Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling

The normal inverse Gaussian distribution is defined as a variance-mean mixture of a normal distribution with the inverse Gaussian as the mixing distribution. The distribution determines an homogeneous Lévy process, and this process is representable through subordination of Brownian motion by the inverse Gaussian process. The canonical, Lévy type, decomposition of the process is determined. As a preparation for developments in the latter part of the paper the connection of the normal inverse Gaussian distribution to the classes of generalized hyperbolic and inverse Gaussian distributions is briefly reviewed. Then a discussion is begun of the potential of the normal inverse Gaussian distribution and Lévy process for modelling and analysing statistical data, with particular reference to extensive sets of observations from turbulence and from finance. These areas of application imply a need for extending the inverse Gaussian Lévy process so as to accommodate certain, frequently observed, temporal dependence structures. Some extensions, of the stochastic volatility type, are constructed via an observation-driven approach to state space modelling. At the end of the paper generalizations to multivariate settings are indicated.     

Absolute Moments of Generalized Hyperbolic Distributions and Approximate Scaling of Normal Inverse Gaussian Lévy Processes

Abstract.  Expressions for (absolute) moments of generalized hyperbolic and normal inverse Gaussian (NIG) laws are given in terms of moments of the corresponding symmetric laws. For the (absolute) moments centred at the location parameter μ explicit expressions as series containing Bessel functions are provided. Furthermore, the derivatives of the logarithms of absolute μ -centred moments with respect to the logarithm of time are calculated explicitly for NIG Lévy processes. Computer implementation of the formulae obtained is briefly discussed. Finally, some further insight into the apparent scaling behaviour of NIG Lévy processes is gained.     

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     

Poisson-mixed Inverse Gaussian Regression Model and Its Application

In this article, we have developed a Poisson-mixed inverse Gaussian (PMIG) distribution. The mixed inverse Gaussian distribution is a mixture of the inverse Gaussian distribution and its length-biased counterpart. A PMIG regression model is developed and the maximum likelihood estimation of the parameters is studied. A dataset dealing with the number of hospital stays among the elderly population is analyzed by using the PMIG and the PIG (Poisson-inverse Gaussian) regression models and it has been shown that the PMIG model fits the data better than the PIG model.     

The Volatility of Realized Volatility

In recent years, with the availability of high-frequency financial market data modeling realized volatility has become a new and innovative research direction. The construction of “observable” or realized volatility series from intra-day transaction data and the use of standard time-series techniques has lead to promising strategies for modeling and predicting (daily) volatility. In this article, we show that the residuals of commonly used time-series models for realized volatility and logarithmic realized variance exhibit non-Gaussianity and volatility clustering. We propose extensions to explicitly account for these properties and assess their relevance for modeling and forecasting realized volatility. In an empirical application for S&P 500 index futures we show that allowing for time-varying volatility of realized volatility and logarithmic realized variance substantially improves the fit as well as predictive performance. Furthermore, the distributional assumption for residuals plays a crucial role in density forecasting.     

The Volatility of Realized Volatility

In recent years, with the availability of high-frequency financial market data modeling realized volatility has become a new and innovative research direction. The construction of “observable” or realized volatility series from intra-day transaction data and the use of standard time-series techniques has lead to promising strategies for modeling and predicting (daily) volatility. In this article, we show that the residuals of commonly used time-series models for realized volatility and logarithmic realized variance exhibit non-Gaussianity and volatility clustering. We propose extensions to explicitly account for these properties and assess their relevance for modeling and forecasting realized volatility. In an empirical application for S&P 500 index futures we show that allowing for time-varying volatility of realized volatility and logarithmic realized variance substantially improves the fit as well as predictive performance. Furthermore, the distributional assumption for residuals plays a crucial role in density forecasting.     

