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.     

Goodness-of-fit testing of a weak memoryless property of life distributions

A life distribution is said to have a weak memoryless property if its conditional probability of survival beyond a fixed time point is equal to its (unconditional) survival probability at that point. Goodness‐of‐fit testing of this notion is proposed in the current investigation, both when the fixed time point is known and when it is unknown but estimable from the data. The limiting behaviour of the proposed test statistic is obtained and the null variance is explicitly given. The empirical power of the test is evaluated for a commonly known alternative using Monte Carlo methods, showing that the test performs well. The case when the fixed time point t₀ equals a quantile of the distribution F gives a distribution‐free test procedure. The procedure works even if t₀ is unknown but is estimable.     

Mixture of linear mixed models using multivariate t distribution

Linear mixed models are widely used when multiple correlated measurements are made on each unit of interest. In many applications, the units may form several distinct clusters, and such heterogeneity can be more appropriately modelled by a finite mixture linear mixed model. The classical estimation approach, in which both the random effects and the error parts are assumed to follow normal distribution, is sensitive to outliers, and failure to accommodate outliers may greatly jeopardize the model estimation and inference. We propose a new mixture linear mixed model using multivariate t distribution. For each mixture component, we assume the response and the random effects jointly follow a multivariate t distribution, to conveniently robustify the estimation procedure. An efficient expectation conditional maximization algorithm is developed for conducting maximum likelihood estimation. The degrees of freedom parameters of the t distributions are chosen data adaptively, for achieving flexible trade-off between estimation robustness and efficiency. Simulation studies and an application on analysing lung growth longitudinal data showcase the efficacy of the proposed approach.     

Models for residual time to AIDS

The distributions of the time from Human Immunodeficiency Virus (HIV) infection to the onset of Acquired Immune Deficiency Syndrome (AIDS) and of the residual time to AIDS diagnosis are important for modeling the growth of the AIDS epidemic and for predicting onset of the disease in an individual. Markers such as CD4 counts carry valuable information about disease progression and therefore about the two survival distributions. Building on the framework set out by Jewell and Kalbfleisch (1992), we study these two survival distributions based on stochastic models for the marker process (X(t)) and a marker-dependent hazard (h()). We examine various plausible CD4 marker processes and marker-dependent hazard functions for AIDS proposed in recent literature. For a random effects plus Brownian motion marker process X(t)=(a+bt+BM(t))⁴, where a has a normal distribution, b<0 is an unknown parameter and BM(t) is Brownian motion, and marker-dependent hazard h(X(t)), we prove that, given CD4 cell count X(t), the residual time to AIDS distribution does not depend on the time since infection t. Using simulation and numerical integration, we find the marginal incubation period distribution, the marginal hazard and the residual time distribution for several combinations of marker processes and marker-dependent hazards. An example using data from the Multicenter AIDS Cohort Study is given. A simple regression model relating the cube root of residual time to AIDS to CD4 count is suggested.     

A Nonparametric Test for Autoregressive Conditional Heteroscedasticity: A Markov-Chain Approach

In this article we propose a nonparametric test for autoregressive conditional heteroscedasticity based on finite-state Markov chains. A simple Monte Carlo experiment suggests that in finite samples it performs comparably to the Lagrange multiplier test under conditional normality and is superior for the t, lognormal, and exponential distributions. As an illustration, we apply both tests to Canadian/U.S. forward foreign exchange data.     

Nonparametric tests for conditional independence using conditional distributions

The concept of causality is naturally defined in terms of conditional distribution, however almost all the empirical works focus on causality in mean. This paper aims to propose a nonparametric statistic to test the conditional independence and Granger non-causality between two variables conditionally on another one. The test statistic is based on the comparison of conditional distribution functions using an L₂ metric. We use Nadaraya–Watson method to estimate the conditional distribution functions. We establish the asymptotic size and power properties of the test statistic and we motivate the validity of the local bootstrap. We ran a simulation experiment to investigate the finite sample properties of the test and we illustrate its practical relevance by examining the Granger non-causality between S&P 500 Index returns and VIX volatility index. Contrary to the conventional t-test which is based on a linear mean-regression, we find that VIX index predicts excess returns both at short and long horizons.     

Testing for Threshold Diffusion

The threshold diffusion model assumes a piecewise linear drift term and a piecewise smooth diffusion term, which constitutes a rich model for analyzing nonlinear continuous-time processes. We consider the problem of testing for threshold nonlinearity in the drift term. We do this by developing a quasi-likelihood test derived under the working assumption of a constant diffusion term, which circumvents the problem of generally unknown functional form for the diffusion term. The test is first developed for testing for one threshold at which the drift term breaks into two linear functions. We show that under some mild regularity conditions, the asymptotic null distribution of the proposed test statistic is given by the distribution of certain functional of some centered Gaussian process. We develop a computationally efficient method for calibrating the p-value of the test statistic by bootstrapping its asymptotic null distribution. The local power function is also derived, which establishes the consistency of the proposed test. The test is then extended to testing for multiple thresholds. We demonstrate the efficacy of the proposed test by simulations. Using the proposed test, we examine the evidence of nonlinearity in the term structure of a long time series of U.S. interest rates.     

