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In this paper, we enhanced change point detection in skew normal distribution models by integrating the EM algorithm's Q-function with the modified information criterion (MIC). The new QMIC framework improves sensitivity and accuracy in detecting changes, outperforming the modified information criterion (MIC) and the traditional Bayesian information criterion (BIC). Due to the complexity of deriving analytic asymptotic distributions, bootstrap simulations were used to determine critical values at various significance levels. Extensive simulations demonstrate that QMIC offers superior detection capabilities. We applied the QMIC method to two stock market datasets, successfully identifying multiple change points, and highlighting its effectiveness for real-world financial data analysis. [ABSTRACT FROM AUTHOR]

In this paper, we enhanced change point detection in skew normal distribution models by integrating the EM algorithm's Q-function with the modified information criterion (MIC). The new QMIC framework improves sensitivity and accuracy in detecting changes, outperforming the modified information criterion (MIC) and the traditional Bayesian information criterion (BIC). Due to the complexity of deriving analytic asymptotic distributions, bootstrap simulations were used to determine critical values at various significance levels. Extensive simulations demonstrate that QMIC offers superior detection capabilities. We applied the QMIC method to two stock market datasets, successfully identifying multiple change points, and highlighting its effectiveness for real-world financial data analysis.

We propose a model for multiple waves of an epidemic that decomposes the health outcome of interest into the sum of scaled skew normal curves. When applied to daily COVID-19 mortality in six regions (Japan, Italy, Belgium, Ontario, Texas, and Peru), this model provides three notable results. First, when fit to data from early 2020 to May 31, 2022, the estimated skew normal curves substantially overlap with the dates of COVID-19 waves in Ontario and Belgium, as determined by their respective health authorities. Second, the asymmetry of the skew normal curves changes over time – they progress from increasing more quickly to decreasing more quickly, indicating changes in the relative speed that daily COVID-19 mortality rises and falls over time. Third, most regions have day-of-the-week effects, which suggests that day-of-the-week effects should be included when modeling daily COVID-19 mortality. We conclude by discussing limitations and possible extensions of this model and its results, including commenting on its applicability to potential future COVID-19 waves. [ABSTRACT FROM AUTHOR]

Vascular-related diseases pose significant public health challenges and are a leading cause of mortality and disability. Understanding the complex structure of the vascular system and its processes is crucial for addressing these issues. Recent advancements in medical imaging technology have enabled the generation of high-resolution 3D images of vascular structures, leading to a diverse array of methods for vascular extraction. While previous research has often assumed a normal distribution of image data, this paper introduces a novel vessel extraction method that utilizes the skew normal distribution for more accurate probability distribution modeling. The proposed method begins with a preprocessing step to enhance vessel structures and reduce noise in Magnetic Resonance Angiography (MRA) images. The skew normal distribution, known for its ability to model skewed data, is then employed to characterize the intensity distribution of vessels. By estimating the parameters of the skew normal distribution using the Expectation-Maximization (EM) algorithm, the method effectively separates vessel pixels from the background and non-vessel regions. To extract vessels, a thresholding technique is applied based on the estimated skew normal distribution parameters. This segmentation process enables accurate vessel extraction, particularly in detecting thin vessels and enhancing the delineation of vascular edges with low contrast. Experimental evaluations on a diverse set of MRA images demonstrate the superior performance of the proposed method compared to previous approaches in terms of accuracy and computational efficiency. The presented vessel extraction method holds promise for improving the diagnosis and treatment of vascular-related diseases. By leveraging the skew normal distribution, it provides accurate and efficient vessel segmentation, contributing to the advancement of vascular imaging in the field of medical image analysis. [ABSTRACT FROM AUTHOR]

