Skew-t Distribution Research Articles

The Representative Points of Generalized Alpha Skew-t Distribution and Applications

Assuming the underlying statistical distribution of data is critical in information theory, as it impacts the accuracy and efficiency of communication and the definition of entropy. The real-world data are widely assumed to follow the normal distribution. To better comprehend the skewness of the data, many models more flexible than the normal distribution have been proposed, such as the generalized alpha skew-t (GAST) distribution. This paper studies some properties of the GAST distribution, including the calculation of the moments, and the relationship between the number of peaks and the GAST parameters with some proofs. For complex probability distributions, representative points (RPs) are useful due to the convenience of manipulation, computation and analysis. The relative entropy of two probability distributions could have been a good criterion for the purpose of generating RPs of a specific distribution but is not popularly used due to computational complexity. Hence, this paper only provides three ways to obtain RPs of the GAST distribution, Monte Carlo (MC), quasi-Monte Carlo (QMC), and mean square error (MSE). The three types of RPs are utilized in estimating moments and densities of the GAST distribution with known and unknown parameters. The MSE representative points perform the best among all case studies. For unknown parameter cases, a revised maximum likelihood estimation (MLE) method of parameter estimation is compared with the plain MLE method. It indicates that the revised MLE method is suitable for the GAST distribution having a unimodal or unobvious bimodal pattern. This paper includes two real-data applications in which the GAST model appears adaptable to various types of data.

Robust Classification via Finite Mixtures of Matrix Variate Skew-t Distributions

Robust Classification via Finite Mixtures of Matrix Variate Skew-t Distributions

Early prediction of gestational diabetes mellitus using first trimester maternal serum pregnancy-associated plasma protein-a: A cross-sectional study.

The oral glucose tolerance test with 75 g glucose is commonly regarded as the gold standard (GS) for the detection of gestational diabetes mellitus (GDM). However, one limitation of this test is its administration in the late second trimester of pregnancy in some countries (e.g., Iran). This study aimed to evaluate the accuracy of pregnancy-associated plasma protein-A (PAPP-A) for predicting GDM in the early first trimester of pregnancy using a novel statistical modeling technique. The study population consisted of 344 pregnant women who participated in the first trimester screening program for GDM. Maternal serum PAPP-A levels were measured between 11 and 13 gestational weeks for all participants. A Bayesian latent profile model (LPM) under the skew-t (ST) distribution was employed to estimate the diagnostic accuracy measures of PAPP-A in the absence of GS test outcomes. The mean (standard deviation) age of the participants was 28.87 ± 5.20 years. The median (interquartile range (IQR)) PAPP-A MoM was 0.91 (0.69-1.34). Utilizing the LPM under the ST distribution while adjusting for covariates, the sensitivity, specificity, and area under the receiver operating characteristic curve (AUC) of PAPP-A were 92% (95% credible interval [CrI]: 0.89, 0.98), 81% (95% CrI: 0.76, 0.91), and 0.91 (95% CrI: 0.83, 0.97), respectively. Notably, the pregnant women with GDM had significantly lower PAPP-A values ( -0.52, 95% CrI [-0.61, -0.46]). Generally, our findings confirmed that PAPP-A could serve as a potential screening tool for the identification of GDM in the early stages of pregnancy.

Volatility spillovers among economic policy uncertainty, energy and carbon markets—The quantile time-frequency perspective

As global uncertainty and trade tensions escalate, the world's economic stability faces growing risks. This study examines the volatility spillovers among economic policy uncertainty (EPU), energy, and carbon markets. We assume that returns in energy and carbon markets follow a Skew-t distribution and estimate volatility using an asymmetric EGARCH model. Then, we utilize methods developed by Diebold and Yilmaz (2012) and Barunik and Křehlík (2018), as well as a new quantile connectedness approach to investigate the directional and dynamic risk associations across different time frequencies and market environments. Our study first reveals that long-term volatility spillovers dominate normal periods, whereas medium-term spillovers become more pronounced in extreme cases. Second, EPU is a net receiver of risk in the short term but shifts to a risk transmitter in the medium and long term. Third, the solar energy market, a representative of new energy, drives spillover effects, while the carbon market is particularly susceptible to external influences. Fourth, the total risk spillovers exhibit a dynamic time-varying characteristic and are susceptible to extreme events. Finally, we observe an asymmetric distribution of risk spillovers among markets during extreme events. These findings offer valuable implications for policymakers and investors.

