Skew-normal Distribution Research Articles

Tractable skew-normal approximations via matching

Many approximate Bayesian inference methods assume a particular parametric form for approximating the posterior distribution. A Gaussian distribution provides a convenient density for such approaches; examples include the Laplace, penalized quasi-likelihood, Gaussian variational, and expectation propagation methods. Unfortunately, these all ignore potential posterior skewness. The recent work of Durante et al. [Skewed Bernstein-von Mises theorem and skew-modal approximations; 2023. ArXiv preprint arXiv:2301.03038.] addresses this using skew-modal (SM) approximations, and is theoretically justified by a skewed Bernstein-von Mises theorem. However, the SM approximation can be impractical to work with in terms of tractability and storage costs, and uses only local posterior information. We introduce a variety of matching-based approximation schemes using the standard skew-normal distribution to resolve these issues. Experiments were conducted to compare the performance of this skew-normal matching method (both as a standalone approximation and as a post-hoc skewness adjustment) with the SM and existing Gaussian approximations. We show that for small and moderate dimensions, skew-normal matching can be much more accurate than these other approaches. For post-hoc skewness adjustments, this comes at very little cost in additional computational time.

New Vessel Extraction Method by Using Skew Normal Distribution for MRA Images

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.

G.ridge: An R Package for Generalized Ridge Regression for Sparse and High-Dimensional Linear Models

Ridge regression is one of the most popular shrinkage estimation methods for linear models. Ridge regression effectively estimates regression coefficients in the presence of high-dimensional regressors. Recently, a generalized ridge estimator was suggested that involved generalizing the uniform shrinkage of ridge regression to non-uniform shrinkage; this was shown to perform well in sparse and high-dimensional linear models. In this paper, we introduce our newly developed R package “g.ridge” (first version published on 7 December 2023) that implements both the ridge estimator and generalized ridge estimator. The package is equipped with generalized cross-validation for the automatic estimation of shrinkage parameters. The package also includes a convenient tool for generating a design matrix. By simulations, we test the performance of the R package under sparse and high-dimensional settings with normal and skew-normal error distributions. From the simulation results, we conclude that the generalized ridge estimator is superior to the benchmark ridge estimator based on the R package “glmnet”. Hence the generalized ridge estimator may be the most recommended estimator for sparse and high-dimensional models. We demonstrate the package using intracerebral hemorrhage data.

Approximate parameter estimation and mis-specification analysis of degradation model with asymmetric random effects

Asymmetric distributions such as the skew-normal (SN) distribution have been used for modeling the unit-to-unit variation of the degradation rate of products, and its superiority for model fitting and lifetime estimation has been demonstrated in literature. However, there still exist two issues that need to be further studied. That is, how to simply estimate the parameters of the degradation model with asymmetric random effects and whether the asymmetry or skewness affects the remaining useful life (RUL) prediction. Driven by these two issues, the parameter estimation and RUL prediction of the degradation model with general random effects are first introduced and a two-step maximum likelihood estimation (MLE) method for parameter estimation is proposed, which could greatly simplify the parameter estimation of the degradation model with asymmetric random effects. Then, the SN distribution is used as an example to analyze the second issue regarding the effect of model mis-specification on RUL prediction based on regression models and Wiener processes with different asymmetric random effects in theory. Finally, the parameter estimation and mis-specification analysis of degradation models with other asymmetric random effects are conducted through numerical examples and two case studies. The results demonstrate that the proposed two-step MLE method is satisfactory and the effect of asymmetry on RUL prediction is negligible. Therefore, there is no need to consider the asymmetry of random effects for RUL prediction.

Skew Filtering for Online State Estimation and Control

Process control can become challenging when the measurements are affected by irregular noise. Classical approaches utilize Gaussian methods to alleviate the sensory noise. However, many industries involve skewed noise in their processes. While the closed skew-normal (CSN) distribution generalizes a Gaussian distribution with additional parameters, its dimension increases during recursive estimation, making it impractical. Even though there are some techniques for the solution, they are typically too complicated or inaccurate for higher-dimensional problems. This study proposes a novel online optimization scheme to reduce the dimensionality of a CSN distribution while considering the properties of the complete empirical distribution. Since the objective function used during the optimization step considers the geometry of the metric space, the proposed scheme achieves higher accuracy without sacrificing computational efficiency. The proposed filter is applied to two pilot-scale experiments. The results indicate it is beneficial for recursive state estimation in the presence of skewed noise.

