Skew-normal Distribution Research Articles

Parameter recovery of skew multidimensional multiple-group item response models (MGM-MIRT): a comparison of MCMC algorithms and assessment of effects of interest

Multidimensional item response theory (MIRT) models are powerful tools to analyse item response data. Most of them rely on the assumption that the latent traits are normally distributed. However, when such assumption does not hold, the item and the latent traits estimates can be inaccurate. This paper deals, in a very detailed way, through simulation studies, with the utility of the class of the MIRT models developed by Padilla et al. [Multidimensional multiple-group IRT models with skew normal latent trait distributions. J Multivariate Anal 2018;167:250–268], where the multivariate skew-normal distribution under the centred parameterization is employed for modelling the latent traits. The impact of important factors of interest on the parameter recovery is investigated. Such factors are the number of subjects, number of items, number of groups and asymmetry level. In a general way, the results show that our approach outperforms some traditional models when the latent traits follow asymmetric distributions. In addition, different Monte Carlo Markov Chain (MCMC) algorithms for parameter estimation are compared, by considering two schemes of data augmentation and a proposal to speed up their convergence, based on approaches found in the literature, for binary regression models. Guidelines are provided for the correct using of Bayesian inferential methods for the MIRT models.

Grain boundary segregation spectrum in basal-textured Mg alloys: From solute decoration to structural transition

Mg alloys are promising lightweight structural materials due to their low density and excellent mechanical properties. However, their limited formability and ductility necessitate improvements in these properties, specifically through texture modification via grain boundary segregation. While significant efforts have been made, the segregation behavior in Mg polycrystals, particularly with basal texture, remains largely unexplored. In this study, we performed atomistic simulations to investigate grain boundary segregation in dilute and concentrated solid solution Mg–Al alloys. We computed the segregation energy spectrum of basal-textured Mg polycrystals, highlighting the contribution from specific grain boundary sites, such as junctions, and identified a newly discovered bimodal distribution which is distinct compared to the conventional skew-normal distribution found in randomly-oriented polycrystals. Using a hybrid molecular dynamics/Monte Carlo approach, we simulated segregation behavior at finite temperatures, identifying grain boundary structural transitions, particularly the varied fraction and morphology of topologically close-packed grain boundary phases when changing thermodynamic variables. The outcomes of this study offer crucial insights into basal-textured grain boundary segregation and phase formation, which can be extended to other relevant Mg alloys containing topologically close-packed intermetallics.

Mathematical Formalization and Applications to Data with Excess of Zeros and Ones of the Unit-Proportional Hazard Inflated Models

In this study, we model the rate or proportion of a specific phenomenon using a set of known covariates. To fit the regression model, which explains the phenomenon within the intervals (0,1), [0,1), (0,1], or [0,1], we employ a logit link function. This approach ensures that the model’s predictions remain within the appropriate range of zero to one. In cases of inflation at zero, one, or both, the logit link function is similarly applied to model the dichotomous Bernoulli-type variable with a multinomial response. The findings demonstrate that the model yields a non-singular information matrix, ensuring valid statistical inference. This ensures the invertibility of the information matrix, allowing for hypothesis testing based on likelihood statistics regarding the parameters in the model. This is not possible with other asymmetric models, such as those derived from the skew-normal distribution, which has a singular information matrix at the boundary of the skewness parameter. Finally, empirical results show the model’s effectiveness in analyzing proportion data with inflation at zero and one, proving its robustness and practicality for analyzing bounded data in various fields of research.

Scale mixtures of multivariate centered skew-normal distributions

Scale mixtures of multivariate centered skew-normal distributions

A novel constrained skew-Gaussian filter and its application to maneuverable reentry vehicle tracking