On testing exponentiality against new better than used of specified age

R.M. Hollander, D.H. Park and F. Proschan [A class of life distributions for aging, J. Amer. Statist. Assoc. 81 (1986) 91–95] introduced the concept of the larger class of life distributions called new better than used of specified age. In practice, one might be interested in the new better than used behaviour at an unknown but estimable age t₀. Here, we investigate the testing of new better than used of specified age t₀ (NBU-t₀) alternatives. A class of test statistics for testing NBU-t₀ (t₀ is known) based on a U-statistic whose kernel depends on sub-sample minima is proposed. A member of the class of tests proposed by N. Ebrahimi and M. Habbibullah [Testing whether the survival distribution is new better than used of specified age, Biometrika 77 (1990) 212–215] for this problem belongs to the class of tests proposed here. The distributional properties of the class of test statistics are studied. The performances of a few members of the proposed class of tests are studied in terms of Pitman asymptotic relative efficiency. The Pitman ARE values show that the members of the class perform well in comparison with the N. Ebrahimi and M. Habbibullah [Testing whether the survival distribution is new better than used of specified age, Biometrika 77 (1990) 212–215] tests. The proposed class of tests is shown to be consistent for NBU-t₀ alternatives.     

Bayesian analysis of time series Poisson data

This paper provides a practical simulation-based Bayesian analysis of parameter-driven models for time series Poisson data with the AR(1) latent process. The posterior distribution is simulated by a Gibbs sampling algorithm. Full conditional posterior distributions of unknown variables in the model are given in convenient forms for the Gibbs sampling algorithm. The case with missing observations is also discussed. The methods are applied to real polio data from 1970 to 1983.     

Conditional and unconditional tests with historical controls

Score statistics utilizing historical control data have been proposed to test for increasing trend in tumour occurrence rates in laboratory carcinogenicity studies. Novel invariance arguments are used to confirm, under slightly weaker conditions, previously established asymptotic distributions (mixtures of normal distributions) of tests unconditional on the tumor response rate in the concurrent control group. Conditioning on the control response rate, an ancillary statistic, leads to a new conditional limit theorem in which the test statistic converges to an unknown random variable. Because of this, a subasymptotic approximation to the conditional limiting distribution is also considered. The adequacy of these large-sample approximations in finite samples is evaluated using computer simulation. Bootstrap methods for use in finite samples are also proposed. The application of the conditional and unconditional tests is illustrated using bioassay data taken from the literature. The results presented in this paper are used to formulate recommendations for the use of tests for trend with historical controls in practice.     

From statistical power to statistical assurance: It's time for a paradigm change in clinical trial design

A well-designed clinical trial requires an appropriate sample size with adequate statistical power to address trial objectives. The statistical power is traditionally defined as the probability of rejecting the null hypothesis with a pre-specified true clinical treatment effect. This power is a conditional probability conditioned on the true but actually unknown effect. In practice, however, this true effect is never a fixed value. Thus, we discuss a newly proposed alternative to this conventional statistical power: statistical assurance, defined as the unconditional probability of rejecting the null hypothesis. This kind of assurance can then be obtained as an expected power where the expectation is based on the prior probability distribution of the unknown treatment effect, which leads to the Bayesian paradigm. In this article, we outline the transition from conventional statistical power to the newly developed assurance and discuss the computations of assurance using Monte Carlo simulation-based approach.     

Order statistics from trivariate normal and -distributions in terms of generalized skew-normal and skew- distributions

We consider here a generalization of the skew-normal distribution, GSN(λ₁,λ₂,ρ), defined through a standard bivariate normal distribution with correlation ρ, which is a special case of the unified multivariate skew-normal distribution studied recently by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561–574]. We then present some simple and useful properties of this distribution and also derive its moment generating function in an explicit form. Next, we show that distributions of order statistics from the trivariate normal distribution are mixtures of these generalized skew-normal distributions; thence, using the established properties of the generalized skew-normal distribution, we derive the moment generating functions of order statistics, and also present expressions for means and variances of these order statistics.Next, we introduce a generalized skew-t_ν distribution, which is a special case of the unified multivariate skew-elliptical distribution presented by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561–574] and is in fact a three-parameter generalization of Azzalini and Capitanio's [2003. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J. Roy. Statist. Soc. Ser. B 65, 367–389] univariate skew-t_ν form. We then use the relationship between the generalized skew-normal and skew-t_ν distributions to discuss some properties of generalized skew-t_ν as well as distributions of order statistics from bivariate and trivariate t_ν distributions. We show that these distributions of order statistics are indeed mixtures of generalized skew-t_ν distributions, and then use this property to derive explicit expressions for means and variances of these order statistics.     

