Student-t Distribution Research Articles

Rectified Gaussian Scale Mixtures and the Sparse Non-Negative Least Squares Problem.

In this paper, we develop a Bayesian evidence maximization framework to solve the sparse non-negative least squares problem (S-NNLS). We introduce a family of probability densities referred to as the Rectified Gaussian Scale Mixture (R-GSM), to model the sparsity enforcing prior distribution for the signal of interest. The R-GSM prior encompasses a variety of heavy-tailed distributions such as the rectified Laplacian and rectified Student-t distributions with a proper choice of the mixing density. We utilize the hierarchical representation induced by the R-GSM prior and develop an evidence maximization framework based on the Expectation-Maximization (EM) algorithm. Using the EM-based method, we estimate the hyper-parameters and obtain a point estimate for the solution of interest. We refer to this proposed method as rectified Sparse Bayesian Learning (R-SBL). We provide four EM-based R-SBL variants that offer a range of options to trade-off computational complexity to the quality of the E-step computation. These methods include the Markov Chain Monte Carlo EM, linear minimum mean square estimation, approximate message passing and a diagonal approximation. Using numerical experiments, we show that the proposed R-SBL method outperforms existing S-NNLS solvers in terms of both signal and support recovery, and is very robust against the structure of the design matrix.

On the Heavy-Tail Behavior of the Distributionally Robust Newsvendor

Since the seminal work of Scarf (1958) [A min-max solution of an inventory problem, Studies in the Mathematical Theory of Inventory and Production, pages 201-209] on the newsvendor problem with ambiguity in the demand distribution, there has been a growing interest in the study of the distributionally robust newsvendor problem. The optimal order quantity is computed by accounting for the worst possible distribution from a set of demand distributions that is characterized by partial information, such as moments. The model is criticized at times for being overly conservative since the worst-case distribution is discrete with a few support points. However, it is the order quantity from the model that is typically of practical relevance. A simple observation shows that the optimal order quantity in Scarf’s model with known first and second moment is also optimal for a heavy-tailed censored student-t distribution with degrees of freedom 2. In this paper, we generalize this “heavy- tail optimality” property of the distributionally robust newsvendor to a more general ambiguity set where information on the first and the nth moment is known, for any real number n > 1. We provide a characterization of the optimal order quantity under this ambiguity set by showing that for high critical ratios, the order quantity is optimal for a regularly varying distribution with an approximate power law tail with tail index n. We illustrate the applicability of the model by calibrating the ambiguity set from data and comparing the performance of the order quantities computed via various methods in a dataset.

Volatility dependences of stock markets with structural breaks

ABSTRACTWe develop a Vector Heterogeneous Autoregression model with Continuous Volatility and Jumps (VHARCJ) where residuals follow a flexible dynamic heterogeneous covariance structure. We employ the Bayesian data augmentation approach to match the realised volatility series based on high-frequency data from six stock markets. The structural breaks in the covariance are captured by an exogenous stochastic component that follows a three-state Markov regime-switching process. We find that the stock markets have higher volatility dependence during turmoil periods and that breakdowns in volatility dependence can be attributed to the increase in market volatilities. We also find positive correlations between the Asian stock markets, the European stock market, and the UK stock market. The US stock market has positive correlations with all other markets for most of the sample periods, indicating the leading position of US stock market in the global stock markets. In addition, the proposed three-state VHARCJ model with Dynamic Conditional Correlation (DCC) and break structure under student-t distribution has a superior density forecast performance as compared to the competing models. The forecast models with structural breaks outperform those without structural breaks based on the log predicted likelihood, the log Bayesian factor, and the root mean square loss function.

