Modelling stochastic volatility using generalized t distribution

In modelling financial return time series and time-varying volatility, the Gaussian and the Student-t distributions are widely used in stochastic volatility (SV) models. However, other distributions such as the Laplace distribution and generalized error distribution (GED) are also common in SV modelling. Therefore, this paper proposes the use of the generalized t (GT) distribution whose special cases are the Gaussian distribution, Student-t distribution, Laplace distribution and GED. Since the GT distribution is a member of the scale mixture of uniform (SMU) family of distribution, we handle the GT distribution via its SMU representation. We show this SMU form can substantially simplify the Gibbs sampler for Bayesian simulation-based computation and can provide a mean of identifying outliers. In an empirical study, we adopt a GT–SV model to fit the daily return of the exchange rate of Australian dollar to three other currencies and use the exchange rate to US dollar as a covariate. Model implementation relies on Bayesian Markov chain Monte Carlo algorithms using the WinBUGS package.     

A generalized skew two-piece skew-elliptical distribution

We present a new generalized family of skew two-piece skew-elliptical (GSTPSE) models and derive some its statistical properties. It is shown that the new family of distribution may be written as a mixture of generalized skew elliptical distributions. Also, a new representation theorem for a special case of GSTPSE-distribution is given. Next, we will focus on t kernel density and prove that it is a scale mixture of the generalized skew two-piece skew normal distributions. An explicit expression for the central moments as well as a recurrence relations for its cumulative distribution function and density are obtained. Since, this special case is a uni-/bimodal distribution, a sufficient condition for each cases is given. A real data set on heights of Australian females athletes is analysed. Finally, some concluding remarks and open problems are discussed.     

Gamma mixture of generalized error distribution

Abstract

A new symmetric heavy-tailed distribution, namely gamma mixture of generalized error distribution is defined by scaling generalized error distribution with gamma distribution, its probability density function, k-moment, skewness and kurtosis are derived. After tedious calculation, we also give the Fisher information matrix, moment estimators and maximum likelihood estimators for the parameters of gamma mixture of generalized error distribution. In order to evaluate the effectiveness of the point estimators and the stability of Fisher information matrix, extensive simulation experiments are carried out in three groups of parameters. Additionally, the new distribution is applied to Apple Inc. stock (AAPL) data and compared with normal distribution, F-S skewed standardized t distribution and generalized error distribution. It is found that the new distribution has better fitting effect on the data under the Akaike information criterion (AIC). To a certain extent, our results enrich the probability distribution theory and develop the scale mixture distribution, which will provide help and reference for financial data analysis.     

Robust finite mixture modeling of multivariate unrestricted skew-normal generalized hyperbolic distributions

In this paper, we introduce an unrestricted skew-normal generalized hyperbolic (SUNGH) distribution for use in finite mixture modeling or clustering problems. The SUNGH is a broad class of flexible distributions that includes various other well-known asymmetric and symmetric families such as the scale mixtures of skew-normal, the skew-normal generalized hyperbolic and its corresponding symmetric versions. The class of distributions provides a much needed unified framework where the choice of the best fitting distribution can proceed quite naturally through either parameter estimation or by placing constraints on specific parameters and assessing through model choice criteria. The class has several desirable properties, including an analytically tractable density and ease of computation for simulation and estimation of parameters. We illustrate the flexibility of the proposed class of distributions in a mixture modeling context using a Bayesian framework and assess the performance using simulated and real data.

     

A hierarchical Bayesian regression model for the uncertain functional constraint using screened scale mixtures of Gaussian distributions

This paper considers a hierarchical Bayesian analysis of regression models using a class of Gaussian scale mixtures. This class provides a robust alternative to the common use of the Gaussian distribution as a prior distribution in particular for estimating the regression function subject to uncertainty about the constraint. For this purpose, we use a family of rectangular screened multivariate scale mixtures of Gaussian distribution as a prior for the regression function, which is flexible enough to reflect the degrees of uncertainty about the functional constraint. Specifically, we propose a hierarchical Bayesian regression model for the constrained regression function with uncertainty on the basis of three stages of a prior hierarchy with Gaussian scale mixtures, referred to as a hierarchical screened scale mixture of Gaussian regression models (HSMGRM). We describe distributional properties of HSMGRM and an efficient Markov chain Monte Carlo algorithm for posterior inference, and apply the proposed model to real applications with constrained regression models subject to uncertainty.     