Vascular-related diseases pose significant public health challenges and are a leading cause of mortality and disability. Understanding the complex structure of the vascular system and its processes is crucial for addressing these issues. Recent advancements in medical imaging technology have enabled the generation of high-resolution 3D images of vascular structures, leading to a diverse array of methods for vascular extraction. While previous research has often assumed a normal distribution of image data, this paper introduces a novel vessel extraction method that utilizes the skew normal distribution for more accurate probability distribution modeling. The proposed method begins with a preprocessing step to enhance vessel structures and reduce noise in Magnetic Resonance Angiography (MRA) images. The skew normal distribution, known for its ability to model skewed data, is then employed to characterize the intensity distribution of vessels. By estimating the parameters of the skew normal distribution using the Expectation-Maximization (EM) algorithm, the method effectively separates vessel pixels from the background and non-vessel regions. To extract vessels, a thresholding technique is applied based on the estimated skew normal distribution parameters. This segmentation process enables accurate vessel extraction, particularly in detecting thin vessels and enhancing the delineation of vascular edges with low contrast. Experimental evaluations on a diverse set of MRA images demonstrate the superior performance of the proposed method compared to previous approaches in terms of accuracy and computational efficiency. The presented vessel extraction method holds promise for improving the diagnosis and treatment of vascular-related diseases. By leveraging the skew normal distribution, it provides accurate and efficient vessel segmentation, contributing to the advancement of vascular imaging in the field of medical image analysis.

This paper proposes an observation-driven finite mixture model for clustering high-dimension data. A simple algorithm using static hidden variables statically clusters the data into separate model components. The model accommodates normal and skew-normal distributed mixtures with time-varying component means, covariance matrices and skewness coefficient. These parameters are estimated using the EM algorithm and updated with the Generalized Autoregressive Scale (GAS) approach. Our proposed model is preferably clustered using a skew-normal distribution rather than a normal distribution when dealing with real data that may be skewed and asymmetrical. Finally, our proposed model will be evaluated using a simulation study and the results will be discussed using a real data set. [ABSTRACT FROM AUTHOR]

Despite the progress made by numerous contributions in recent decades on uncertainty in hydrological simulation, there are still knowledge gaps in estimating uncertainty sources, especially associated with precipitation. The aim of this study was to determine the precipitation uncertainty through an error model based on the skew normal distribution function and to evaluate the effect of its propagation towards the simulated flow with the TETIS distributed hydrological model in a poorly instrumented tropical Andean basin. The results show the performance of the hydrological model is more sensitive to the location of the meteorological station used than to the number of stations employed in a real case with scarce information. Implementing the Bayesian approach for the study of uncertainty in input data such as precipitation is essential for its quantification, improving the knowledge of how this source of error propagates to the results of the hydrological simulation. [ABSTRACT FROM AUTHOR]

Most item response theory (IRT) models for dichotomous responses are based on probit or logit link functions which assume a symmetric relationship between the probability of a correct response and the latent traits of individuals taking a test. This assumption restricts the use of those models to the case in which all items behave symmetrically. On the other hand, asymmetric models proposed in the literature impose that all the items in a test behave asymmetrically. This assumption is inappropriate for great majority of tests which are, in general, composed of both symmetric and asymmetric items. Furthermore, a straightforward extension of the existing models in the literature would require a prior selection of the items' symmetry/asymmetry status. This paper proposes a Bayesian IRT model that accounts for symmetric and asymmetric items in a flexible but parsimonious way. That is achieved by assigning a finite mixture prior to the skewness parameter, with one of the mixture components being a point mass at zero. This allows for analyses under both model selection and model averaging approaches. Asymmetric item curves are designed through the centred skew normal distribution, which has a particularly appealing parametrization in terms of parameter interpretation and computational efficiency. An efficient Markov chain Monte Carlo algorithm is proposed to perform Bayesian inference and its performance is investigated in some simulated examples. Finally, the proposed methodology is applied to a data set from a large‐scale educational exam in Brazil. [ABSTRACT FROM AUTHOR]

We consider a new approach for estimating non‐Gaussian undirected graphical models. Specifically, we model continuous data from a class of multivariate skewed distributions, whose conditional dependence structure depends on both a precision matrix and a shape vector. To estimate the graph, we propose a novel estimation method based on nodewise regression: we first fit a linear model, and then fit a one component projection pursuit regression model to the residuals obtained from the linear model, and finally threshold appropriate quantities. Theoretically, we establish error bounds for each nodewise regression and prove the consistency of the estimated graph when the number of variables diverges with the sample size. Simulation results demonstrate the strong finite sample performance of our new method over existing methods for estimating Gaussian and non‐Gaussian graphical models. Finally, we demonstrate an application of the proposed method on observations of physicochemical properties of wine. [ABSTRACT FROM AUTHOR]