Gamma-Generalized Inverse Gaussian Class of Distributions with Applications

In this article, a new family of probability distributions with domain in ?⁺ is introduced. This class can be considered as a natural extension of the exponential-inverse Gaussian distribution in Bhattacharya and Kumar (1986 Bhattacharya , S. K. , Kumar , S. ( 1986 ). E-IG model in life testing . Calcutta Statist. Assoc. Bull. 35 : 85 – 90 . [Google Scholar]) and Frangos and Karlis (2004 Frangos , N. , Karlis , D. ( 2004 ). Modelling losses using an exponential-inverse Gaussian distribution . Insur. Math. Econo. 35 : 53 – 67 .[Crossref], [Web of Science ®] , [Google Scholar]). This new family is obtained through the mixture of gamma distribution with generalized inverse Gaussian distribution. We also show some important features such as expressions of probability density function, moments, etc. Special attention is paid to the mixture with the inverse Gaussian distribution, as a particular case of the generalized inverse Gaussian distribution. From the exponential-inverse Gaussian distribution two one-parameter family of distributions are obtained to derive risk measures and credibility expressions. The versatility of this family has been proven in numerical examples.     

An Easily Implemented Generalised Inverse Gaussian Generator

Using the ‘ratio’ method an easily implemented algorithm is derived for the generalised inverse Gaussian distribution. Computer timings and efficiency calculations show that the procedure is fast over a wide range of distribution parameter values.     

A Stochastic Volatility Model With Conditional Skewness

We develop a discrete-time affine stochastic volatility model with time-varying conditional skewness (SVS). Importantly, we disentangle the dynamics of conditional volatility and conditional skewness in a coherent way. Our approach allows current asset returns to be asymmetric conditional on current factors and past information, which we term contemporaneous asymmetry. Conditional skewness is an explicit combination of the conditional leverage effect and contemporaneous asymmetry. We derive analytical formulas for various return moments that are used for generalized method of moments (GMM) estimation. Applying our approach to S&P500 index daily returns and option data, we show that one- and two-factor SVS models provide a better fit for both the historical and the risk-neutral distribution of returns, compared to existing affine generalized autoregressive conditional heteroscedasticity (GARCH), and stochastic volatility with jumps (SVJ) models. Our results are not due to an overparameterization of the model: the one-factor SVS models have the same number of parameters as their one-factor GARCH competitors and less than the SVJ benchmark.     

Kolmogorov-Smirnov Type Test-Statistics For The Gamma,Erlang-2 And The Inverse Gaussian Distributions When The Parameters Are Unknown.

Critical values are presented for the Kolmogorov-Smirnov type test statistics for the following three cases: (i) the gamma distribution when both the scale and the shape parameters are not known, (ii) the scale parameter of the gamma distribution is not known and (iii) the inverse Gaussian distribution when both the parameters are unknown. This study was motivated by the necessity to fit the gamma, the Erlang-2 and the inverse Gaussian distributions to the interpurchase times of individuals for coffee in marketing research.     

Markov Switching in GARCH Processes and Mean-Reverting Stock-Market Volatility

This article introduces four models of conditional heteroscedasticity that contain Markov-switching parameters to examine their multiperiod stock-market volatility forecasts as predictions of options-implied volatilities. The volatility model that best predicts the behavior of the options-implied volatilities allows the Student-t degrees-of-freedom parameter to switch such that the conditional variance and kurtosis are subject to discrete shifts. The half-life of the most leptokurtic state is estimated to be a week, so expected market volatility reverts to near-normal levels fairly quickly following a spike.     

Analysis of Bivariate Survival Data using Shared Inverse Gaussian Frailty Model

In this article, we consider shared frailty model with inverse Gaussian distribution as frailty distribution and log-logistic distribution (LLD) as baseline distribution for bivariate survival times. We fit this model to three real-life bivariate survival data sets. The problem of analyzing and estimating parameters of shared inverse Gaussian frailty is the interest of this article and then compare the results with shared gamma frailty model under the same baseline for considered three data sets. Data are analyzed using Bayesian approach to the analysis of clustered survival data in which there is a dependence of failure time observations within the same group. The variance component estimation provides the estimated dispersion of the random effects. We carried out a test for frailty (or heterogeneity) using Bayes factor. Model comparison is made using information criteria and Bayes factor. We observed that the shared inverse Gaussian frailty model with LLD as baseline is the better fit for all three bivariate data sets.     