The state of the system generally satisfies specific constraints imposed by material properties or physical laws, so the application of these constraints can improve the accuracy of state estimation. In this paper, a novel recursive filter referred as constrained high-degree cubature skew-Gaussian filter (CHCSGF) is proposed, which achieves soft-constrained state estimation by compressing the probability density of unconstrained states with constraint information. First, the probability density of the state under inequality soft constraints is modeled as a skew-Gaussian (SG) distribution, rather than truncated or single Gaussian distributions. Then, a recursive constrained SG filter is developed to handle inequality soft constraints in linear systems. Addressing nonlinear challenges, a 5th-degree spherical-radial cubature approximation method is presented to numerically calculate SG-weighted integrals for the nonlinear transformation of SG distribution. Finally, the CHCSGF algorithm is proposed using this method to tackle nonlinear filtering problems. The CHCSGF is applied to reentry trajectory tracking to improve estimation accuracy by dealing with heat flow, dynamic pressure and overload constraints during reentry flight. Simulation results demonstrate that the CHCSGF achieves higher estimation accuracy than unconstrained methods under nonlinear inequality soft constraints, and is robust to the constraints with a prior error. Compared to particle filter and moving horizon estimation, the computational complexity of CHCSGF is significantly reduced.

The information matrix of the bivariate extended skew-normal distribution

AbstractFor the extended skew-normal distribution, which represents an extension of the normal (or Gaussian) distribution, we focus on the properties of the log-likelihood function and derived quantities in the bivariate case. Specifically, we derive explicit expressions for the score function and the information matrix, in the observed and the expected form; these do not appear to have been examined before in the literature. Ensuing numerical work illustrates some relevant aspects of the expected information matrix.

Bayesian Approach for Estimating the Parameters of the Time-To-Failure Distribution and Its Percentiles Under Power Degradation Model

The degradation models are often applied on the degradation data for studying time-to-failure distribution. In this study, the Bayesian approach is applied on the power degradation model for estimating the parameters of the time-to-failure distribution and its percentiles. Two different distributions are assumed for the degradation parameter of the model. The degradation parameter is firstly assumed to follow the skew-normal distribution with three jointly independently distributed parameters such that the gamma prior is assumed for the shape parameter, while the scale and the location parameters are assumed uniform. The second distribution assumed for the degradation parameter is the log-logistic distribution with two jointly independent random parameters where the shape parameter is assumed gamma, while the scale parameter is assumed uniform. Based on the Gibbs sampling method carried out under the JAGS platform, the models considered are applied on the simulated data and the NASA turbofan Jet engine dataset and the results found are compared. In modeling the time-to-failure distribution, it is shown that based on the simulated data and real data, the Bayesian approach for the power degradation model with the skew-normal degradation parameter outperformed the Bayesian approach for the power degradation model with the log-logistic degradation parameter.

Multivariate skew-normal distribution for modelling skewed spatial data

Multivariate spatial data are commonly modelled using the shared spatial component and multivariate intrinsic conditional autoregressive (MICAR) models where the spatial random variables are assumed to be normally distributed. However, the normality assumption may not be always right as the spatially structured component may show non-normal distributions. We present, multivariate skew-normal spatial distribution in the modelling of multivariate conditional autoregressive models. Simulations and an application to estimate district HIV rates in South Africa are used for illustrating the capabilities of the proposed multivariate skewed spatial model. The estimation is done in a Bayesian framework. A comparison between our suggested approach and the common MICAR model is made using conditional predictive ordinate (CPO). The CPO values indicate that our suggested approach is better than the MICAR model for predicting the outcome variables of both the simulated and HIV data.

A New Multimodal Modification of the Skew Family of Distributions: Properties and Applications to Medical and Environmental Data

The skew distribution has the characteristic of appropriately modeling asymmetric unimodal data. However, in practice, there are several cases in which the data present more than one mode. In the literature, it is possible to find a large number of authors who have studied extensions based on the skew distribution to model this type of data. In this article, a new family is introduced, consisting of a multimodal modification to the family of skew distributions. Using the methodology of the weighted version of a function, we perform the product of the density function of a family of skew distributions with a polynomial of degree 4, thus obtaining a more flexible model that allows modeling data sets, whose distribution contains at most three modes. The density function, some properties, moments, skewness coefficients, and kurtosis of this new family are presented. This study focuses on the particular cases of skew-normal and Laplace distributions, although it can be applied to any other distribution. A simulation study was carried out, to study the behavior of the model parameter estimates. Illustrations with real data, referring to medicine and environmental data, show the practical performance of the proposed model in the two particular cases presented.

