Student-t Distribution Research Articles

Failure rate of Birnbaum–Saunders distributions: Shape, change-point, estimation and robustness

The Birnbaum–Saunders (BS) distribution has been largely studied and applied. A random variable with BS distribution is a transformation of another random variable with standard normal distribution. Generalized BS distributions are obtained when the normally distributed random variable is replaced by another symmetrically distributed random variable. This allows us to obtain a wide class of positively skewed models with lighter and heavier tails than the BS model. Its failure rate admits several shapes, including the unimodal case, with its change-point being able to be used for different purposes. For example, to establish the reduction in a dose, and then in the cost of the medical treatment. We analyze the failure rates of generalized BS distributions obtained by the logistic, normal and Student-t distributions, considering their shape and change-point, estimating them, evaluating their robustness, assessing their performance by simulations, and applying the results to real data from different areas.

DYNAMIC CHANGES IN PARAMETERS OF ORAL FLUID VISCOSITY AND pH IN CHILDREN WITH RECURRENT ACUTE RESPIRATORY VIRAL INFECTIONS IN THE COURSE OF PREVENTIVE TREATMETN

A large number of common etiological factors, among which there are frequent recurrent acute respiratory viral infections, are known to contribute to the development of dental caries. The aim of this study is to investigate the dynamics of the parameters of oral fluid viscosity and pH in children with recurrent acute respiratory viral infections in the course of preventive treatment. The study included 80 children divided into 4 groups of 20 people in each. The viscosity of the oral fluid was assessed by the method of T. L. Redinova. The oral fluid pH was evaluated using the universal indicator paper. The findings obtained were statistically processed by the Student's t-distribution, the results were considered reliable at p ≤ 0.05. The incidence rate of acute respiratory viral infections was determined by studying medical records and by questioning the parents. The effectiveness of therapeutic and preventive measures was assessed in 1 and 6 months after the first visit to dentist’s office. The parameters of oral fluid viscosity were within the normal range in all groups under the study. The children of groups 3 and 4, who were found to more often suffer from acute respiratory viral infections demonstrated that in 6 months following the completion of the preventive treatment we elaborated this value remained lower than during the first examination. The study of the oral fluid pH in the children of the examined groups showed its positive changes in all cases. During the second examination, an increase in the parameters ​​was observed in all the test groups, but these changes were significant in the 4 group of children (6.90 ± 0.05 units), who kept the complex of preventive measured we designed. During the 3rd check-up in 6 months, the result remained stable in the children of the 3rd and 4th test groups (6.79 ± 0.05 units and 6.87 ± 0.06 units, respectively). We can suggest that significant improvement in the values of oral fluid viscosity and pH in all the test groups is due to regular oral care and good oral hygiene, use of therapeutic and prophylactic means as well as additional hygiene products.

Modeling and Forecasting the Volatility of Gasoline Prices Using Symmetric and Asymmetric GARCH Models in Erbil City

The paper aims to compare the performance of several univariate symmetric and asymmetric GARCH volatility models in modeling and forecasting the volatility of daily Gasoline prices in Erbil city. This paper chooses the GARCH, GARCH-M, TGARCH, E-GARCH and Power GARCH model to analyze the daily return of Gasoline under three different error distributions: normal distribution, student-t distribution and generalized error distribution and then compare the results and choose the appropriate model to forecast the volatility. The sample is divided into two subsamples. The first subsample is called in-sample data set (Training sample) used to estimate the ARMA-GARCH models for underlying data and the second subsample is called out-sample data set (Testing sample) used to investigate the performance of volatility forecasting. As a result of analyses, we conclude that the best model fits the volatility of Gasoline returns series is AR(2)-Power GARCH(2,1,1) non-linear asymmetric model with innovation student-t distribution (d.f =10), and has better forecasting performance than others models. This result is important in many fields of finance such as investment decisions, asset pricing, portfolio allocation and risk management. Keywords : Conditional Variance; Volatility Clustering; Symmetric and Asymmetric GARCH Models; Error Distribution; Volatility Forecasting; Root Mean Square Error (RMSE)

