW random fields and a simulation study is performed to evaluate the likelihood estimation of their parameters.

We consider the problem of inference about SKEWness parameter in SKEW-NORMAL DISTRIBUTION and it's difficulties. An approximately maximum likelihood estimator that is based on some prior information is proposed. Finally the efficiency of proposed estimator is assessed both theoretically and by a simulation study.

In this paper, we discuss a generalization of Balakrishnan SKEW NORMAL DISTRIBUTION with two parameters that contains the SKEW-NORMAL, the Balakrishnan SKEW-NORMAL and the two-parameter generalized SKEW-NORMAL DISTRIBUTIONs as special cases. Furthermore, we establish some useful properties and two extensions of this DISTRIBUTION.

Objective Investors typically seek to strike the optimal balance between potential returns and associated risks in their trades. Various models have been presented to choose the optimal portfolio using different approaches. one of these methods is based on the statistical DISTRIBUTION of asset return. In these methods, the type of DISTRIBUTION of returns is first identified, and a suitable portfolio selection method is then applied based on this identified DISTRIBUTION type. This study compares the effectiveness of the mean-absolute deviation-entropy model utilizing both SKEW-NORMAL DISTRIBUTION and SKEW-Laplace-NORMAL DISTRIBUTION for constructing an optimal portfolio in the Tehran Stock Exchange over 36 months from April 2018 to March 2020.   Methods The data used in this study comprises the monthly returns of 181 companies listed on the Tehran Stock Exchange. These returns were gathered from a statistical population of 338 members utilizing Morgan's table and Cochran’s formula. After fitting density functions for SKEW-NORMAL and SKEW-Laplace-NORMAL DISTRIBUTIONs to the returns, maximum likelihood estimates were obtained using the Stats package and the optim Function in R software. The reliability of these estimates was then checked using bootstrap sampling with 1,000 repetitions. Subsequently, relationships corresponding to the mathematical expectation of return DISTRIBUTION and the objective function representing the risk of absolute deviation were estimated using numerical methods. Therefore, this paper aimed to propose a multi-objective optimization model, namely a mean-absolute deviation-entropy model for portfolio optimization by using a goal-programming approach based on SKEW-NORMAL DISTRIBUTION and SKEW-Laplace-NORMAL DISTRIBUTION. The objective functions of the model were to maximize the mean return, minimize the absolute deviation, and maximize the entropy of the portfolio.   Results It can be inferred from the observed values ​​of the descriptive statistics of the monthly stock returns corresponding to the stock exchange symbols that some stocks have different SKEWness and kurtosis values ​​compared to the NORMAL DISTRIBUTION. For example, The symbol "Shepna" exhibits negative SKEWness, indicating a left-SKEWed DISTRIBUTION. Similarly, the DISTRIBUTION of the "Basama" symbol exceeds the NORMAL DISTRIBUTION. These instances suggest that the NORMAL DISTRIBUTION is inadequate for describing monthly return DISTRIBUTIONs. Instead, DISTRIBUTIONs with parameters should be employed to account for SKEWness and kurtosis. According to the obtained results, the model utilizing the SKEW- Laplace- NORMAL DISTRIBUTION has a higher performance ratio than the model based on the SKEW-NORMAL DISTRIBUTION.   Conclusion The reason for this superiority, where the model utilizing the SKEW-Laplace-NORMAL DISTRIBUTION outperforms the model based on the SKEW-NORMAL DISTRIBUTION, is the incorporation of both SKEWness and kurtosis criteria within the former. Additionally, upon analyzing the descriptive statistics of the symbols, it's evident that the kurtosis of most stock symbols is substantial. Therefore, integrating a combination of higher-order moments (SKEWness and kurtosis) along with entropy leads to enhanced performance.

Spatial generalized linear mixed models are usually used for modeling non-Gaussian and discrete spatial responses. In these models, spatial correlation of the data can be considered via latent variables. Estimation of the latent variables at the sampled locations, the model parameters and the prediction of the latent variables at un-sampled locations are of the most important interest in SGLMM. Often the NORMAL assumption for latent variables is considered just for convenient in practice. Although this assumption simplifies the calculations, in practice, it is not necessarily true or possible to be tested. In this paper, a CLOSED SKEW NORMAL DISTRIBUTION is proposed for the spatial latent variables. This DISTRIBUTION includes the NORMAL DISTRIBUTION and also remains CLOSED under linear conditioning and marginalization. In these models, likelihood function cannot usually be given in a CLOSED form and maximum likelihood estimations may be computationally prohibitive. In this paper, for maximum likelihood estimation of the model parameters and predictions of latent variables, an approximate algorithm is introduced that is faster than the former method. The performance of the proposed model and algorithm are illustrated through a simulation study.

In this paper we consider several generalizations of the SKEW t-NORMAL DISTRIBUTION, and some of their properties. Also, we represent several theorems for constructing each generalized SKEW t-NORMAL DISTRIBUTION. Next, we illustrate the application of the proposed DISTRIBUTION studying the ratio of two heavy metals, Nickel and Vanadium, associated with crude oil in Shadgan wetland in the southwest of Iran at the head of the Persian Gulf.

The stochastic data envelopment analysis (SDEA) was developed considering the value of inputs and outputs as random variables. Therefore, statistical DISTRIBUTIONs play an important role in this regard. The SKEW-NORMAL (SN) DISTRIBUTION is a family of probability density functions that is frequently used in practical situations. In this paper, we assume that the input and output variables are SKEW-NORMALly distributed. With introducing asymmetric error structure for random variables of SN DISTRIBUTION, a stochastic BCC model is provided. The proposed model includes BCC model assuming a NORMAL DISTRIBUTION of data as well. Finally, the proposed model is used in a numerical example.