Tempered stable Ornstein– Uhlenbeck processes: A practical view

We study the one-dimensional Ornstein–Uhlenbeck (OU) processes with marginal law given by tempered stable and tempered infinitely divisible distributions. We investigate the transition law between consecutive observations of these processes and evaluate the characteristic function of integrated tempered OU processes with a view toward practical applications. We then analyze how to draw a random sample from this class of processes by considering both the classical inverse transform algorithm and an acceptance–rejection method based on simulating a stable random sample. Using a maximum likelihood estimation method based on the fast Fourier transform, we empirically assess the simulation algorithm performance.     

Infinite Variation Tempered Stable Ornstein–Uhlenbeck Processes with Discrete Observations

We investigate transition law between consecutive observations of Ornstein–Uhlenbeck processes of infinite variation with tempered stable stationary distribution. Thanks to the Markov autoregressive structure, the transition law can be written in the exact sense as a convolution of three random components; a compound Poisson distribution and two independent tempered stable distributions, one with stability index in (0, 1) and the other with index in (1, 2). We discuss simulation techniques for those three random elements. With the exact transition law and proposed simulation techniques, sample paths simulation proves significantly more efficient, relative to the known approximative technique based on infinite shot noise series representation of tempered stable Lévy processes.     

Alternative Estimators for Randomized Response Techniques in Multi-Character Surveys

Tail estimates are developed for power law probability distributions with exponential tempering, using a conditional maximum likelihood approach based on the upper-order statistics. Tempered power law distributions are intermediate between heavy power-law tails and Laplace or exponential tails, and are sometimes called “semi-heavy” tailed distributions. The estimation method is demonstrated on simulated data from a tempered stable distribution, and for several data sets from geophysics and finance that show a power law probability tail with some tempering.     

The normal tempered stable regression model

Abstract

This paper deals with the statistical studies of the normal tempered stable model defined by Barndorff-Nielsen and Shephard. It represents the natural extension of the normal inverse Gaussian one introduced by Barndorff-Nielsen. We basically use the Monte-Carlo’s approximation in order to simulate this distribution. We introduce a linear regression model with normal tempered stable error. We apply this model for the analyzing of the daily logarithm returns data on CAC40 index. The parameters estimation results show that this model better deals with long tailed distribution which is the case for the CAC40 logarithm returns.     

Tempered Mittag-Leffler Lévy processes

In this article, we introduce tempered Mittag-Leffler Lévy processes (TMLLP). TMLLP is represented as tempered stable subordinator delayed by a gamma process. Its probability density function and Lévy density are obtained in terms of infinite series and Mittag-Leffler function, respectively. Asymptotic forms of the tails and moments are given. A step-by-step procedure of the parameters estimation and simulation of sample paths is given. We also provide main results available for Mittag-Leffler Lévy processes (MLLP) and some extensions which are not available in a collective way in a single article. Our results generalize and complement the results available on Mittag-Leffler distribution and MLLP in several directions. Further, the asymptotic forms of the moments of the first-exit times of the TMLLP are also discussed.     

Accurate and efficient numerical calculation of stable densities via optimized quadrature and asymptotics

Stable distributions are an important class of infinitely divisible probability distributions, of which two special cases are the Cauchy distribution and the normal distribution. Aside from a few special cases, the density function for stable distributions has no known analytic form and is expressible only through the variate’s characteristic function or other integral forms. In this paper, we present numerical schemes for evaluating the density function for stable distributions, its gradient, and distribution function in various parameter regimes of interest, some of which had no preexisting efficient method for their computation. The novel evaluation schemes consist of optimized generalized Gaussian quadrature rules for integral representations of the density function, complemented by asymptotic expansions near various values of the shape and argument parameters. We report several numerical examples illustrating the efficiency of our methods. The resulting code has been made available online.     

Fractionally integrated GARCH model with tempered stable distribution: a simulation study

With the growing availability of high-frequency data, long memory has become a popular topic in finance research. Fractionally Integrated GARCH (FIGARCH) model is a standard approach to study the long memory of financial volatility. The original specification of FIGARCH model is developed using Normal distribution, which cannot accommodate fat-tailed properties commonly existing in financial time series. Traditionally, the Student-t distribution and General Error Distribution (GED) are used instead to solve that problem. However, a recent study points out that the Student-t lacks stability. Instead, the Stable distribution is introduced. The issue of this distribution is that its second moment does not exist. To overcome this new problem, the tempered stable distribution, which retains most attractive characteristics of the Stable distribution and has defined moments, is a natural candidate. In this paper, we describe the estimation procedure of the FIGARCH model with tempered stable distribution and conduct a series of simulation studies to demonstrate that it consistently outperforms FIGARCH models with the Normal, Student-t and GED distributions. An empirical evidence of the S&P 500 hourly return is also provided with robust results. Therefore, we argue that the tempered stable distribution could be a widely useful tool for modelling the high-frequency financial volatility in general contexts with a FIGARCH-type specification.     