MSEPBurr Distribution: Properties and Parameter Estimation

MSNBurr and MSTBurr distribution have been developed as Neo-Normal distributions that represent a relaxation of normality. The difference between them is that the MSTBurr’s peak is below MSNBurr’s. In this paper, we propose a MSEPBurr distribution with its peak could be not only lower but also high-er than MSNBurr. Furthermore, we study several properties of MSEPBurr, such as mean, variance, skewness, kurtosis, and quantile. The MSEPBurr parameters are estimated by using the Bayesian approach with the BUGS language implementation for its computation. We employ simulation study and use existing data to illustrate the application of the regression model. In real data, we notice that MSEPBurr has similar performance with MSNBurr and MSTBurr that they outperform Normal and Student-t distribution in Australian athlete data because their skewness can accommodate long left tail excellently. However, their performance is less than the Student-t model in chemical reaction rate data because their skewness can not accommodate long right tail perfectly. Although in general their perfor-mance is the same, we observe that the MSEPBurr performs better than the MSNBurr and the MSTBurr in some simulated data.

Comparison Between Apple and AT&T (Wilshire 500)

This study presents the comparison between APPLE and AT&T by using the Capital Asset Pricing Model (CAPM). This process has improved asset pricing abilities and allow to discovery of the “one true” market returns, hence the CAPM can be used to find the both company performance with the overall index. The market portfolio returns required for CAPM to be accurate are then calculated and compared with its performance for the past 10 years (1/4/2009-29/12/2017)of each both companies Apple and AT&T comparing their performance to the overall index market of the whole world Wilshire 500. The overall best distribution to use for CAPM market return is the student t-distribution. This study also contributes to literature in that it finds the market proxies used in other studies to discredit the CAPM are inefficient and adversely affect the result.

Analysis of the Duration–Hardness Ratio Plane of Gamma-Ray Bursts Using Skewed Distributions

Abstract The two widely accepted classes of gamma-ray bursts (GRBs), short and long, are with confidence ascribed to mergers of compact objects and collapse of massive stars, respectively. A third, intermediate/soft class, remains putative. Its existence was claimed based on univariate and bivariate analyses of GRB observables modeled with Gaussian distributions. This, however, may not be the appropriate approach, as it has already been shown that the univariate distributions of durations are better described by mixtures of two skewed components rather than three Gaussian ones. This paper investigates whether data in the duration–hardness ratio plane is better modeled by mixtures of skewed bivariate distributions than by normal ones. The archival data set of the Compton Gamma-Ray Observatory/BATSE and Fermi/Gamma-ray Burst Monitor data from the most recent catalog release are examined. The preferred model is chosen based on two information criteria, Akaike and Bayesian. It is found that the best description is given by a two-component mixture of skewed Student-t distributions, which outperforms any other model considered. This implies that the distribution of the studied parameters is intrinsically skewed, introducing spurious Gaussian components, and hence the third class is unlikely to be a real phenomenon. Its existence, based on statistical inference, is therefore rejected as unnecessary to explain the observations.

Stock Market Integration of Pakistan with Its Trading Partners: A Multivariate DCC-GARCH Model Approach

A decade after the global financial crisis, the developments in stock market integration have increased the stability and liquidity of markets, and decreased the diversification benefits for investors. International trade is an important determinant of stock market interdependence. The objective of this study is to analyze the co-movements and the portfolio diversification between the stock markets of Pakistan and its top trading partners, namely China, Indonesia, Malaysia, the United Kingdom, and the United States. We employed Dynamic Conditional Covariance (DCC)-Generalized Autoregressive Conditional Heteroscedasticity (GARCH) methodology with student t-distribution to examine time-varying correlation and volatilities of stock markets of Pakistan and its trading partners. We used Morgan Stanley capital international (MSCI) daily returns data of developed and emerging markets for the period 2005 to 2018. The results of the study highlighted that stock markets of Pakistan and its trading partners were closely integrated during the financial crisis of 2008, while the integration among stock markets decreased substantially after the period of financial crises. Furthermore, the results showed the slow decay process. Therefore, it is a positive sign for the Pakistani and international investors to diversify their portfolio among the stock markets of Pakistan and its trading partners.