Generalized Inverse Gaussian Distributions and their Wishart Connections

The matrix generalized inverse Gaussian distribution (MGIG) is shown to arise as a conditional distribution of components of a Wishart distributio n. In the special scalar case, the characterization refers to members of the class of generalized inverse Gaussian distributions (GIGs) and includes the inverse Gaussian distribution among others     

Improved auxiliary mixture sampling for hierarchical models of non-Gaussian data

The article considers Bayesian analysis of hierarchical models for count, binomial and multinomial data using efficient MCMC sampling procedures. To this end, an improved method of auxiliary mixture sampling is proposed. In contrast to previously proposed samplers the method uses a bounded number of latent variables per observation, independent of the intensity of the underlying Poisson process in the case of count data, or of the number of experiments in the case of binomial and multinomial data. The bounded number of latent variables results in a more general error distribution, which is a negative log-Gamma distribution with arbitrary integer shape parameter. The required approximations of these distributions by Gaussian mixtures have been computed. Overall, the improvement leads to a substantial increase in efficiency of auxiliary mixture sampling for highly structured models. The method is illustrated for finite mixtures of generalized linear models and an epidemiological case study.     

Bayesian Inference in Generalized Error and Generalized Student-t Regression Models

This study takes up inference in linear models with generalized error and generalized t distributions. For the generalized error distribution, two computational algorithms are proposed. The first is based on indirect Bayesian inference using an approximating finite scale mixture of normal distributions. The second is based on Gibbs sampling. The Gibbs sampler involves only drawing random numbers from standard distributions. This is important because previously the impression has been that an exact analysis of the generalized error regression model using Gibbs sampling is not possible. Next, we describe computational Bayesian inference for linear models with generalized t disturbances based on Gibbs sampling, and exploiting the fact that the model is a mixture of generalized error distributions with inverse generalized gamma distributions for the scale parameter. The linear model with this specification has also been thought not to be amenable to exact Bayesian analysis. All computational methods are applied to actual data involving the exchange rates of the British pound, the French franc, and the German mark relative to the U.S. dollar.     

Locally asymptotically optimal tests for AR(p) against diagonal bilinear dependence

The problem of testing linear AR(p1) against diagonal bilinear BL(p1, 0; p2, p2) dependence is considered. Emphasis is put on local asymptotic optimality and the nonspecification of innovation densities. The tests we are deriving are asymptotically valid under a large class of densities, and locally asymptotically most stringent at some selected density f. They rely on generalized versions of residual autocorrelations (the spectrum), and generalized versions of the so-called cubic autocorrelations (the bispectrum). Local powers are explicitly provided. The local power of the Gaussian Lagrange multipliers method follows as a particular case.     

On the Sampling Distributions of the Estimated Process Loss Indices with Asymmetric Tolerances

The inverse Gaussian distribution provides a flexible model for analyzing positive, right-skewed data. The generalized variable test for equality of several inverse Gaussian means with unknown and arbitrary variances has satisfactory Type-I error rate when the number of samples (k) is small (Tian, 2006). However, the Type-I error rate tends to be inflated when k goes up. In this article, we propose a parametric bootstrap (PB) approach for this problem. Simulation results show that the proposed test performs very satisfactorily regardless of the number of samples and sample sizes. This method is illustrated by an example.     

Optimal locally robust M-estimates of regression

First, we show that many robust estimates of regression which depend only on the regression residuals (including M-, S-, Tau-, least median of squares-, least trimmed of squares- and some R-estimates) have infinite gross-error-sensitivity. More precisely, we show that the maximum-bias function of a large class of estimates, called residual admissible in Yohai and Zamar (Ann. Statist. 21, 1993, 1824–1842), is of order √ε near zero. Based on this finding we define a new robustness measure for estimates with B_T(ε) = o(ε^β), the contamination sensitivity of order β, which extends Hampel's gross error sensitivity for estimates with unbounded influence. We compute this measure for regression M-estimates with a general scale and show that β = 0.5 in this case. Then we solve a Hampel-like optimality problem, namely, one of minimizing the asymptotic variance subject to a bound on the contamination sensitivity of order β = 0.5, for estimates in this class. Finally, we show that a certain least α-quantile estimate has the smallest contamination sensitivity of order 0.5 among all residual admissible estimates. In the Gaussian case α = 0.683.     