Infrared temperature distribution characteristics and state assessment method of sandstone under tension and compression stress

Abnormalities in infrared radiation temperature (IRT) often accompany the loading In this investigation, stress experiments and splitting tests were conducted on sandstone to investigate the IRT characteristics of sandstone under various loading conditions. The results shows that the Maximum Infrared Radiation Temperature (MAXIRT) increases by up to 5 °C, while in splitting tests, the minimum infrared radiation temperature (MINIRT) decreases by approximately 1.5 °C. Compared to the average infrared radiation temperature (AIRT), MAXIRT is more sensitive to compressive stress, whereas (MINIRT) is more responsive to tensile stress. The correlation between IRT indicators and stress were further analyzed. The findings reveal a positive correlation between IRT and compressive stress. At low stress levels, the correlation coefficient fluctuates between −0.4 and 0.6, generally showing low correlation. As stress increases, the correlation coefficient rises, reaching above 0.8, indicating a high correlation. Tensile stress exhibits a negative correlation, with a consistent trend. Additionally, a statistical analysis of the frequency distribution of IRT under different stress conditions was conducted, followed by hypothesis testing. The results demonstrate that the probability distribution of IRT during loading follows a skewed normal distribution. The changes in skewness can be divided into three stages: initial fluctuation, stable variation, and abrupt failure. Under compressive stress, a right-skewed distribution is observed, with skewness exceeding 3.5 before fracture and reaching above 10 near fracture. Under tensile stress, a left-skewed distribution is noted, with skewness reaching −4.5 near fracture. These findings contribute to the assessment of rock stress states and provide early warning information for predicting rock failure.

Joint modeling of mixed skewed longitudinal responses using convolution of normal and log-normal distributions: a Bayesian approach

This paper investigates the joint modeling of mixed ordinal and continuous longitudinal responses using a random effects model and applying a conditional approach. For the ordinal responses, a latent variable model with a logistic distribution is employed. To address skewness in the data, the model incorporates normal and log-normal convolution (NLNC) for both the error term and the random effects in the longitudinal model. Parameter estimation is carried out within a Bayesian framework using Gibbs sampling. The performance of the proposed model is evaluated through simulation studies, comparing it to joint models of mixed ordinal and continuous longitudinal responses assuming skew-normal or normal distributions. The results indicate that joint models using skew-normal or normal distributions can lead to biased parameter estimates, whereas the NLNC joint model performs better overall. Additionally, the proposed method is applied to analyze data from the British Household Panel Survey (BHPS), using life satisfaction and annual income as the correlated ordinal and continuous longitudinal responses, respectively, with annual income showing significant skewness. The results demonstrate that the proposed model provides the best fit for capturing the substantial skewness in the data.

Multivariate doubly truncated moments for generalized skew-elliptical distributions with applications

ABSTRACT In this paper, we focus on multivariate doubly truncated first two moments of generalized skew-elliptical distributions. This class of distributions includes many useful distributions, such as skew-normal, skew Student- t , skew-logistic and skew-Laplace-normal distributions, as special cases. The formulas of multivariate doubly truncated covariance (MDTCov) for generalized skew-elliptical distributions are also given. Further, we compute multivariate doubly truncated expectations (MDTEs) and MDTCovs for 2 -variate skew-normal, skew-Student- t , skew-logistic and skew-Laplace-normal distributions, and use Monte-Carlo method to simulate and compare with the above results. As applications, the results of multivariate tail conditional expectation (MTCE) and multivariate tail covariance (MTCov) for generalized skew-elliptical distributions are derived. In addition, an optimal problem involving MDTE and MDTCov risk measures is proposed. Finally, we use real data to fit skew-normal distribution and to discuss MTCEs and MTCovs of logarithm of adjusted prices for two portfolios consisting of three companies from S&P (Standard & Poor’s) sectors.