Summary: We propose a multivariate normality test against skew normal distributions using higher‐order log‐likelihood derivatives, which is asymptotically equivalent to the likelihood ratio but only requires estimation under the null. Numerically, it is the supremum of the univariate skewness coefficient test over all linear combinations of the variables. We can simulate its exact finite sample distribution for any multivariate dimension and sample size. Our Monte Carlo exercises confirm its power advantages over alternative approaches. Finally, we apply it to the joint distribution of US city sizes in two consecutive censuses finding that non‐normality is very clearly seen in their growth rates. [ABSTRACT FROM AUTHOR]

In this paper, we use the likelihood ratio test to detect changes in the parameters of the skew slash distribution. Simulations have been conducted under different scenarios to investigate the performance of the proposed method. In the end, the real data applications are studied. [ABSTRACT FROM AUTHOR]

Independence and normality of observations for each level of the classification variables are two fundamental assumptions in traditional analysis of variance (ANOVA), whereas in many real applications the data violate seriously from these assumptions. Accordingly, in these situations using this traditional theory leads to unappealing results. We consider time series ANOVA by assuming a skew normal distribution for innovations. We provide iterative closed forms for the maximum likelihood estimators and construct asymptotic confidence intervals for them. A simulation study and a real data example are used to evaluate the efficiency and applicability of the proposed model for analyzing skew-symmetric time series data. [ABSTRACT FROM AUTHOR]

Weighted distribution is a valuable method for constructing flexible models and analyzing data sets. In this paper, a new weighted distribution of skew-normal is introduced with four parameters. The proposed model is a generalized version of several distributions, such as normal, bimodal normal, skew-normal, and skewed bimodal normal-normal. This weighted model is form-invariant under the proposed weight function. The basic characteristics of the model are expressed. A method has been used to generate data from the model. The maximum likelihood estimations of parameters are given and evaluated using a simulation study. The model is fitted to the three real data sets. The advantage of the proposed model has been shown on the rival distributions using appropriate criteria. [ABSTRACT FROM AUTHOR]

In this article, we propose a modified max-cumulative sum (CUSUM) procedure for detecting changes in parameters of skew normal distribution. The corresponding false alarms frequency and the postchange detection delay are investigated. Asymptotic behaviors of detection delay and theoretical optimality of the detection procedure have been established. Simulations have been conducted to show the performance of the proposed method and compare it to the other existing methods including CUSUM. Real data are given to illustrate the detection procedure. [ABSTRACT FROM AUTHOR]

Grain-size distribution (GSD) is the superposition of multiple subpopulations from different sedimentary processes, conveying the indispensable message of original sediments, and the frequency curve may show a bimodal character. Traditional methods of sedimentological grain-size analysis use a single probability density function (PDF) to fit their frequency profile. In practice, however, the sandstone and small grain compositions do not necessarily come from the same distribution family. The mixture distribution model is based on mixing PDFs from two different distribution families and provides a novel technique for decomposing GSD data from bimodal sediments. Taking 131 bimodal sediment GSD data from an offshore oil field in China as an example, 262 subpopulations are separated and extracted by using the mixture distribution model. Calculating the mean, sorting, skewness, and kurtosis of all samples by Folk-Ward formula, while these of subpopulations are calculated by moment method. The average coefficient of determination and fitting residual between 131 fitted GSD data and original samples are 0.9783 and 0.25, which indicates that the fitting quality is excellent. The variance of coefficient of determination and fitting residual for these 131 samples are 0.0001 and 0.1326 respectively, illustrating that the fit results fluctuated little and the fit was stable. The results demonstrate that the mixture distribution model is effective in fitting and unmixing the sediment GSD frequency data. [ABSTRACT FROM AUTHOR]

The class of skew normal distributions, introduced by Azzalini (1985), which is an asymmetric distribution and allows the presence of skewness. In this paper, we propose the pivotal quantity approach to construct the confidence interval for the mean, prediction interval for the mean of the future sample, and tolerance interval for the quantile. The fiducial distribution is also studied. Moreover, the performances of all the proposed confidence intervals are investigated through the Monte Carlo simulation. The pivotal quantity is a common method for calculating confidence intervals, which is used to construct confidence intervals in this paper. And the convergence of the obtained confidence interval is illustrated by the figures. Finally, a real data is used to explain proposed intervals in real life. [ABSTRACT FROM AUTHOR]