Large Skew-t Copula Models and Asymmetric Dependence in Intraday Equity Returns

Skew-t copula models are attractive for the modeling of financial data because they allow for asymmetric and extreme tail dependence. We show that the copula implicit in the skew-t distribution of Azzalini and Capitanio allows for a higher level of pairwise asymmetric dependence than two popular alternative skew-t copulas. Estimation of this copula in high dimensions is challenging, and we propose a fast and accurate Bayesian variational inference (VI) approach to do so. The method uses a generative representation of the skew-t distribution to define an augmented posterior that can be approximated accurately. A stochastic gradient ascent algorithm is used to solve the variational optimization. The methodology is used to estimate skew-t factor copula models with up to 15 factors for intraday returns from 2017 to 2021 on 93 U.S. equities. The copula captures substantial heterogeneity in asymmetric dependence over equity pairs, in addition to the variability in pairwise correlations. In a moving window study we show that the asymmetric dependencies also vary over time, and that intraday predictive densities from the skew-t copula are more accurate than those from benchmark copula models. Portfolio selection strategies based on the estimated pairwise asymmetric dependencies improve performance relative to the index.

A Two-Parameter Inverted Exponentiated Skewed Student-T Distribution: Theory and Application

In this study, a new two-parameter generalization of the skew-t distribution called the Inverted Exponentiated Skew t distribution, has been proposed. Some properties of the model like order statistics, entropy, asymptotic behavior, moments, characteristic function, and quantile function were derived. Parameter estimates using maximum likelihood estimation were consistent and the behavior of the estimates with increase in sample size was studied using Montecarlo simulation. The flexibility of the Inverted Exponentiated Skew Student t model was demonstrated by application to four financial datasets and the results revealed that the Inverted Exponentiated Skew Student t model yielded a better fit

Economic policy uncertainty and capital flows' tail risk in China

Since the financial crisis, uncertainty has been a focus of attention in the financial sector. As the two-way opening-up of China's financial market continues to accelerate, special attention must be paid to the risks and challenges arising from cross-border capital flows to maintain financial market stability in China. This study uses predictive quantile regression and skewed t-distribution fitting method to examine the impact of Economic Policy Uncertainty (EPU) on the full probability distribution and tail risks of capital flows. Additionally, we apply the “Capital Flows at Risk (CFaR)” analysis framework from the new paradigm of macroeconomic research to the situation in China, to study the impact of EPU on CFaR. We find that EPU significantly affects the predictive distribution of capital flows. When EPU is rising, the downside risk of capital flows, that is, the left-tail risk of capital flows, will increase; however, when EPU is declining, the upside risk of capital flows, that is, the right-tail risk of capital flows, will increase. In recent years, the scale of China's CFaR has shown a fluctuating upward trend, and EPU has increased the scale of CFaR in the country. This study is significant for stabilizing international capital flows and preventing and defusing the external financial shocks of cross-border capital inflows and outflows.

Flexible Bayesian semiparametric mixed-effects model for skewed longitudinal data

BackgroundIn clinical trials and epidemiological research, mixed-effects models are commonly used to examine population-level and subject-specific trajectories of biomarkers over time. Despite their increasing popularity and application, the specification of these models necessitates a great deal of care when analysing longitudinal data with non-linear patterns and asymmetry. Parametric (linear) mixed-effect models may not capture these complexities flexibly and adequately. Additionally, assuming a Gaussian distribution for random effects and/or model errors may be overly restrictive, as it lacks robustness against deviations from symmetry.MethodsThis paper presents a semiparametric mixed-effects model with flexible distributions for complex longitudinal data in the Bayesian paradigm. The non-linear time effect on the longitudinal response was modelled using a spline approach. The multivariate skew-t distribution, which is a more flexible distribution, is utilized to relax the normality assumptions associated with both random-effects and model errors.ResultsTo assess the effectiveness of the proposed methods in various model settings, simulation studies were conducted. We then applied these models on chronic kidney disease (CKD) data and assessed the relationship between covariates and estimated glomerular filtration rate (eGFR). First, we compared the proposed semiparametric partially linear mixed-effect (SPPLM) model with the fully parametric one (FPLM), and the results indicated that the SPPLM model outperformed the FPLM model. We then further compared four different SPPLM models, each assuming different distributions for the random effects and model errors. The model with a skew-t distribution exhibited a superior fit to the CKD data compared to the Gaussian model. The findings from the application revealed that hypertension, diabetes, and follow-up time had a substantial association with kidney function, specifically leading to a decrease in GFR estimates.ConclusionsThe application and simulation studies have demonstrated that our work has made a significant contribution towards a more robust and adaptable methodology for modeling intricate longitudinal data. We achieved this by proposing a semiparametric Bayesian modeling approach with a spline smoothing function and a skew-t distribution.

Finite mixture of regression models for censored data based on the skew-t distribution

Finite mixture of regression models for censored data based on the skew-t distribution

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.

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.

Semiparametric multivariate joint model for skewed-longitudinal and survival data: A Bayesian approach.