Modelling financial time series based on heavy-tailed market microstructure models with scale mixtures of normal distributions

ABSTRACTThis paper presents a type of heavy-tailed market microstructure models with the scale mixtures of normal distributions (MM-SMN), which include two specific sub-classes, viz. the slash and the Student-t distributions. Under a Bayesian perspective, the Markov Chain Monte Carlo (MCMC) method is constructed to estimate all the parameters and latent variables in the proposed MM-SMN models. Two evaluating indices, namely the deviance information criterion (DIC) and the test of white noise hypothesis on the standardised residual, are used to compare the MM-SMN models with the classic normal market microstructure (MM-N) model and the stochastic volatility models with the scale mixtures of normal distributions (SV-SMN). Empirical studies on daily stock return data show that the MM-SMN models can accommodate possible outliers in the observed returns by use of the mixing latent variable. These results also indicate that the heavy-tailed MM-SMN models have better model fitting than the MM-N model, and the market microstructure model with slash distribution (MM-s) has the best model fitting. Finally, the two evaluating indices indicate that the market microstructure models with three different distributions are superior to the corresponding stochastic volatility models.

A multivariate tail covariance measure for elliptical distributions

This paper introduces a multivariate tail covariance (MTCov) measure, which is a matrix-valued risk measure designed to explore the tail dispersion of multivariate loss distributions. The MTCov is the second multivariate tail conditional moment around the MTCE, the multivariate tail conditional expectation (MTCE) risk measure. Although MTCE was recently introduced in Landsman et al. (2016a), in this paper we essentially generalize it, allowing for quantile levels to obtain the different values corresponded to each risk. The MTCov measure, which is also defined for the set of different quantile levels, allows us to investigate more deeply the tail of multivariate distributions, since it focuses on the variance–covariance dependence structure of a system of dependent risks. As a natural extension, we also introduced the multivariate tail correlation matrix (MTCorr). The properties of this risk measure are explored and its explicit closed-form expression is derived for the elliptical family of distributions. As a special case, we consider the normal, Student-t and Laplace distributions, prevalent in actuarial science and finance. The results are illustrated numerically with data of some stock returns.

On designing a new cumulative sum Wilcoxon signed rank chart for monitoring process location.

In this paper, ranked set sampling is used for developing a non-parametric location chart which is developed on the basis of Wilcoxon signed rank statistic. The average run length and some other characteristics of run length are used as the measures to assess the performance of the proposed scheme. Some selective distributions including Laplace (or double exponential), logistic, normal, contaminated normal and student’s t-distributions are considered to examine the performance of the proposed Wilcoxon signed rank control chart. It has been observed that the proposed scheme shows superior shift detection ability than some of the competing counterpart schemes covered in this study. Moreover, the proposed control chart is also implemented and illustrated with a real data set.

D iCE: Diastereomeric in Silico Chiral Elucidation, Expanded DP4 Probability Theory Method for Diastereomer and Structural Assignment.

NMR chemical shift prediction at the B3LYP/cc-pVDZ level of theory was used to develop a highly accurate probability theory algorithm for the determination of the stereochemistry of diastereomers as well as the regiochemistry. DFT-GIAO calculations were performed for each conformer using geometry optimization and a CPCM solvent model. Boltzmann averaged shielding constants were converted to chemical shifts for 1H and 13C, using the generalized linear scaling terms determined in four different solvents for 1H and 13C and extended to 15N in DMSO. The probability theory algorithm, D iCE, was based on the DP4 method and developed for 1H, 13C, and 15N NMR using individual and combined probability data. The chemical shift calculation errors were fitted to a Student's t-distribution for 1H and 13C and a normal distribution for 15N. The application yielded a high accuracy for structural assignment with a low computational cost.

A Fast Distributed Variational Bayesian Filtering for Multisensor LTV System With Non-Gaussian Noise.