We present a proposal to deal with the non-normality issue in the context of regression models with measurement errors when both the response and the explanatory variable are observed with error. We extend the normal model by jointly modelling the unobserved covariate and the random errors by a finite mixture of scale mixture of skew-normal distributions. This approach allows us to model data with great flexibility, accommodating skewness, heavy tails, and multi-modality. The main virtue of considering measurement error models under the class of scale mixtures of skew-normal distributions is that they have a nice hierarchical representation which allows easy implementation of inference. In order to illustrate the usefulness of the proposed method some simulation studies are presented and a real dataset (Systemic lupus erythematosus) is analyzed. [ABSTRACT FROM AUTHOR]

Here, we first propose three-parameter model and call it as the compound zero-truncated Poisson normal (ZTP-N) distribution. The model is based on the random sum of N independent Gaussian random variables, where N is a zero truncated Poisson random variable. The proposed ZTP-N distribution is a very flexible probability distribution function. The probability density function can take variety of shapes. It can be both positively and negatively skewed, moreover, normal distribution can be obtained as a special case. It can be unimodal, bimodal as well as multimodal also. It has three parameters. An efficient EM type algorithm has been proposed to compute the maximum likelihood estimators of the unknown parameters. We further propose a four-parameter bivariate distribution with continuous and discrete marginals, and discuss estimation of unknown parameters based on the proposed EM type algorithm. Some simulation experiments have been performed to see the effectiveness of the proposed EM type algorithm, and one real data set has been analyzed for illustrative purposes. [ABSTRACT FROM AUTHOR]

In this paper, we consider various methods for evaluating the cdf of the Skew Normal distribution. This distribution arises in the stochastic frontier model because it is the distribution of the composed error, which is the sum (or difference) of a Normal and a Half-Normal random variable. The cdf must be evaluated in models in which the composed error is linked to other errors using a Copula, in some methods of goodness of fit testing, or in the likelihood of models with sample selection bias. We investigate the accuracy of the evaluation of the cdf using expressions based on the bivariate Normal distribution, and also using simulation methods and some approximations. We find that the expressions based on the bivariate Normal distribution are quite accurate in the central portion of the distribution, and we propose several new approximations that are accurate in the extreme tails. By a simulated example, we show that the use of approximations instead of the theoretical exact expressions may be critical in obtaining meaningful and valid estimation results. [ABSTRACT FROM AUTHOR]

The accuracy of vehicle speed measured by a speedometer is analysed. The stress on the application of skew normal distribution is laid. The accuracy of measured vehicle speed depends on many error sources: construction of speedometer, measurement method, model inadequacy to real physical process, transferring information signal, external conditions, production process technology etc. The errors of speedometer are analysed in a complex relation to errors of the speed control gauges, whose functionality is based on the Doppler effect. Parameters of the normal distribution and skew normal distribution were applied in the errors analysis. It is shown that the application of maximum permissible errors to control the measuring results of vehicle speed gives paradoxical results when, in the case of skew normal distribution, the standard deviations of higher vehicle speeds are smaller than the standard deviations of lower speeds. In the case of normal distribution a higher speed has a greater standard deviation. For the speed measurements by Doppler speed gauges it is suggested to calculate the vehicle weighted average speed instead of the arithmetic average speed, what will correspond to most real dynamic changes of the vehicle speed parameters.

Premature mortality is often a neglected component of overall deaths, and the most difficult to identify. However, it is important to estimate its prevalence. Following Pearson's theory about mortality components, a definition of premature deaths and a parametric model to study its transformations are introduced. The model is a mixture of three distributions: a Half Normal for the first part of the death curve and two Skew Normals to fit the remaining pieces. One advantage of the model is the possibility of obtaining an explicit equation to compute life expectancy at birth and to break it down into mortality components. We estimated the mixture model for Sweden, France, East Germany and Czech Republic. In addition, to the well-known reduction in infant deaths, and compression and shifting trend of adult mortality, we were able to study the trend of the central part of the distribution of deaths in detail. In general, a right shift of the modal age at death for young adults is observed; in some cases, it is also accompanied by an increase in the number of deaths at these ages: in particular for France, in the last twenty years, premature mortality increases. [ABSTRACT FROM AUTHOR]