Gaussian Copula Regression Modeling for Marker Classification Metrics with Competing Risk Outcomes

Decisions regarding competing risks are usually based on a continuous-valued marker toward predicting a cause-specific outcome. The classification power of a marker can be summarized using the time-dependent receiver operating characteristic curve and the corresponding area under the curve (AUC). This paper proposes a Gaussian copula-based model to represent the joint distribution of the continuous-valued marker, the overall survival time, and the cause-specific outcome. Then, it is used to characterize the time-varying ROC curve in the context of competing risks. Covariate effects are incorporated by linking linear components to the skewed normal distribution for the margin of the marker, a parametric proportional hazards model for the survival time, and a logit model for the cause of failure. Estimation is carried out using maximum likelihood, and a bootstrap technique is implemented to obtain confidence intervals for the AUC. Information-criteria strategies are employed to find a parsimonious model. The performance of the proposed model is evaluated in simulation studies, considering different sample sizes and censoring distributions. The methods are illustrated with the reanalysis of a prostate cancer clinical trial. The joint regression strategy produces a straightforward and flexible model of the time-dependent ROC curve in the presence of competing risks, enhancing the understanding of how covariates may affect the discrimination of a marker.

Classification Methods for the Serological Status Based on Mixtures of Skew-Normal and Skew-t Distributions

Gaussian mixture models are widely employed in serological data analysis to discern between seropositive and seronegative individuals. However, serological populations often exhibit significant skewness, making symmetric distributions like Normal or Student-t distributions unreliable. In this study, we propose finite mixture models based on Skew-Normal and Skew-t distributions for serological data analysis. Although these distributions are well established in the literature, their application to serological data needs further exploration, with emphasis on the determination of the threshold that distinguishes seronegative from seropositive populations. Our previous work proposed three methods to estimate the cutoff point when the true serological status is unknown. This paper aims to compare the three cutoff techniques in terms of their reliability to estimate the true threshold value. To attain this goal, we conducted a Monte Carlo simulation study. The proposed cutoff points were also applied to an antibody dataset against four SARS-CoV-2 virus antigens where the true serological status is known. For this real dataset, we also compared the performance of our estimated cutoff points with the ROC curve method, commonly used in situations where the true serological status is known.

Implication of site-specific segregation on grain boundary structural transition and deformation response in nanocrystalline Ni-Nb alloy

In this study, atomistic simulation technique is utilized to examine the impact of grain boundary (GB) character (disorientation angle), structure (boundary-free volume) and energy (boundary excess energy) on site-specific Nb solute segregation behaviour and structural transition in nanocrystalline (NC) Ni. Further, the role of site-specific solute segregation on deformation response in NC Ni is also studied. Towards this, hybrid Monte Carlo/molecular dynamics simulations is performed to obtain the equilibrium structure in a Ni-Nb binary alloy at 500 K and 1200 K. The spectral nature of GB atoms is explored in NC Ni, and found that the GBs have a skew-normal distribution for excess atomic volume and excess atomic energy. Furthermore, comprehensive segregation energy calculations are performed for each GB site, revealing the preferential affinity of Nb towards GB sites in NC Ni. The study also examined interfacial solute excess for each boundary as a function of GB energy and Voronoi volume. The findings indicate that the GB energy decreased as the solute excess increased, whereas the Voronoi volume demonstrated an increment in response to the solute excess. Moreover, simulations conducted at 1200 K reveal that not all GBs within the microstructure are transformed to amorphous complexions; rather some GBs remain ordered. The simulated tensile tests conducted at 500 K and 1200 K reveal that the strengthening in Ni-Nb enhances with the increase in dopant concentration. Additionally, the simulated shear test conducted at 1200 K shows that the deformation in Ni-5Nb occurs more uniformly than pure NC Ni due to formation of thicker amorphous GBs in the former one.

Generalized asymmetric mixture normal distribution: properties and applications

A new class of skew generalized normal distribution as a generalization of the skew-normal distribution of Arellano-Valle (Commun. Statist. Theor. Meth., 2004) and Kumar and Anusree (Commun. Statist. Theor. Meth., 2015) are considered. Several statistical properties of the proposed class of distributions are investigated and the location-scale extension of the model is obtained. Further, the estimation of its parameters by the method of maximum likelihood estimation is discussed and the distribution has been fitted to specific real-life data sets to illustrate the model’s usefulness. In addition, a brief simulation study is conducted for examining the performance of likelihood estimators of the parameters of the proposed model.