Goodness-of-fit testing under long memory

This paper discusses the problem of fitting a distribution function to the marginal distribution of a long memory moving average process. Because of the uniform reduction principle, unlike in the i.i.d. set up, classical tests based on empirical process are relatively easy to implement. More importantly, we discuss fitting the marginal distribution of the error process in location, scale, location–scale and linear regression models. An interesting observation is that in the location model, location–scale model, or more generally in the linear regression models with non-zero intercept parameter, the null weak limit of the first order difference between the residual empirical process and the null model is degenerate at zero, and hence it cannot be used to fit an error distribution in these models for the large samples. This finding is in sharp contrast to a recent claim of Chan and Ling (2008) that the null weak limit of such a process is a continuous Gaussian process. This note also proposes some tests based on the second order difference for the location case. Another finding is that residual empirical process tests in the scale problem are robust against not knowing the scale parameter.     

Random projections for Bayesian regression

This article deals with random projections applied as a data reduction technique for Bayesian regression analysis. We show sufficient conditions under which the entire d-dimensional distribution is approximately preserved under random projections by reducing the number of data points from n to \(k\in O({\text {poly}}(d/\varepsilon ))\) in the case \(n\gg d\). Under mild assumptions, we prove that evaluating a Gaussian likelihood function based on the projected data instead of the original data yields a \((1+O(\varepsilon ))\)-approximation in terms of the \(\ell _2\) Wasserstein distance. Our main result shows that the posterior distribution of Bayesian linear regression is approximated up to a small error depending on only an \(\varepsilon \)-fraction of its defining parameters. This holds when using arbitrary Gaussian priors or the degenerate case of uniform distributions over \(\mathbb {R}^d\) for \(\beta \). Our empirical evaluations involve different simulated settings of Bayesian linear regression. Our experiments underline that the proposed method is able to recover the regression model up to small error while considerably reducing the total running time.     

A class of finite mixture of quantile regressions with its applications

Mixture of linear regression models provide a popular treatment for modeling nonlinear regression relationship. The traditional estimation of mixture of regression models is based on Gaussian error assumption. It is well known that such assumption is sensitive to outliers and extreme values. To overcome this issue, a new class of finite mixture of quantile regressions (FMQR) is proposed in this article. Compared with the existing Gaussian mixture regression models, the proposed FMQR model can provide a complete specification on the conditional distribution of response variable for each component. From the likelihood point of view, the FMQR model is equivalent to the finite mixture of regression models based on errors following asymmetric Laplace distribution (ALD), which can be regarded as an extension to the traditional mixture of regression models with normal error terms. An EM algorithm is proposed to obtain the parameter estimates of the FMQR model by combining a hierarchical representation of the ALD. Finally, the iterated weighted least square estimation for each mixture component of the FMQR model is derived. Simulation studies are conducted to illustrate the finite sample performance of the estimation procedure. Analysis of an aphid data set is used to illustrate our methodologies.     

Value-at-risk forecasting based on Gaussian mixture ARMA–GARCH model

In this paper, we develop a new forecasting algorithm for value-at-risk (VaR) based on ARMA–GARCH (autoregressive moving average–generalized autoregressive conditional heteroskedastic) models whose innovations follow a Gaussian mixture distribution. For the parameter estimation, we employ the conditional least squares and quasi-maximum-likelihood estimator (QMLE) for ARMA and GARCH parameters, respectively. In particular, Gaussian mixture parameters are estimated based on the residuals obtained from the QMLE of GARCH parameters. Our algorithm provides a handy methodology, spending much less time in calculation than the existing resampling and bias-correction method developed in Hartz et al. [Accurate value-at-risk forecasting based on the normal-GARCH model, Comput. Stat. Data Anal. 50 (2006), pp. 3032–3052]. Through a simulation study and a real-data analysis, it is shown that our method provides an accurate VaR prediction.     