Likelihood-based estimation for Gaussian MRFs

Markov random fields (MRFs) express spatial dependence through conditional distributions, although their stochastic behavior is defined by their joint distribution. These joint distributions are typically difficult to obtain in closed form, the problem being a normalizing constant that is a function of unknown parameters. The Gaussian MRF (or conditional autoregressive model) is one case where the normalizing constant is available in closed form; however, when sample sizes are moderate to large (thousands to tens of thousands), and beyond, its computation can be problematic. Because the conditional autoregressive (CAR) model is often used for spatial-data modeling, we develop likelihood-inference methodology for this model in situations where the sample size is too large for its normalizing constant to be computed directly. In particular, we use simulation methodology to obtain maximum likelihood estimators of mean, variance, and spatial-depencence parameters (including their asymptotic variances and covariances) of CAR models.     

A Class of Multivariate Bilateral Selection t Distributions and Its Properties

This article proposes a class of multivariate bilateral selection t distributions useful for analyzing non-normal (skewed and/or bimodal) multivariate data. The class is associated with a bilateral selection mechanism, and it is obtained from a marginal distribution of the centrally truncated multivariate t. It is flexible enough to include the multivariate t and multivariate skew-t distributions and mathematically tractable enough to account for central truncation of a hidden t variable. The class, closed under linear transformation, marginal, and conditional operations, is studied from several aspects such as shape of the probability density function, conditioning of a distribution, scale mixtures of multivariate normal, and a probabilistic representation. The relationships among these aspects are given, and various properties of the class are also discussed. Necessary theories and two applications are provided.     

A New Procedure to Monitor the Mean of a Quality Characteristic

The Shewhart, Bonferroni-adjustment, and analysis of means (ANOM) control charts are typically applied to monitor the mean of a quality characteristic. The Shewhart and Bonferroni procedure are utilized to recognize special causes in production process, where the control limits are constructed by assuming normal distribution for known parameters (mean and standard deviation), and approximately normal distribution regarding to unknown parameters. The ANOM method is an alternative to the analysis of variance method. It can be used to establish the mean control charts by applying equicorrelated multivariate non central t distribution. In this article, we establish new control charts, in phases I and II monitoring, based on normal and t distributions having as a cause a known (or unknown) parameter (standard deviation). Our proposed methods are at least as effective as the classical Shewhart methods and have some advantages.     

Large-Sample Inference for the Epsilon-Skew-t Distribution

The main object of this article is to discuss maximum likelihood inference for the epsilon-skew-t distribution. Special cases of this distribution include the epsilon-skew-Cauchy and the epsilon-skew-normal distributions. We derive the information matrix for the maximum likelihood estimators. The approach is applied to a data set presenting significant amount of skewness and heavy tails. In the application we consider the epsilon-skew-t distribution with known and unknown degrees of freedom parameter, showing great flexibility in adjusting to skew data with heavy tails.     

Sampling-based Inference of Time Deformation Models with Heavy Tail Distributions

This article focuses on simulation-based inference for the time-deformation models directed by a duration process. In order to better capture the heavy tail property of the time series of financial asset returns, the innovation of the observation equation is subsequently assumed to have a Student-t distribution. Suitable Markov chain Monte Carlo (MCMC) algorithms, which are hybrids of Gibbs and slice samplers, are proposed for estimation of the parameters of these models. In the algorithms, the parameters of the models can be sampled either directly from known distributions or through an efficient slice sampler. The states are simulated one at a time by using a Metropolis-Hastings method, where the proposal distributions are sampled through a slice sampler. Simulation studies conducted in this article suggest that our extended models and accompanying MCMC algorithms work well in terms of parameter estimation and volatility forecast.     

The use of Jeffreys priors for the Student-t distribution

The Jeffreys-rule prior and the marginal independence Jeffreys prior are recently proposed in Fonseca et al. [Objective Bayesian analysis for the Student-t regression model, Biometrika 95 (2008), pp. 325–333] as objective priors for the Student-t regression model. The authors showed that the priors provide proper posterior distributions and perform favourably in parameter estimation. Motivated by a practical financial risk management application, we compare the performance of the two Jeffreys priors with other priors proposed in the literature in a problem of estimating high quantiles for the Student-t model with unknown degrees of freedom. Through an asymptotic analysis and a simulation study, we show that both Jeffreys priors perform better in using a specific quantile of the Bayesian predictive distribution to approximate the true quantile.     

Maximal type test statistics based on conditional processes

A general methodology is presented for non-parametric testing of independence, location and dispersion in multiple regression. The proposed testing procedures are based on the concepts of conditional distribution function, conditional quantile, and conditional shortest t-fraction. Techniques involved come from empirical process and extreme-value theory. The asymptotic distributions are standard Gumbel.     

Analysis of simple step-stress model in presence of competing risks

ABSTRACT

In this article, we consider a simple step-stress life test in the presence of exponentially distributed competing risks. It is assumed that the stress is changed when a pre-specified number of failures takes place. The data is assumed to be Type-II censored. We obtain the maximum likelihood estimators of the model parameters and the exact conditional distributions of the maximum likelihood estimators. Based on the conditional distribution, approximate confidence intervals (CIs) of unknown parameters have been constructed. Percentile bootstrap CIs of model parameters are also provided. Optimal test plan is addressed. We perform an extensive simulation study to observe the behaviour of the proposed method. The performances are quite satisfactory. Finally we analyse two data sets for illustrative purposes.