Quantification and comparison of hierarchy in Public Transport Networks

Network hierarchy describes the relative arrangement of network elements and reflects its fundamental structure. We propose a multi-dimensional topology-based method for quantifying and comparing the extent to which different Public Transport Networks (PTNs) exhibit a hierarchical structure. The proposed method considers the uneven distribution of node importance with different definitions (e.g., degree centrality and betweenness centrality) in a PTN, the clustering of nodes and the node connection patterns. We apply the developed method on 63 high-capacity PTNs worldwide using General Transit Feed Specification (GTFS) data. In addition to global indicators, we use the goodness-of-fit between the probability density function of local indicators and a skew-normal distribution to quantify the extent of PTN hierarchy. Results show that the scale-free network structure and preferential attachment do not vary much across PTNs. In contrast, stop accessibility and traffic intermediacy vary considerably across PTNs as reflected by the closeness centrality and betweenness centrality distributions. Lastly, metro systems exhibit a more hierarchical structure than their tram and Bus Rapid Transit (BRT) counterparts. This work makes a first step towards a better mapping and comparison of different PTNs, which can assist academics and practitioners in better (re)designing and planning the PTNs of the future.

A data-driven energy management strategy for plug-in hybrid electric buses considering vehicle mass uncertainty

Conventional energy management strategies (EMSs) for hybrid electric vehicles are devised assuming the vehicle mass remains constant under dynamic driving conditions. However, the EMSs cannot adapt to different load conditions due to the dynamic change of vehicle mass. Investigating the characteristics of vehicle mass change plays significant roles in energy optimization, thereby improving overall efficiency and mobility. To this end, we propose an adaptive EMS for a plug-in hybrid electric bus (PHEB) based on artificial neutral network-Pontryagin minimum principle (ANN-PMP) by considering mass distribution characteristics. Firstly, the skew-normal distribution characteristics of PHEB mass are analyzed, and the distribution characteristic spectrum of vehicle mass is obtained based on the Monte Carlo method. Secondly, the influence of mass uncertainty on the PMP is analyzed, and then the ANN-PMP is devised by updating a dynamic co-state with ANN. Finally, an enhanced ANN-PMP (so-called ANN-PMP-CS) is proposed by combining the ANN-PMP strategy obtained by mass distribution training and CS (charge sustaining strategy) to include the no-load or full-load cases. The simulation results demonstrate that the ANN-PMP can adapt to mass change while ensuring that the final state-of-charge (SOC) convergence to the target value. We also observe that the fuel economy of ANN-PMP-CS is similar to that of the widely-used dynamic programming (DP) strategy. Compared with the charge depletion-charge sustaining (CD-CS) strategy, the fuel economy can be improved by about 46.93 % on average.

Squared normal model and its generalization for the analysis of skewed positive data

To model skewed positive data with high kurtosis, this paper proposes, a new squared normal (SQN) distribution, which is constructed by squaring the normal random variable with non-zero mean and non-unit variance. The SQN distribution can be regarded as an extension of the chi-squared distribution, which cannot be applied to directly modeling real-life data. Some distributional properties, parameter estimation methods and hypothesis testings are investigated for the SQN distribution itself and the corresponding regression model. Furthermore, a generalization to a squared skew-normal (SSN) distribution is proposed by incorporating an additional skew parameter, improving the flexibility of the model. Finally, simulation experiments are conducted and a real data set is analyzed to demonstrate the proposed methodologies.

Parameter estimation of the multivariate unrestricted skew-normal distribution using ECM algorithm

The probability density function of the multivariate unrestricted skew-normal (SUN) distribution, corresponding to a screened normal density, allow to modeling skewness and kurtosis in data in terms of a skewness parameter vector and a truncation parameter matrix. These parameters are related to the shape and heavy-tails of the density. In this article, we present the Expectation/Conditional Maximization (ECM) algorithm for the SUN distribution based on a hierarchical stochastic representation. In addition, behavior of ECM algorithm’s steps is measured using an information theoretic approach based on Jeffrey’s divergence and related homogeneity test. Usefulness of the proposed method is illustrated by an application to Chilean economic perception data.

Multimodal Alpha Skew Normal Distribution: A New Distribution to Model Skewed Multimodal Observations

Multimodal Alpha Skew Normal Distribution: A New Distribution to Model Skewed Multimodal Observations

A multivariate skew-normal-Tukey-[formula omitted] distribution

We introduce a new family of multivariate distributions by taking the component-wise Tukey-h transformation of a random vector following a skew-normal distribution with an alternative parameterization. The proposed distribution is named the skew-normal-Tukey-h distribution and is an extension of the skew-normal distribution for handling heavy-tailed data. We compare this proposed distribution to the skew-t distribution, which is another extension of the skew-normal distribution for modeling tail-thickness, and demonstrate that when there are substantial differences in marginal kurtosis, the proposed distribution is more appropriate. Moreover, we derive many appealing stochastic properties of the proposed distribution and provide a methodology for the estimation of the parameters that can be applied to large dimensions. Using simulations, as well as a wine and a wind speed data application, we illustrate how to draw inferences based on the multivariate skew-normal-Tukey-h distribution.