For multisensor linear time-varying system with non-Gaussian measurement noise, how to design distributed robust estimator to increase the accuracy and robustness to outliers at a relatively low computation and communication cost is a fundamental task. This paper proposes a fast distributed variational Bayesian (VB) filtering algorithm to recursively estimate the state and noise distribution over three conventional sensor networks: 1) incremental-based; 2) diffusion-based; and 3) consensus-based. To be specific, the non-Gaussian measurement noise of each sensor is modeled as Student- t distribution, and the system state and the parameters of the distribution are estimated via VB approach in each iteration step. An interaction scheme is then added to obtain the global optimal parameter by fusing the local optimal parameters over incremental, diffusion, and consensus communication topology. An efficient sensor selection criterion under these topologies based on the Cramér-Rao lower bound is proposed to reduce the communication and computation burden. Compared with the existing centralized VB filtering algorithms, the proposed algorithm in this paper can extensively increase the robustness to node or link failure at a lower computation cost with acceptable estimation performance and communication load. The theoretic results and simulation results are given to show the efficiency of our proposed algorithm.

An empirical application of a stochastic volatility model with GH skew Student's t-distribution to the volatility of Latin-American stock returns

Using daily stocks returns data of a set of Latin-American countries (Argentina, Brazil, Chile, Mexico and Peru) for the sample period 1996:01–2013:12, we estimate a stochastic volatility model incorporating both leverage effects and skewed heavy-tailed disturbances through of the GH Skew Student's t-distribution based on Bayesian estimation method proposed by Nakajima and Omori (2012). Two alternative models are estimated, one using an alternative Skew Student's t-distribution and the other using a symmetric Student's t-distribution. The results suggest the presence of leverage effects in all markets except for Peru where the evidence is unclear. In addition, there is evidence of asymmetries and heavy tails in the Argentina and S&P500 markets while in the other countries there is no robust evidence of such characteristics. Using the Bayes factor, the results indicate that the SVGHSkewt model dominates the other two models for the cases of Peru, Argentina, Brazil and S&P500 whereas the simple SVt model is preferred for the markets of Mexico and Chile. Similar findings are obtained after performing a robustness analysis regarding the priors of the parameters associated with the skewness and the tails of the distribution.

Forecasting the Volatility of Ethiopian Birr/Euro Exchange Rate Using Garch-Type Models

This paper provides a robust analysis of volatility forecasting of Euro-ETB exchange rate using weekly data spanning the period January 3, 2000–December 2, 2015. The forecasting performance of various GARCH-type models is investigated based on forecasting performance criteria such as MSE and MAE based tests, and alternative measures of realized volatility. To our knowledge, this is the first study that focuses on Euro-ETB exchange rate using high frequency data, and a range of econometric models and forecast performance criteria. The empirical results indicate that the Euro-ETB exchange rate series exhibits persistent volatility clustering over the study period. We document evidence that ARCH (8), GARCH (1, 1), EGARCH (1, 1) and GJR-GARCH (2, 2) models with normal distribution, student’s-t distribution and GED are the best in-sample estimation models in terms of the volatility behavior of the series. Amongst these models, GJR-GARCH (2, 2) and GARCH (1, 1) with students t-distribution are found to perform best in terms of one step-ahead forecasting based on realized volatility calculated from the underlying daily data and squared weekly first differenced of the logarithm of the series, respectively. A one-step-ahead forecasted conditional variance of weekly Euro-ETB exchange rate portrays large spikes around 2010 and it is evident that weekly Euro-ETB exchange rate are volatile. This large spikes indicates that devaluation of Ethiopian birr against the Euro. This volatility behavior may affects the International Foreign Investment and trade balance of the country. Therefore, GJR-GARCH (2, 2) with student’s t-distribution is the best model both interms of the stylized facts and forecasting performance of the volatility of Ethiopian Birr/Euro exchange rate among others.

Robust Student's-t mixture modelling via Markov random field and its application in image segmentation

Finite mixture model has been widely applied to image segmentation. However, the technique does not consider the spatial information in images that leads to unsatisfactory results for image segmentation. To address this problem, in this paper, a Student's-t mixture model is proposed for image segmentation based on Markov random field (MRF). There are three advantages in the proposed model. Firstly, a representation of spatial relationships among pixels is given. Secondly, Student's t-distribution is chosen to be the component function of the proposed model instead of the Gaussian distribution because of its heavy tail. Thirdly, to deduce the parameters of the proposed model, a gradient descent method is applied during the inference process. Comprehensive experiments are carried out on greyscale noisy images and real-world colour images. The experimental results have shown the effectiveness and robustness of the proposed model.