Joint models and statistical inference for longitudinal and survival data have been an active area of statistical research and have mostly coupled a longitudinal biomarker-based mixed-effects model with normal distribution and an event time-based survival model. In practice, however, the following issues may standout: (i) Normality of model error in longitudinal models is a routine assumption, but it may be unrealistically violating data features of subject variations. (ii) Data collected are often featured by the mixed types of multiple longitudinal outcomes which are significantly correlated, ignoring their correlation may lead to biased estimation. Additionally, a parametric model specification may be inflexible to capture the complicated patterns of longitudinal data. (iii) Missing observations in the longitudinal data are often encountered; the missing measures are likely to be informative (nonignorable) and ignoring this phenomenon may result in inaccurate inference. Multilevel item response theory (MLIRT) models have been increasingly used to analyze the multiple longitudinal data of mixed types (ie, continuous and categorical) in clinical studies. In this article, we develop an MLIRT-based semiparametric joint model with skew-t distribution that consists of an extended MLIRT model for the mixed types of multiple longitudinal data and a Cox proportional hazards model, linked through random-effects. A Bayesian approach is employed for joint modeling. Simulation studies are conducted to assess performance of the proposed models and method. A real example from primary biliary cirrhosis clinical study is analyzed to estimate parameters in the joint model and also evaluate sensitivity of parameter estimates for various plausible nonignorable missing data mechanisms.

Return volatility, correlation, and hedging of green and brown stocks: Is there a role for climate risk factors?

We examine the effects of three monthly climate risk factors, climate policy uncertainty (CPU), climate change news (CCN), and negative climate change news (NCCN), on the long-run volatilities and correlation of daily green and brown energy stock returns, and perform a hedging analysis. Given that our dataset combines daily and monthly data, we apply mixed data sampling models such as GARCH-MIDAS and DCC-MIDAS. To deal with volatility clustering, asymmetric effects, and negative skewness in innovations, which characterize our dataset, we use those models in asymmetric form with a bivariate skew-t distribution. Firstly, the GARCH-MIDAS models indicate that climate risk has a significant impact on the long-run volatility of brown energy stocks. Secondly, the DCC-MIDAS models reveal that the long-run correlation of green-brown stock returns decreases with the climate risk, suggesting a negative effect and hedging opportunities. Thirdly, the hedging analysis shows that incorporating a climate risk factor, especially NCCN, into the long-run component of dynamic correlation significantly improves the hedging performance between green and brown energy stock indices. The results are robust to an out-of-sample analysis under various refitting window sizes. They matter to portfolio and risk managers for energy transition and portfolio decarbonization.

Likelihood-based inference for the multivariate skew-[formula omitted] regression with censored or missing responses

Skew-t regression models have been widely used to model and analyze asymmetric heavy-tailed data. Moreover, observations in this kind of data can be missing or subject to some upper and/or lower detection limits because of the restriction of the experimental apparatus. We propose a novel robust regression model for multiple censored or missing data based on the multivariate skew-t distribution for such data structures. This approach allows us to model data with great flexibility, simultaneously accommodating heavy tails and skewness. We develop an analytically simple yet efficient EM-type algorithm to conduct maximum likelihood estimation of the parameters. The algorithm has closed-form expressions at the E-step that rely on formulas for the mean and variance of truncated multivariate Student’s-t, skew-t, and extended skew-t distributions. Furthermore, a general information-based method for approximating the asymptotic covariance matrix of the estimators is also presented. Results obtained from the analysis of both simulated and real datasets are reported to demonstrate the effectiveness of the proposed method.

Measuring systemic risk with high-frequency data: A realized GARCH approach

This paper incorporates high-frequency information to measure systemic risk. Under the Multivariate Realized GARCH framework, we compute the CoVaR measure using a multivariate skew-t distribution. Using 5-minute data of 98 U.S. financial institutions from 2000 to 2022, we show the empirical improvement of the high-frequency measurement. We also investigate the relationship between institutions’ systemic risk contributions and firm-level characteristics. Our empirical findings suggest that firm size and leverage are positively related to institutions’ contributions to systemic risk.

The Linear Skew-t Distribution and Its Properties

The aim of this expository paper is to present the properties of the linear skew-t distribution, which is a specific example of a symmetry modulated-distribution. The skewing function remains the distribution function of Student’s t, but its argument is simpler than that used for the standard skew-t. The linear skew-t offers different insights, for example, different moments and tail behavior, and can be simpler to use for empirical work. It is shown that the distribution may be expressed as a hidden truncation model. The paper describes an extended version of the distribution that is analogous to the extended skew-t. For certain parameter values, the distribution is bimodal. The paper presents expressions for the moments of the distribution and shows that numerical integration methods are required. A multivariate version of the distribution is described. The bivariate version of the distribution may also be bimodal. The distribution is not closed under marginalization, and stochastic ordering is not satisfied. The properties of the distribution are illustrated with numerous examples of the density functions, table of moments and critical values. The results in this paper suggest that the linear skew-t may be useful for some applications, but that it should be used with care for methodological work.