The class of skew normal distributions, introduced by Azzalini (1985), which is an asymmetric distribution and allows the presence of skewness. In this paper, we propose the pivotal quantity approach to construct the confidence interval for the mean, prediction interval for the mean of the future sample, and tolerance interval for the quantile. The fiducial distribution is also studied. Moreover, the performances of all the proposed confidence intervals are investigated through the Monte Carlo simulation. The pivotal quantity is a common method for calculating confidence intervals, which is used to construct confidence intervals in this paper. And the convergence of the obtained confidence interval is illustrated by the figures. Finally, a real data is used to explain proposed intervals in real life.

In this paper, a change constrained optimization programming problem is studied under the assumption that the model coefficients in the inequalities defined as random variables are independent and assumed to be Normal, t; Non Normal Skew distributions; Skew Normal and Skew t distributions. The Hulkursar method transform the stochastic programming problem into a non-linear deterministic problem is used in the study. The most common distribution in CCSP is the Normal Distribution; but the real world problems always may not include normality. Therefore; in the practice stage, an application that the aij technologic coefficient and the bi right side values in the inequalities have both Normal, t; Skew Normal and Skew t distributions is given. Finally the obtained results have been compared. [ABSTRACT FROM AUTHOR]

This paper presents the R package EMMIXcskew for the fitting of the canonical fundamental skew t-distribution (CFUST) and finite mixtures of CFUST distributions (FMCFUST) via maximum likelihood (ML). The CFUST distribution provides a flexible family to model non-normal data, with parameters for capturing skewness and heavy-tails in the data. It formally encompasses the normal, t, and skew normal distributions as special and/or limiting cases. A few other versions of the skew t-distributions are also nested within the CFUST distribution. In this paper, an expectation-maximization (EM) algorithm is described for computing the ML estimates of the parameters of the FM-CFUST model, and different strategies for initializing the algorithm are discussed and illustrated. The methodology is implemented in the EMMIXcskew package, and examples are presented using two real datasets. The EMMIXcskew package contains functions to fit the FM-CFUST model, including procedures for generating different initial values. Additional features include random sample generation and contour visualization in 2D and 3D.

In this paper, we propose a goodness-of-fit test based on the energy statistic for skew normality. Simulations indicate that the Type-I error of the proposed test can be controlled reasonably well for given nominal levels. Power comparisons to other existing methods under different settings show the advantage of the proposed test. Such a test is applied to two real data sets to illustrate the testing procedure. [ABSTRACT FROM AUTHOR]

In this paper, we consider a random vector X = X 1 , X 2 following a multivariate Skew Normal distribution and we provide an explicit formula for the expected value of X conditioned to the event X ≤ X ¯ , with X ¯ ∈ R 2 . Such a conditional expectation has an intuitive interpretation in the context of risk measures. [ABSTRACT FROM AUTHOR]

Profile monitoring is a technique to test the stability of the relationship between a response variable and explanatory variables over time. The most relevant linear profile monitoring methods have been constructed using the normality assumption. However, the normality assumption could be violated in many quality control applications. In this study, we consider a situation in which the random errors in a linear profile model follow a skew‐normal distribution. The skew‐normal distribution is a generalized version of the normal distribution. A new Shewhart‐type chart and exponentially weighted moving average (EWMA) chart, named the ShewhartR and EWMAR charts, respectively, are constructed based on residuals to monitor the parameters of linear profile model. The simulation results show that the multivariate EWMA chart is sensitive to the normality assumption and that the proposed ShewhartR and EWMAR charts have good ability to detect big and small‐to‐moderate process shifts, respectively. An example using photo mask techniques in semiconductor manufacturing is provided to illustrate the applications of the ShewhartR and EWMAR charts. [ABSTRACT FROM AUTHOR]

In the stochastic frontier model, the composed error is the sum (or difference) of a normal and a half normal random variable. Often the composed error is linked to other errors using a copula, and evaluation of the copula requires evaluation of the cdf of the composed error. There is no analytical expression for this cdf, though there are several approximations. We propose a computationally efficient simulation based method of evaluation and use it to evaluate the accuracy of these approximations. We also derive the exact cdf of the composed error for the special case that the stochastic frontier relative variance parameter λ equals one, and we use this expression to investigate the accuracy of our evaluations and the existing approximations. [ABSTRACT FROM AUTHOR]