Robust Student's-t mixture modelling via Markov random field and its application in image segmentation

Finite mixture model has been widely applied to image segmentation. However, the technique does not consider the spatial information in images that leads to unsatisfactory results for image segmentation. To address this problem, in this paper, a Student's-t mixture model is proposed for image segmentation based on Markov random field (MRF). There are three advantages in the proposed model. Firstly, a representation of spatial relationships among pixels is given. Secondly, Student's t-distribution is chosen to be the component function of the proposed model instead of the Gaussian distribution because of its heavy tail. Thirdly, to deduce the parameters of the proposed model, a gradient descent method is applied during the inference process. Comprehensive experiments are carried out on greyscale noisy images and real-world colour images. The experimental results have shown the effectiveness and robustness of the proposed model.

Estimating GARCH Modeling Using Metropolis-Hastings Method in R

This paper mainly talks about a popular approach of volatility of a GARCH-type model in R, while the disturbances are independent and have identical Student-t distribution. It uses the Metropolis-Hastings method to perform the computations and gives the programs in details in R.

Bayesian data analysis to quantify the uncertainty of intact rock strength

One of the main difficulties in the geotechnical design process lies in dealing with uncertainty. Uncertainty is associated with natural variation of properties, and the imprecision and unpredictability caused by insufficient information on parameters or models. Probabilistic methods are normally used to quantify uncertainty. However, the frequentist approach commonly used for this purpose has some drawbacks. First, it lacks a formal framework for incorporating knowledge not represented by data. Second, it has limitations in providing a proper measure of the confidence of parameters inferred from data. The Bayesian approach offers a better framework for treating uncertainty in geotechnical design. The advantages of the Bayesian approach for uncertainty quantification are highlighted in this paper with the Bayesian regression analysis of laboratory test data to infer the intact rock strength parameters σci and mi used in the Hoek–Brown strength criterion. Two case examples are used to illustrate different aspects of the Bayesian methodology and to contrast the approach with a frequentist approach represented by the nonlinear least squares (NLLS) method. The paper discusses the use of a Student's t-distribution versus a normal distribution to handle outliers, the consideration of absolute versus relative residuals, and the comparison of quality of fitting results based on standard errors and Bayes factors. Uncertainty quantification with confidence and prediction intervals of the frequentist approach is compared with that based on scatter plots and bands of fitted envelopes of the Bayesian approach. Finally, the Bayesian method is extended to consider two improvements of the fitting analysis. The first is the case in which the Hoek–Brown parameter, a, is treated as a variable to improve the fitting in the triaxial region. The second is the incorporation of the uncertainty in the estimation of the direct tensile strength from Brazilian test results within the overall evaluation of the intact rock strength.

Identification of robust probabilistic slow feature regression model for process data contaminated with outliers

Modeling of high dimensional dynamic process is considered as a challenging task. In this regard, probabilistic Slow Feature Analysis (PSFA), a dynamic latent variable model, is proven to be a useful tool which extracts temporally correlated dynamic features from the high-dimensional raw measurements. The extracted latent Slow Features (SFs) can capture process variations which are useful in developing dynamic models. Often times industrial data is affected by outliers, and modeling such data could result in inferior prediction performance. To deal with such scenarios, we propose a robust PSFA (RPSFA) based regression model that models outliers in the observation data using the Student's t-distribution. To estimate the parameters in RPSFA and to extract reduced dimension of SFs, we employ Expectation-Maximization (EM) algorithm under the Maximum Likelihood Estimation (MLE) framework considering SFs as hidden variables. To estimate the hidden SFs we propose a weighted gain Kalman filter based approach as the Normal distribution assumption of the observations is no longer valid. The validity and merits of the proposed approach are demonstrated though a simulated example, an industrial application and an experimental study.