Tuning tempered transitions

The method of tempered transitions was proposed by Neal (Stat. Comput. 6:353–366, 1996) for tackling the difficulties arising when using Markov chain Monte Carlo to sample from multimodal distributions. In common with methods such as simulated tempering and Metropolis-coupled MCMC, the key idea is to utilise a series of successively easier to sample distributions to improve movement around the state space. Tempered transitions does this by incorporating moves through these less modal distributions into the MCMC proposals. Unfortunately the improved movement between modes comes at a high computational cost with a low acceptance rate of expensive proposals. We consider how the algorithm may be tuned to increase the acceptance rates for a given number of temperatures. We find that the commonly assumed geometric spacing of temperatures is reasonable in many but not all applications.     

Stochastic volatility modelling in continuous time with general marginal distributions: Inference,prediction and model selection

We compare results for stochastic volatility models where the underlying volatility process having generalized inverse Gaussian (GIG) and tempered stable marginal laws. We use a continuous time stochastic volatility model where the volatility follows an Ornstein–Uhlenbeck stochastic differential equation driven by a Lévy process. A model for long-range dependence is also considered, its merit and practical relevance discussed. We find that the full GIG and a special case, the inverse gamma, marginal distributions accurately fit real data. Inference is carried out in a Bayesian framework, with computation using Markov chain Monte Carlo (MCMC). We develop an MCMC algorithm that can be used for a general marginal model.     

Sampling from multimodal distributions using tempered transitions

I present a new Markov chain sampling method appropriate for distributions with isolated modes. Like the recently developed method of simulated tempering, the tempered transition method uses a series of distributions that interpolate between the distribution of interest and a distribution for which sampling is easier. The new method has the advantage that it does not require approximate values for the normalizing constants of these distributions, which are needed for simulated tempering, and can be tedious to estimate. Simulated tempering performs a random walk along the series of distributions used. In contrast, the tempered transitions of the new method move systematically from the desired distribution, to the easily-sampled distribution, and back to the desired distribution. This systematic movement avoids the inefficiency of a random walk, an advantage that is unfortunately cancelled by an increase in the number of interpolating distributions required. Because of this, the sampling efficiency of the tempered transition method in simple problems is similar to that of simulated tempering. On more complex distributions, however, simulated tempering and tempered transitions may perform differently. Which is better depends on the ways in which the interpolating distributions are deceptive.     

Simulation of right and left truncated gamma distributions by mixtures

We study the properties of truncated gamma distributions and we derive simulation algorithms which dominate the standard algorithms for these distributions. For the right truncated gamma distribution, an optimal accept–reject algorithm is based on the fact that its density can be expressed as an infinite mixture of beta distribution. For integer values of the parameters, the density of the left truncated distributions can be rewritten as a mixture which can be easily generated. We give an optimal accept–reject algorithm for the other values of the parameter. We compare the efficiency of our algorithm with the previous method and show the improvement in terms of minimum acceptance probability. The algorithm proposed here has an acceptance probability which is superior to e/4.     

SUBSAMPLING THE JOHANSEN TEST WITH STABLE INNOVATIONS

By taking into account the thick-tail property of the errors, cointegration analysis in vector error-correction models with infinite-variance stable errors is a natural generalization of cointegration analysis in error-correction models with normally distributed errors. We study the Johansen test for cointegrated systems under symmetric stable innovations with discrete spectral measures. The results show that the distributions of the Johansen test statistics under these innovations involve nuisance parameters. To overcome the problem of nuisance parameters, we implement a nonparametric subsampling procedure. We document some subsampling simulation results and demonstrate in an empirical example how the test can be used in practice.     

Behaviour of the posterior error rate with partial prior information in auditing

Most of the Bayesian literature on statistical techniques in auditing has focused on assessing appropriate prior density using parameters such as interest, error rate and the mean of the error amount. Frequently, prior beliefs and mathematical tractable reasons are jointly used to assess prior distributions. As a robust Bayesian approach, we propose to replace the prior distribution with a set of prior distributions compatible with auditor's beliefs. We show how an auditor may draw the behaviour of the posterior error rate, using only partial prior information (quartiles of the prior distribution for the error rate O and, very often, the prior distribution is assumed to be unimodal). An example is pursued in depth.     