Symmetric elliptical distributions have been intensively used in data modeling and robustness studies. The area of applications was considerably widened after transforming elliptical distributions into the skew elliptical ones that preserve several good properties of the corresponding symmetric distributions and increase possibilities of data modeling. We consider three-parameter p-variate skew t-distribution where p-vector μ is the location parameter, Σ:p×p is the positive definite scale parameter, p-vector α is the skewness or shape parameter, and the number of degrees of freedom ν is fixed. Special attention is paid to the two-parameter distribution when μ=0 that is useful for construction of the skew t-copula. Expressions of the parameters are presented through the moments and parameter estimates are found by the method of moments. Asymptotic normality is established for the estimators of Σ and α. Convergence to the asymptotic distributions is examined in simulation experiments.

The accuracy of vehicle speed measured by a speedometer is analysed. The stress on the application of skew normal distribution is laid. The accuracy of measured vehicle speed depends on many error sources: construction of speedometer, measurement method, model inadequacy to real physical process, transferring information signal, external conditions, production process technology etc. The errors of speedometer are analysed in a complex relation to errors of the speed control gauges, whose functionality is based on the Doppler effect. Parameters of the normal distribution and skew normal distribution were applied in the errors analysis. It is shown that the application of maximum permissible errors to control the measuring results of vehicle speed gives paradoxical results when, in the case of skew normal distribution, the standard deviations of higher vehicle speeds are smaller than the standard deviations of lower speeds. In the case of normal distribution a higher speed has a greater standard deviation. For the speed measurements by Doppler speed gauges it is suggested to calculate the vehicle weighted average speed instead of the arithmetic average speed, what will correspond to most real dynamic changes of the vehicle speed parameters. [ABSTRACT FROM AUTHOR]

One of the most important property of the mixture normal distributions-model is its flexibility to accommodate various types of distribution functions (df's). We show that the mixture of the skew normal distribution and its reverse, after adding a location parameter to the skew normal distribution, and adding the same location parameter with different sign to its reverse is a family of df's that contains all the possible types of df's. Besides, it has a very remarkable wide range of the indices of skewness and kurtosis. Computational techniques using EM-type algorithms are employed for iter-atively computing maximum likelihood estimates of the model parameters. Moreover, an application with a body mass index real data set is presented. [ABSTRACT FROM AUTHOR]

In this note we propose a newly formulated skew exponential power distribution that behaves substantially better than previously defined versions. This new model performs very well in terms of the large sample behavior of the maximum likelihood estimation procedure when compared to the classically defined four parameter model defined by Azzalini. More recently, approaches to defining a skew exponential power distribution have used five or more parameters. Our approach improves upon previous attempts to extend the symmetric power exponential family to include skew alternatives by maintaining a minimum set of four parameters corresponding directly to location, scale, skewness and kurtosis. We illustrate the utility of our proposed model using translational and clinical data sets. [ABSTRACT FROM AUTHOR]

The sample mean and sample variance are commonly used statistics in our study. In this paper, distributions of the sample mean and sample variance from a skew normal population are derived under closed skew normal (CSN) settings. The noncentral closed skew chi-square distribution is defined, and the distribution of quadratic forms is discussed. Our results generalize the corresponding results given under skew normal settings. Several examples are given for illustration of our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the asymptotic distributions of functionals of the sample covariance matrix and the sample mean vector obtained under the assumption that the matrix of observations has a matrix‐variate location mixture of normal distributions. The central limit theorem is derived for the product of the sample covariance matrix and the sample mean vector. Moreover, we consider the product of the inverse sample covariance matrix and the mean vector for which the central limit theorem is established as well. All results are obtained under the large‐dimensional asymptotic regime, where the dimension p and the sample size n approach infinity such that p/n→c ∈ [0, + ∞) when the sample covariance matrix does not need to be invertible and p/n→c ∈ [0,1) otherwise. [ABSTRACT FROM AUTHOR]