Moments of Student's t-distribution: A Unified Approach

In this paper, we derive the closed form formulae for moments of Student's t-distribution in the one dimensional case as well as in higher dimensions through a unified probability framework. Interestingly, the closed form expressions for the moments of Student's t-distribution can be written in terms of the familiar Gamma function, Kummer's confluent hypergeometric function, and the hypergeometric function.

A New Robust Kalman Filter for SINS/DVL Integrated Navigation System

A new robust strap-down inertial navigation system (SINS) and Doppler velocity log (DVL) integrated navigation algorithm are proposed in this paper with a focus on suppressing the process uncertainty and measurement outliers induced by severe manoeuvering. In the proposed algorithm, the one-step predicted probability density function is modeled as Student's t-distribution to deal with the heavy-tailed process noise, and hierarchical Gaussian state-space model for SINS/DVL integrated navigation algorithm is constructed. To detect and eliminate the measurement outliers, each measurement is marked by a binary indicator variable modeled as a beta-Bernoulli distribution. The variational Bayesian approach is used to jointly estimate state vector, auxiliary random variable, scale matrix, Bernoulli variable, and beta variable. The experimental results illustrate that the proposed algorithm has better robustness and navigation accuracy to deal with process uncertainty and measurement outliers than existing state-of-the-art algorithms.

The Labeled Multi-Bernoulli Filter for Jump Markov Systems Under Glint Noise

This paper proposes a novel labeled multi-Bernoulli (LMB) filter for jump Markov systems (JMS) to track the multiple maneuvering objects under glint noise. By modeling the glint noise as a Student's t-distribution and using the variational Bayesian method to acquire the approximate state distribution, we present an efficient implementation of the LMB filter with joint prediction and update for JMS. Simulation results illustrate that the proposed filter outperforms the existing filters for multi-object tracking under glint noise.

Geological Context and Statistical Assessment of the Impacts of Sugarloaf Twin Disasters, in Western Sierra Leone

In the event of natural disasters, Sierra Leone like any other country loses lives as well as properties and natural resources. Despite notable occurrences of small disasters in the country, only on August 14th sugarloaf landslide and the resulting flash flood (twin disasters) which affected Regent and other communities in 2017 have captured both national and international attention due to their alarming impacts. To obtain a better knowledge on the nature of the disasters, this research considered the geological context of the landslide and focused on statistical analysis of the impacts of the twin disasters using Statistical Package for Social Sciences (SPSS). Geological field-based approach and self-administered questionnaires of both closed and open-ended questions, personal interviews and observations were adopted as research techniques. Descriptive statistics and qualitative analysis were used to analyze the data of the research. Relevant literature from books, journals, and newspapers was also consulted, including Radio, Television discussions and the internet. Slope destabilization that caused the Landslide was linked to the geology of sub-surface formation (highly weathered olivine gabbro), lineaments and the rate of weathering. Study showed remarkable economic, social and political impacts as a consequence of the twin disasters. Study also revealed that families assisted the most in both financial and non-financial ways to the victims, and renowned misappropriation of donated disaster funds. All variables tested using student t-distribution and chi-square test of independence accepted the hypothesis. Information obtained from this work could be fundamental for policy makers and development practitioners (Osuteye & Leck, 2017).

Measuring portfolio risk of non-energy commodity using time-varying vine copula

In literature, many empirical studies have used the vine copula to measure and analyse the risk of stock indices, energetic products or crypto currencies portfolio (e.g., Zhang et al., 2014; Shahzad et al., 2018; Boako et al., 2019). These studies used mainly the VaR and ES with different versions of vine copula. However, few works have focused on a portfolio composed of non-energy commodities. Our aim in this work is to model the dependence between non-energy commodities returns by modelling standardised residuals obtained from univariate GARCH model by different versions of time varying vine copula, to quantify risk of N-dimensional non-energy commodity portfolio by VaR and ES and to compare the predictive performance of this method with traditional and competitive traditional univariate VaR methods. Empirical results suggest that risk quantifies generated by AR-GARCH vine copula methods with student-t distribution are sufficiently accurate at both low and high confidence levels. Given these results, we recommend the application of vine copula method to understanding the non-energy commodity behaviours which are very important to investors, producers, consumers, and policy makers.