The Generalized Birnbaum–Saunders Distribution and Its Theory,Methodology, and Application

In this paper, we discuss the class of generalized Birnbaum–Saunders distributions, which is a very flexible family suitable for modeling lifetime data as it allows for different degrees of kurtosis and asymmetry and unimodality as well as bimodality. We describe the theoretical developments on this model including properties, transformations and related distributions, lifetime analysis, and shape analysis. We also discuss methods of inference based on uncensored and censored data, diagnostics methods, goodness-of-fit tests, and random number generation algorithms for the generalized Birnbaum–Saunders model. Finally, we present some illustrative examples and show that this distribution fits the data better than the classical Birnbaum–Saunders model.     

Identifiability of Finite Mixtures of Elliptical Distributions

Abstract.  We present general results on the identifiability of finite mixtures of elliptical distributions under conditions on the characteristic generators or density generators. Examples include the multivariate t -distribution, symmetric stable laws, exponential power and Kotz distributions. In each case, the shape parameter is allowed to vary in the mixture, in addition to the location vector and the scatter matrix. Furthermore, we discuss the identifiability of finite mixtures of elliptical densities with generators that correspond to scale mixtures of normal distributions.     

Asymptotic Likelihood Based Inference for Co-integrated Homogenous Gaussian Diffusions

In this paper we consider inference for a multivariate Gaussian homogenous diffusion which is co-integrated, i.e. admits a representation in terms of stable relations (ergodic diffusions) plus Brownian motions. We show that inference on co-integration rank (the number of stable relations) in continuous time can be based on existing asymptotic distributions from discrete time co-integration analysis. Likewise the asymptotic distributions of the co-integration parameters are shown to be mixed Gaussian. Special attention is given to the parametrization of the drift terms. It is shown that the asymptotic distribution of the co-integration rank test statistic does not depend on the level of the process as a result of the chosen parametrization.     

Flexible Class of Skew-Symmetric Distributions

Abstract.  We propose a flexible class of skew-symmetric distributions for which the probability density function has the form of a product of a symmetric density and a skewing function. By constructing an enumerable dense subset of skewing functions on a compact set, we are able to consider a family of distributions, which can capture skewness, heavy tails and multimodality systematically. We present three illustrative examples for the fibreglass data, the simulated data from a mixture of two normal distributions and the Swiss bills data.     

On the Predictive Distributions of Outcome Gains in the Presence of an Unidentified Parameter

In this article we describe methods for obtaining the predictive distributions of outcome gains in the framework of a standard latent variable selection model. Although most previous work has focused on estimation of mean treatment parameters as the method for characterizing outcome gains from program participation, we show how the entire distributions associated with these gains can be obtained in certain situations. Although the out-of-sample outcome gain distributions depend on an unidentified parameter, we use the results of Koop and Poirier to show that learning can take place about this parameter through information contained in the identified parameters via a positive definiteness restriction on the covariance matrix. In cases where this type of learning is not highly informative, the spread of the predictive distributions depends more critically on the prior. We show both theoretically and in extensive generated data experiments how learning occurs, and delineate the sensitivity of our results to the prior specifications. We relate our analysis to three treatment parameters widely used in the evaluation literature—the average treatment effect, the effect of treatment on the treated, and the local average treatment effect—and show how one might approach estimation of the predictive distributions associated with these outcome gains rather than simply the estimation of mean effects. We apply these techniques to predict the effect of literacy on the weekly wages of a sample of New Jersey child laborers in 1903.     

On Max-Multiscaling Distributions as Extended Max-Semistable Ones

Abstract

We introduce max-multiscaling distributions as solutions to a functional equation which, in a natural way, extends the one fulfilled by max-semistable distributions. We establish that strictly max-multiscaling distributions are products of at most two max-semistable distributions. Next, we show how to obtain these solutions as limit laws of normalized maximum of suitable independent sequences of random variables when sample size has geometric growth.     

Extreme Value Distributions for the Skew-Symmetric Family of Distributions

We derive the extreme value distribution of the skew-symmetric family, the probability density function of the latter being defined as twice the product of a symmetric density and a skewing function. We show that, under certain conditions on the skewing function, this extreme value distribution is the same as that for the symmetric density. We illustrate our results using various examples of skew-symmetric distributions as well as two data sets.