The skew normal (SN) distribution of Azzalini (Scand J Stat 12:171–178, 1985) is one of the widely used probability distributions for modelling skewed data. In this article, we introduce a general class of skewed distributions based on mean mixtures of normal distributions, which includes the SN distribution as a special case. Some properties of this new class, such as expressions for mean, variance, skewness and kurtosis coefficients and characteristic function, are derived. Also, estimates of the model parameters are first obtained by the method of moments. Two special cases of this new class are studied in detail. It is shown that the range of skewness and kurtosis coefficients for the special cases is wider than that of the SN distribution, and in addition, unlike the SN distribution, one of these models is an infinitely divisible distribution. For carrying out the maximum likelihood (ML) estimation, an ECM algorithm is developed. This algorithm is analytically simple because closed-form expressions of conditional expectations in the E-step as well as the updating estimators in the CM-step are in explicit form. The observed information matrix is provided for approximating the asymptotic covariance matrix of the ML estimators of the parameters. The usefulness of the proposed distribution is illustrated through simulated as well as two real data sets. Next, a new extension of regression models is constructed by assuming the proposed distributions for the error term. Finally, a multivariate version of the proposed model is discussed. [ABSTRACT FROM AUTHOR]

In this article, the new family of multivariate skew slash distribution is defined. According to the definition, a stochastic representation of the multivariate skew slash distribution is derived. The first four moments and measures of skewness and kurtosis of a random vector with the multivariate skew slash distribution are obtained. The distribution of quadratic forms for the multivariate skew slash distribution and the non central skew slash χ2 distribution are studied. Maximum likelihood inference and real data illustration are discussed. In the end, the potential extension of multivariate skew slash distribution is discussed. [ABSTRACT FROM AUTHOR]

The assumption of the error normality in the regression model was often questioned especially in cases where there was an outlier, which causes the behavior of asymmetric data. To overcome this, without data transformation, we could use skew distribution. This distribution was very important and applicable in various fields of science such as finance, economics, actuarial science, medicine, biology, investment. Skew Normal distributions had been proven to have a convenient for calculating bias in data with asymmetric behavior. This study aims to model SUR with Skew Normal error using Bayesian approach applied to East Java GRDP data. This study would compared two types of models, namely models with Normal distributed errors and models with Skew Normal distributed errors. The result of parameter estimation with Bayesian approach shows that SUR Skew Normal model was more suitable for East Java GRDP modeling rather than using normal error model. This was based on their smaller Root of Mean Square Error (RMSE), Mean Absolute Error (MAE) and Mean Absolute Percentage Error (MAPE) value. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of estimating the location and scale parameters of the skew normal distribution introduced by Azzalini. For this distribution, the classic maximum likelihood estimators(MLEs) do not take explicit forms. We approximate the likelihood equations and derive explicit estimators of the parameters. The bias and variance of the estimators are investigated and Monte Carlo simulation studies show that the estimators are as efficient as the classic MLEs. We demonstrate that the probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic normality are unsatisfactory, especially when the sample size is small. The use of unconditional simulated percentage points of these quantities is suggested. Finally, a numerical example is used to illustrate the proposed inference methods. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear regression models when the model errors follow a slash skew-elliptical distribution, are considered. In the special case of nonlinear regression models under slash skew-t distribution, we present some distributional properties, and to estimate their parameters, we use an EM-type algorithm. Also, to find the estimation errors, we derive the observed information matrix analytically. To describe the influence of the observations on the ML estimates, we use a sensitivity analysis. Finally, we conduct some simulation studies and a real data analysis to show the performance of the proposed model. [ABSTRACT FROM AUTHOR]

We introduce two families of continuous distribution functions with not-necessarily symmetric densities, which contain a parent distribution as a special case. The two families proposed depend on two parameters and are presented as an alternative to the skew normal distribution and other proposals in the statistical literature. The density functions of these new families are given by a closed expression which allows us to easily compute probabilities, moments and related quantities. The second family can exhibit bimodality and its standardized fourth central moment (kurtosis) can be lower than that of the Azzalini skew normal distribution. Since the second proposed family can be bimodal we fit two well-known data set with this feature as applications. We concentrate attention on the case in which the normal distribution is the parent distribution but some consideration is given to other parent distributions, such as the logistic distribution.