Bayesian bridge-randomized penalized quantile regression estimation for linear regression model with AP(q) perturbation

Bridge penalized regression has many desirable statistical properties such as unbiasedness, sparseness as well as ‘oracle’. In Bayesian framework, bridge regularized penalty can be implemented based on generalized Gaussian distribution (GGD) prior. In this paper, we incorporate Bayesian bridge-randomized penalty and its adaptive version into the quantile regression (QR) models with autoregressive perturbations to conduct Bayesian penalization estimation. Employing the working likelihood of the asymmetric Laplace distribution (ALD) perturbations, the Bayesian joint hierarchical models are established. Based on the mixture representations of the ALD and generalized Gaussian distribution (GGD) priors of coefficients, the hybrid algorithms based on Gibbs sampler and Metropolis-Hasting sampler are provided to conduct fully Bayesian posterior estimation. Finally, the proposed Bayesian procedures are illustrated by some simulation examples and applied to a real data application of the electricity consumption.     

Regression Properties for Asymmetric Generalized Scale Mixtures of Multivariate Gaussian Variables

Analytical properties of regression and the variance–covariance matrix of asymmetric generalized scale mixture of multivariate Gaussian variables are presented. The analysis includes an in-depth analytical investigation of the first two conditional moments of the mixing variable. Exact computable expressions for the prediction and the conditional variance are presented for the generalized hyperbolic distribution using the inversion theorem for Fourier transforms. An application to financial log returns is demonstrated via the classical Euler approximation. The methodology is illustrated by analyzing the regression of intraday log returns for CISCO against the corresponding data from S&P 500.     

Estimation,prediction and interpretation of NGG random effects models: an application to Kevlar fibre failure times

We propose a class of Bayesian semiparametric mixed-effects models; its distinctive feature is the randomness of the grouping of observations, which can be inferred from the data. The model can be viewed under a more natural perspective, as a Bayesian semiparametric regression model on the log-scale; hence, in the original scale, the error is a mixture of Weibull densities mixed on both parameters by a normalized generalized gamma random measure, encompassing the Dirichlet process. As an estimate of the posterior distribution of the clustering of the random-effects parameters, we consider the partition minimizing the posterior expectation of a suitable class of loss functions. As a merely illustrative application of our model we consider a Kevlar fibre lifetime dataset (with censoring). We implement an MCMC scheme, obtaining posterior credibility intervals for the predictive distributions and for the quantiles of the failure times under different stress levels. Compared to a previous parametric Bayesian analysis, we obtain narrower credibility intervals and a better fit to the data. We found that there are three main clusters among the random-effects parameters, in accordance with previous frequentist analysis.     

Estimation of a scale parameter in mixture models with unknown location

Estimation of the scale parameter in mixture models with unknown location is considered under Stein's loss. Under certain conditions, the inadmissibility of the “usual” estimator is established by exhibiting better estimators. In addition, robust improvements are found for a specified submodel of the original model. The results are applied to mixtures of normal distributions and mixtures of exponential distributions. Improved estimators of the variance of a normal distribution are shown to be robust under any scale mixture of normals having variance greater than the variance of that normal distribution. In particular, Stein's (Ann. Inst. Statist. Math. 16 (1964) 155) and Brewster's and Zidek's (Ann. Statist. 2 (1974) 21) estimators obtained under the normal model are robust under the t model, for arbitrary degrees of freedom, and under the double-exponential model. Improved estimators for the variance of a t distribution with unknown and arbitrary degrees of freedom are also given. In addition, improved estimators for the scale parameter of the multivariate Lomax distribution (which arises as a certain mixture of exponential distributions) are derived and the robustness of Zidek's (Ann. Statist. 1 (1973) 264) and Brewster's (Ann. Statist. 2 (1974) 553) estimators of the scale parameter of an exponential distribution is established under a class of modified Lomax distributions.     

On the bias and mean-square error of order-restricted maximum likelihood estimators

The inverse Gaussian family (IG) (μ,λ) is a versatile family for modelling nonnegative right-skewed data. In this paper, we propose robust methods for testing homogeneity of the scale-like parameters λ_i from k independent IG populations subject to order restrictions. Robustness of the procedures is examined for a variety of IG-symmetric alternatives including lognormal and the recently introduced contaminated inverse Gaussian populations. Our study shows that these inference procedures for the inverse Gaussian scale-like parameters and their properties exhibit striking similarities to those of the scale parameters of the normal distribution.