Long-range memory, distributional variation and randomness of bitcoin volatility

We investigate the nonlinear patterns of volatility in seven Bitcoin markets. In particular, we explore the fractional long-range dependence in conjunction with the potential inherent stochasticity of volatility time series under four diverse distributional assumptions, i.e., Normal, Student-t, Generalized Error (GED), and t-Skewed distribution. Our empirical findings signify the existence of long-range memory in Bitcoin market volatility, irrespectively of distributional inference. The same applies to entropy measurement, which indicates a high degree of randomness in the estimated series. As Bitcoin markets are highly disordered and risky, they cannot be considered suitable for hedging purposes. Our results provide strong evidence against the efficient market hypothesis.

On mixture autoregressive conditional heteroskedasticity

We consider mixture univariate autoregressive conditional heteroskedastic models, both with Gaussian or Student t-distributions, which were proposed in the literature for modeling nonlinear time series. We derive sufficient conditions for second order stationarity of these processes. Then we propose an algorithm in matrix form for the estimation of model parameters, and derive a formula in closed form for the asymptotic Fisher information matrix. Our results are proved by using the theory of time series models with Markov changes in regime. An illustrative example of the theoretical results and a real application on financial data complete the paper.

'Parallel Universe' or 'Proven Future'? The Language of Dependent Means t-Test Interpretations

Of the three kinds of two-mean comparisons which judge a test statistic against a critical value taken from a Student t-distribution, one – the repeated measures or dependent-means application – is distinctive because it is meant to assess the value of a parameter which is not part of the natural order. This absence forces a choice between two interpretations of a significant test result and the meaning of the test hypothesis. The parallel universe view advances a conditional, backward-looking conclusion. The more practical proven future interpretation is a non-conditional proposition about what will happen if an intervention is (now) applied to each population element. Proven future conclusions are subject to the corrupting influence of time-displacement, which include the effects of learning, development, and history. These two interpretations are explored, and a proposal for new conceptual categories and nomenclature is given to distinguish them, applicable to other repeated measures procedures derived from the general linear model including ANOVA.

Some Inferences on Skew-t Distribution of 2 Degrees of Freedom

Understanding the properties of Skew distributions is important in many statistical applications. The Student-t distribution is widely used in statistics. Skew-t distribution is a useful model to describe data with heavy tails. In this paper several distributional properties of Skew-t distribution are given. Based on the distributional properties some characterizations of the Skew-t distribution are given.

AWARENESS OF PATIENTS’ BILLS OF RIGHTS AMONG MEDICAL STAFF IN MASHHAD MEDICAL UNIVERSITY` TEACHING HOSPITALS

Introduction: Respecting patients’ rights practice and keeping patient satisfied is one of most important principles in ethical medicine. Increasing Awareness and respecting patients’ rights result in better health care considering human and ethical rights. The Ministry of Health and Medical Education in Iran published a patients’ bill of rights and mandated it be posted conspicuously for the patients. With regard to necessity of Patients’ bill of rights, patients’ role in decision making and respecting their right, this study aimed to investigate awareness of patients’ bills of rights among medical staff in Mashhad medical university` teaching hospitals
 Methods: The current analytical cross- sectional study was conducted on 129 medical staff in the year 2014-15 in Mashhad medical university` teaching hospitals. The data were collected using a self-administer questionnaire with 28 questions whose reliability was 80 according to Cronbach's alpha. One-way analysis of variance, Student's t-distribution was used to compare means of Awareness via SPSS Ver. 21.
 Results: The results indicated that awareness of medical staff was M=65/3, SD=67/0. Nursing staff were the most aware and radiology staff were the less aware from patients’ bills of rights.
 Conclusion: Today, observing patients 'rights is one of the most important issues that should be placed at the top of the attention of health care programs, observance of patients’ bills of rights make people feel satisfied with respect for them at health care centers . So it is neceesery to inform patient and professional on patients’ bills of rights by publishing more attention to patients’ bills of rights result on better health care