Robust Complex Probabilistic Slow Feature Analysis in the Presence of Skewed Measurement Noise

Complex slow feature analysis is a feature extraction technique that extracts slow oscillating patterns from the measured data. The measurement noise is usually assumed to follow a Gaussian distribution to obtain a closed-form solution. However, industrial process data is often characterized by measurement issues such as outliers, including asymmetric measurement noise. Such issues reduce the performance of the extracted features if not accounted for explicitly. Therefore, this article proposes a novel robust complex slow feature model to tackle the mentioned issues. In particular, this work considers a Skewed t-distribution for the measurement noise of the complex slow feature model. The parameters of the Skewed t-distribution, especially the degree of freedom and the shape parameters, account for the outliers and the asymmetric nature of the measurement noise. The parameters of the proposed model are jointly estimated using the expectation-maximization algorithm. The efficiency of the approach is demonstrated using simulated and industrial data.

A mixed-effects joint model with skew-t distribution for longitudinal and time-to-event data: A Bayesian approach

A mixed-effects joint model with skew-t distribution for longitudinal and time-to-event data: A Bayesian approach

A Semiparametric Bayesian Joint Modelling of Skewed Longitudinal and Competing Risks Failure Time Data: With Application to Chronic Kidney Disease

In clinical and epidemiological studies, when the time-to-event(s) and the longitudinal outcomes are associated, modelling them separately may give biased estimates. A joint modelling approach is required to obtain unbiased results and to evaluate their association. In the joint model, a subject may be exposed to more than one type of failure event (competing risks). Considering the competing event as an independent censoring of the time-to-event process may underestimate the true survival probability and give biased results. Within the joint model, longitudinal outcomes may have nonlinear (irregular) trajectories over time and exhibit skewness with heavy tails. Accordingly, fully parametric mixed-effect models may not be flexible enough to model this type of complex longitudinal data. In addition, assuming a Gaussian distribution for model errors may be too restrictive to adequately represent within-individual variations and may lack robustness against deviation from distributional assumptions. To simultaneously overcome these issues, in this paper, we presented semiparametric joint models for competing risks failure time and skewed-longitudinal data by using a smoothing spline approach and a multivariate skew-t distribution. We also considered different parameterization approaches in the formulation of joint models and used a Bayesian approach to make the statistical inference. We illustrated the proposed methods by analyzing real data on a chronic kidney disease. To evaluate the performance of the methods, we also carried out simulation studies. The results of both the application and simulation studies revealed that the joint modelling approach proposed in this study performed well when the semiparametric, random-effects parameterization, and skew-t distribution specifications were taken into account.

A Southern Photometric Quasar Catalog from the Dark Energy Survey Data Release 2

We present a catalog of 1.4 million photometrically selected quasar candidates in the southern hemisphere over the ∼5000 deg2 Dark Energy Survey (DES) wide survey area. We combine optical photometry from the DES second data release (DR2) with available near-infrared (NIR) and the all-sky unWISE mid-infrared photometry in the selection. We build models of quasars, galaxies, and stars with multivariate skew-t distributions in the multidimensional space of relative fluxes as functions of redshift (or color for stars) and magnitude. Our selection algorithm assigns probabilities for quasars, galaxies, and stars and simultaneously calculates photometric redshifts (photo-z) for quasar and galaxy candidates. Benchmarking on spectroscopically confirmed objects, we successfully classify (with photometry) 94.7% of quasars, 99.3% of galaxies, and 96.3% of stars when all IR bands (NIR YJHK and WISE W1W2) are available. The classification and photo-z regression success rates decrease when fewer bands are available. Our quasar (galaxy) photo-z quality, defined as the fraction of objects with the difference between the photo-z z p and the spectroscopic redshift z s , ∣Δz∣ ≡ ∣z s − z p ∣/(1 + z s ) ≤ 0.1, is 92.2% (98.1%) when all IR bands are available, decreasing to 72.2% (90.0%) using optical DES data only. Our photometric quasar catalog achieves an estimated completeness of 89% and purity of 79% at r < 21.5 (0.68 million quasar candidates), with reduced completeness and purity at 21.5 < r ≲ 24. Among the 1.4 million quasar candidates, 87,857 have existing spectra, and 84,978 (96.7%) of them are spectroscopically confirmed quasars. Finally, we provide quasar, galaxy, and star probabilities for all (0.69 billion) photometric sources in the DES DR2 coadded photometric catalog.