A Comparison Small Area Estimation for Skewed Data with EBLUP and Hierarchical Bayes Approaching using Rao-Yu Model

Small Area Estimation (SAE) is a method based on modeling for estimating small area parameters, that applies the 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 for its sampling error. However, the researching data namely per capita food expenditure for food crop farmers’ households in Southeast Sulawesi Province has been positively skewed. Applying EBLUP for positively skewed data will produce less accurate estimates. Meanwhile, doing a transformation process will potentially produce biased estimates. The Hierarchical Bayes (HB) approaching often known as Full Bayesian is more flexible regarding normality assumptions and can determine distribution based on data. Because this research will carry out estimates at the district/city level throughout Southeast Sulawesi Province, totaling 17 regions, the efficiency of the estimates can be increased by using the SAE Rao-Yu model which includes area and time random effects. For this reason, this study compares SAE modeling with the EBLUP and HB approaches which are assumed to follow normal, log-normal, and skew-normal distributions. By comparing the Relative Root Mean Square Error (RRMSE), Deviance Information Criterion (DIC), and Coefficient of Variation (CV) values, it is concluded that estimates from the log-normal and skew-normal SAE HB models are more efficient than the SAE EBLUP and normal HB models. However, the SAE estimate that is closer to the direct estimation results is the skew-normal SAE HB.

On testing the skew normal distribution by using Shapiro–Wilk test

The skew normal (SN) family of distributions includes the normal distribution as a particular case as well as a wide variety of skew densities. Because of its tractability and flexibility for modeling both symmetric and skew data sets, the SN distribution has plenty of applications in finance, diverse engineering fields and medicine, among others. In this paper, a data transformation to approximately normal variables is used for testing the null hypothesis that a random sample follows a SN distribution with unknown parameters. Formulae obtained by interpolation methods are provided for approximating the critical values of the proposed procedure for a range of sample sizes, avoiding the use of tables and/or resampling methods. The test preserves the nominal test size and turns out to be competitive in terms of power against existing tests for the same problem. Several data sets are considered in order to illustrate the plausibility of the SN distribution for modeling the probability behavior of real data.

Control of particle transport driven by active noise: strategy of amplification via a periodic potential

We extend our previous studies on a counter-intuitive effect in which a directed transport of a free Brownian particle induced by active fluctuations can be significantly enhanced when the particle is placed in a periodic potential. It is in clear contrast to a common situation when the velocity of the Brownian particle is notably reduced if the periodic potential is switched on. As a model of active fluctuations, we employ white Poissonian shot noise. We reconsider the case of the skew-normal amplitude distribution of shot noise and focus on the impact of statistical characteristics of its amplitude like mean, variance, and skewness on the magnitude of free particle transport enhancement. In particular, we detect intriguing oscillations of the rescaled velocity of the particle as a function of the variance. Our findings can be corroborated experimentally in both biological and artificial microscopic systems.

Lambert W Random Variables and Their Applications in Loss Modelling

Several distributions and families of distributions are proposed to model skewed data, e.g., with skew-normal and related distributions. Lambert W random variables offer an alternative approach in which, instead of constructing a new distribution, a certain transformation is proposed. Such an approach allows the construction of a Lambert W skewed version from any distribution. Here, we choose the Lambert W normal distribution as a natural starting point and include the Lambert W exponential distribution due to the simplicity and shape of the exponential distribution, which, after skewing, may produce a reasonably heavy tail for loss models. In the theoretical part, we focus on the mathematical properties of obtained distributions, including the range of skewness. In the practical part, the suitability of the corresponding Lambert W transformed distributions is evaluated on real insurance data. Finally, the results are compared with those obtained using common loss distributions.

Statistical properties of co-quantiles and their applications to momentum spillovers

This article introduces co-quantiles, a new concept in rank statistics that generalizes quantiles, order statistics, and concomitants. In contrast to conventional order statistics that rank and record the same attribute of a population, or concomitants that consider different attributes observed over the same time period, co-quantiles allow for the ranking and recording of different attributes, and more generally their functions, across different time periods. In the special case of affine co-quantiles (defined formally in the article), we derive explicit expressions for their probability density functions and moments under the assumption that the distribution of the underlying data is multivariate normal. In particular, we establish that the former is a sum of unified skew-normal distributions introduced in Arellano-Valle and Azzalini (2006). The co-quantile results naturally reduce to those for order statistics and concomitants, and generalize those on the distributions of linear combinations and the maxima of vector-valued random variables obtained in Arellano-Valle and Genton (2007, 2008) and on cross-sectional momentum returns obtained in Kwon and Satchell (2018). As an example that has not been analyzed in the financial literature, we consider momentum spill-over returns and establish theoretically that these returns are susceptible to sudden changes in skewness and kurtosis during periods of market uncertainty.