Half-Life Volatility Measure of the Returns of Some Cryptocurrencies

This paper explores the half-life volatility measure of three cryptocurrencies (Bitcoin, Litecoin and Ripple). Two GARCH family models were used (PGARCH (1, 1) and GARCH (1, 1)) with the student-t distribution. It was realised that, the PGARCH (1, 1) was the most appropriate model. Therefore, it was used in determining the half-life of the three returns series. The results revealed that, the half-life was 3 days, 6 days and 4 days for Bitcoin, Litecoin and Ripple respectively. This shows that, the three coins have strong mean reversion and short half-life and that it takes the respective days for volatility in each of coin to return half way back without further volatility.

Selection of Heteroscedastic Models: A Time Series Forecasting Approach

To overcome the weaknesses of in-sample model selection, this study adopted out-of-sample model selection approach for selecting models with improved forecasting accuracies and performances. Daily closing share prices were obtained from Diamond Bank and Fidelity Bank as listed in the Nigerian Stock Exchange spanning from January 3, 2006 to December 30, 2016. Thus, a total of 2713 observations were explored and were divided into two portions. The first which ranged from January 3, 2006 to November 24, 2016, comprising 2690 observations, was used for model formulation. The second portion which ranged from November 25, 2016 to December 30, 2016, consisting of 23 observations, was used for out-of-sample forecasting performance evaluation. Combined linear (ARIMA) and Nonlinear (GARCH-type) models were applied on the returns series with respect to normal and student-t distributions. The findings revealed that ARIMA (2,1,1)-EGARCH (1,1)-norm and ARIMA (1,1,0)-EGARCH (1,1)-norm models selected based on minimum predictive errors throughout-of-sample approach outperformed ARIMA (2,1,1)-GARCH (2,0)-std and ARIMA (1,1,0)-EGARCH (1,1)-std model chosen through in-sample approach. Therefore, it could be deduced that out-of-sample model selection approach was suitable for selecting models with improved forecasting accuracies and performances.

Forecasting and Backtesting of <i>VaR</i> in International Dry Bulk Shipping Market under Skewed Distributions

It is extremely important to model the empirical distributions of dry bulk shipping returns accurately in estimating risk measures. Based on several commonly used distributions and alternative distributions, this paper establishes nine different risk models to forecast the Value-at-Risk (VaR) of dry bulk shipping markets. Several backtests are explored to compare the accuracy of VaR forecasting. The empirical results indicate the risk models based on commonly used distributions have relatively poor performance, while the alternative distributions, i.e. Skewed Student-T (SST) distribution, Skewed Generalized Error Distribution (SGED), and Hyperbolic distribution (HYP) produce more accurate VaR measurement. The empirical results suggest risk managers further consider more flexible empirical distributions when managing extreme risks in dry bulk shipping markets.

Robust Cochlear Modiolar Axis Detection in CT.

The cochlea, the auditory part of the inner ear, is a spiral-shaped organ with large morphological variability. An individualized assessment of its shape is essential for clinical applications related to tonotopy and cochlear implantation. To unambiguously reference morphological parameters, reliable recognition of the cochlear modiolar axis in computed tomography (CT) images is required. The conventional method introduces measurement uncertainties, as it is based on manually selected and difficult to identify landmarks. Herein, we present an algorithm for robust modiolar axis detection in clinical CT images. We define the modiolar axis as the rotation component of the kinematic spiral motion inherent in the cochlear shape. For surface fitting, we use a compact shape representation in a 7-dimensional kinematic parameter space based on extended Plücker coordinates. It is the first time such a kinematic representation is used for shape analysis in medical images. Robust surface fitting is achieved with an adapted approximate maximum likelihood method assuming a Student-t distribution, enabling axis detection even in partially available surface data. We verify the algorithm performance on a synthetic data set with cochlear surface subsets. In addition, we perform an experimental study with four experts in 23 human cochlea CT data sets to compare the automated detection with the manually found axes. Axes found from co-registered high resolution μCT scans are used for reference. Our experiments show that the algorithm reduces the alignment error providing more reliable modiolar axis detection for clinical and research applications.