Stability analysis of plasma waves driven by runaway electrons in tokamak hot plasmas

The local stability analysis of plasma waves driven by runaway electrons (REs) has been performed considering hot plasma Maxwellian background, with electron and ion temperatures of the order of 1 keV. It is shown that hot plasma waves, namely electron plasma waves (EPWs) and ion Bernstein waves (IBWs) can be driven unstable by RE at their coalescence frequency via Cherenkov resonance by RE with energy distribution peaked at about 8 MeV. A skew-normal distribution is used as a model of the RE energy distribution. The EPW and IBW couples of waves occur between any successive ion-cyclotron harmonics frequencies nf ci, above the lower hybrid resonance. At their confluence, the perpendicular group velocity vanishes and significant RF emissions are expected. The frequency gap between two successive confluences is ∼f ci. Groups of RF line emissions, separated by almost constant frequency gap ∼f ci/2 are detected during various quiescent runaway plasma discharges in the FTU tokamak. The analysis of a specific discharge suggests that the frequencies of the line emissions observed and the frequencies occurring at the EPW-IBW confluences are in reasonable agreement. A possible explanation of the line emissions with ∼f ci/2 gap in terms of nonlinear mode coupling is proposed.

Analyzing method comparison data with skew-normal measurement error models: incorporating generalized scale mixtures and varying degrees of freedom

Comparing two measurement methods is vital in various fields, such as medical research, epidemiology, economics, and environmental studies, to determine whether a new measurement method can be used interchangeably with an existing one. Measurement error models (MEMs) are commonly used for this purpose, where the methods have different measuring scales. However, these models often assume normality, which can be problematic when dealing with skewed and heavy-tailed data. To address this issue, we propose the replicated measurement error model (RMEM) under scale mixtures of skew-normal (SMSN) distributions with different levels of skewness and heavy tails of the true covariate and error distributions. Our primary aim is to assess the extent of similarity and agreement between two measurement methods using this model. The expectation conditional maximization (ECM) approach is applied to fit the model. A simulation study is conducted to evaluate the effectiveness and robustness of the proposed methodology and is illustrated by analyzing systolic blood pressure data. The probability of agreement is used further to assess the agreement between the two measurement methods. The findings indicate that the proposed model effectively analyses replicated method comparison data with measurement errors, even when there are outliers, skewness, and heavy-tailedness.

Intrinsic Mass–Richness Relation of Clusters from THE THREE HUNDRED Hydrodynamic Simulations

The main systematics in cluster cosmology is the uncertainty in the mass–observable relation. In this paper, we focus on the most direct cluster observable in optical surveys, i.e., richness, and constrain the intrinsic mass–richness (MR) relation of clusters in THE THREE HUNDRED hydrodynamic simulations with two runs: GIZMO-SIMBA and GADGET-X. We find that modeling the richness at a fixed halo mass with a skewed Gaussian distribution yields a simpler and smaller scatter compared to the commonly used lognormal distribution. Additionally, we observe that baryon models have a significant impact on the scatter while exhibiting no influence on the mass dependence and a slight effect on the amplitude in the MR relation. We select member galaxies based on both stellar mass M ⋆ and absolute magnitude M . We demonstrate that the MR relations obtained from these two selections can be converted to each other by using the M⋆–M relation. Finally, we provide a seven-parameter fitting result comprehensively capturing the dependence of the MR relation on both stellar mass cutoff and redshift.

Multivariate unified skew-[formula omitted] distributions and their properties

The unified skew-t (SUT) is a flexible parametric multivariate distribution that accounts for skewness and heavy tails in the data. A few of its properties can be found scattered in the literature or in a parameterization that does not follow the original one for unified skew-normal (SUN) distributions, yet a systematic study is lacking. In this work, explicit properties of the multivariate SUT distribution are presented, such as its stochastic representations, moments, SUN-scale mixture representation, linear transformation, additivity, marginal distribution, canonical form, quadratic form, conditional distribution, change of latent dimensions, Mardia measures of multivariate skewness and kurtosis, and non-identifiability issue. These results are given in a parameterization that reduces to the original SUN distribution as a sub-model, hence facilitating the use of the SUT for applications. Several models based on the SUT distribution are provided for illustration.