On the Product of Inverse Wishart and Normal Distributions with Applications to Discriminant Analysis and Portfolio Theory

Abstract. In this article we analyse the product of the inverse Wishart matrix and a normal vector. We derive the explicit joint distribution of the components of the product. Furthermore, we suggest several exact tests of general linear hypothesis about the elements of the product. We illustrate the developed techniques on examples from discriminant analysis and from portfolio theory.     

Use of the Likelihood Ratio Test on the Inverse Gaussian Distribution

In the following article, the likelihood ratio test is determined for four tests of hypotheses involving the inverse Gaussian distribution. For three of the hypotheses, the test produces the same statistic as the uniformly most powerful unbiased test.     

Implicit Estimation for the Stochastic Volatility Model

Efficient, accurate, and fast Markov Chain Monte Carlo estimation methods based on the Implicit approach are proposed. In this article, we introduced the notion of Implicit method for the estimation of parameters in Stochastic Volatility models.

Implicit estimation offers a substantial computational advantage for learning from observations without prior knowledge and thus provides a good alternative to classical inference in Bayesian method when priors are missing.

Both Implicit and Bayesian approach are illustrated using simulated data and are applied to analyze daily stock returns data on CAC40 index.     

Varying Dispersion Diagnostics for Inverse Gaussian Regression Models

Homogeneity of dispersion parameters is a standard assumption in inverse Gaussian regression analysis. However, this assumption is not necessarily appropriate. This paper is devoted to the test for varying dispersion in general inverse Gaussian linear regression models. Based on the modified profile likelihood (Cox & Reid, 1987), the adjusted score test for varying dispersion is developed and illustrated with Consumer- Product Sales data (Whitmore, 1986) and Gas vapour data (Weisberg, 1985). The effectiveness of orthogonality transformation and the properties of a score statistic and its adjustment are investigated through Monte Carlo simulations.     

Inference on Overlap for Two Inverse Gaussian Populations: Equal Means Case

The inverse Gaussian distribution is often suited for modeling positive and/or positively skewed data (see Chhikara and Folks, 1989 Chhikara , R. S. , Folks , J. L. ( 1989 ). The Inverse Gaussian Distribution . New York : Marcel Dekker . [Google Scholar]) and presents an interesting alternative to the Gaussian model in such cases. We note here that overlap coefficients and their variants are widely studied in the literature for Gaussian populations (see Mulekar and Mishra, 1994 Mulekar , M. , Mishra , S. N. ( 1994 ). Overlap coefficients of two normal densities: equal means case . J. Japan. Statist. Soc. 24 : 169 – 180 . [Google Scholar], 2000 Mulekar , M. , Mishra , S. N. ( 2000 ). Confidence interval estimation of overlap: equal means case . Computat. Statist. Data Anal. 34 : 121 – 137 .[Crossref], [Web of Science ®] , [Google Scholar], and references therein for further details). This article studies the properties and addresses point estimation for large samples of commonly used measures of overlap when the populations are described by inverse Gaussian distributions. The bias and mean square error properties of the estimators are studied through a simulation study.     

Testing Homogeneity of Inverse Gaussian Scale Parameters Based on Generalized Likelihood Ratio

This article presents a new procedure for testing homogeneity of scale parameters from k independent inverse Gaussian populations. Based on the idea of generalized likelihood ratio method, a new generalized p-value is derived. Some simulation results are presented to compare the performance of the proposed method and existing methods. Numerical results show that the proposed test has good size and power performance.