Bayesian 3D X-ray Computed Tomography with a Hierarchical Prior Model for Sparsity in Haar Transform Domain.

In this paper, a hierarchical prior model based on the Haar transformation and an appropriate Bayesian computational method for X-ray CT reconstruction are presented. Given the piece-wise continuous property of the object, a multilevel Haar transformation is used to associate a sparse representation for the object. The sparse structure is enforced via a generalized Student-t distribution (), expressed as the marginal of a normal-inverse Gamma distribution. The proposed model and corresponding algorithm are designed to adapt to specific 3D data sizes and to be used in both medical and industrial Non-Destructive Testing (NDT) applications. In the proposed Bayesian method, a hierarchical structured prior model is proposed, and the parameters are iteratively estimated. The initialization of the iterative algorithm uses the parameters of the prior distributions. A novel strategy for the initialization is presented and proven experimentally. We compare the proposed method with two state-of-the-art approaches, showing that our method has better reconstruction performance when fewer projections are considered and when projections are acquired from limited angles.

An Estimation of Value at Risk using GARCH Models for the Conventional and Islamic Stock Market in Malaysia

This study attempts to estimate the value-at-risk (VaR) to forecast volatility for both conventional and Islamic stock markets in Malaysia. In particular, the purpose of the article is to investigate whether GARCH models are accurate in the evaluation of VaR in emerging stock markets such as Malaysia.The daily return of the conventional (KLCI) and Islamic (FBMS) stock market are analysed for the period 2000 – 2015. The volatility model of GARCH (1,1), TGARCH (1,1) and CGARCH (1,1) with a normal and and student-t distribution are used to model the conditional variance of the stock market returns. The VaR violations of unconditional coverage and the backtesting procedure of Kupiec test are used to check the reliability and accuracy of the volatility model used for both normal and student-t distribution. Based on the Akaike Information Criterion (AIC), the best model for modelling the conventional and Islamic stock market returns is TGARCH (1,1). The backtesting results showed that for all GARCH models used, the normal distribution gives better forecast VaR compared to the student’s t distribution.

Flexible regression modeling for censored data based on mixtures of student-t distributions

In some applications of censored regression models, the distribution of the error terms departs significantly from normality, for instance, in the presence of heavy tails, skewness and/or atypical observation. In this paper we extend the censored linear regression model with normal errors to the case where the random errors follow a finite mixture of Student-t distributions. This approach allows us to model data with great flexibility, accommodating multimodality, heavy tails and also skewness depending on the structure of the mixture components. We develop an analytically tractable and efficient EM-type algorithm for iteratively computing maximum likelihood estimates of the parameters, with standard errors as a by-product. The algorithm has closed-form expressions at the E-step, that rely on formulas for the mean and variance of the truncated Student-t distributions. The efficacy of the method is verified through the analysis of simulated and real datasets. The proposed algorithm and methods are implemented in the new R package $$\texttt {CensMixReg}$$ .

Multi-object Bayesian filter for jump Markov system under glint noise

Abstract The probability hypothesis density filter under glint noise is efficient for multi-object tracking. However, this filter is inapplicable to tracking the multiple maneuvering objects in case of low detection probability. In order to track the multiple maneuvering objects under glint noise more efficiently, we propose a novel multi-object Bayesian (MOB) filter for jump Markov system under glint noise by applying jump Markov system models and variational Bayesian approach to the MOB filter and develop an implementation of this filter. The developed implementation uses the Student's t-distribution to depict the glint noise, uses the Gaussian-Gamma distribution to represent the predicted state distribution and updated state distribution of each target, and uses the variational Bayesian method to acquire the approximate state distribution of each target. A comparison of the proposed MOB filter with the existing filters demonstrates that the proposed MOB filter is better at tracking the maneuvering objects under glint noise than the existing filters.