Estimation in shape mixtures of skew-normal linear regression models via ECM coupled with Gibbs sampling

Abstract In this paper, we study linear regression models in which the error term has shape mixtures of skew-normal distribution. This type of distribution belongs to the skew-normal (SN) distribution class that can be used for heavy tails and asymmetry data. For the first time, for the classical (non-Bayesian) estimation of the parameters of the SN family, we apply the Markov chains Monte Carlo ECM (MCMC-ECM) algorithm where the samples are generated by Gibbs sampling, denoted by Gibbs-ECM, and also, we extend two other types of the EM algorithm for the above model. Finally, the proposed method is evaluated through a simulation and compared with the Numerical Math-ECM algorithm and Monte Carlo ECM (MC-ECM) using a real data set.

A mixture of logistic skew-normal multinomial models

The logistic normal multinomial distribution is gaining interest in modelling microbiome data. It utilizes a hierarchical structure such that the observed counts conditional on the compositions are assumed to be multinomial random variables and the log-ratio transformed compositions are assumed to be from a Gaussian distribution. While multinomial distribution accounts for the compositional nature of the data, and a Gaussian prior offers flexibility in the structure of covariance matrices, the log-ratio transformed compositions of the microbiome data can be highly skewed, especially at a lower taxonomic level. Thus, a Gaussian distribution may not be an ideal prior for the log-ratio transformed compositions. A novel mixture of logistic skew-normal multinomial (LSNM) distribution is proposed in which a multivariate skew-normal distribution is utilized as a prior for the log-ratio transformed compositions. A variational Gaussian approximation in conjunction with the EM algorithm is utilized for parameter estimation.

An analysis on the accumulation of serotonin indicates the positively correlation between MdT5H5 and serotonin content in apple fruits

Serotonin (5-hydroxytryptamine, 5-HT), a tryptophan derivative, is an important functional component in fruits. Although 5-HT was found in most fruits, the accumulation characteristics of its biosynthesis in apple fruits remain unclear. Here, we conducted a large-scale survey of 148 apple fruits in 5-HT content, and investigated the biosynthesis differences of 5-HT. Our results displayed a broad skewed normal distribution of 5-HT in apple fruits. 5-HT accumulated in all tissues of apple plants, with the highest content in the seeds and the higher content in the pulp than that of the peel. A significant increase of 5-HT at ripening stage was observed in the fruits with high 5-HT content, however, the 5-HT content did not change significantly during fruit development in the fruits with low 5-HT content. The bagging reduced the 5-HT biosynthesis compared to the fruits exposed to light conditions. In addition, the expression level of MdT5H5 was positively correlated with the 5-HT content, suggesting that MdT5H5 may play a key regulatory role in the 5-HT biosynthesis in apple fruits.

SMALL AREA ESTIMATION WITH HIERARCHICAL BAYES FOR CROSS-SECTIONAL AND TIME SERIES SKEWED DATA

Small Area Estimation (SAE) is a method based on modeling for estimating small area parameters, that applies Linear Mixed Model (LMM) as its basic. It is conventionally solved with Empirical Best Linear Unbiased Prediction (EBLUP). The main requirement for LMM to produce high precision estimates is normally distributed. The observation unit is food crop farmer households from Sulawesi Tenggara Province to estimate food and non-food per capita expenditure at the district/city level using SAE that has been positively skewed. Applying EBLUP for positively skewed data will result less accurate estimates. Meanwhile, transformation will be potentially result biased estimates. Therefore, the problem of skewed data and small area level in this research was completed by Hierarchical Bayes (HB) on combination cross-sectional and time series under skew-normal distribution assumption. The results obtained were skew-normal SAE HB model was significantly reducing Relative Root Mean Squared Error (RRMSE) than the direct estimation. It indicates that SAE modeling is able to provide a shrinkage effect on the direct estimation results. But, there is slightly different interpretating between direct estimation and skew-normal SAE HB. It is possible because the modeling used assumption that the autocorrelation coefficient is equal to 1 or known as the random walk effect. However, in reality, Susenas is not a panel data, so unit of observation for each time period may be different. Therefore, further research should be compared it with the skew-normal or another skewed distribution that assumes the autocorrelation coefficient is unknown and should be estimated in the model.