Normal Variance-mean Mixture Research Articles

On Properties of the Hyperbolic Distribution

This paper is set to analytically describe properties of the hyperbolic distribution. This law, along with the variance-gamma distribution, is one of the most popular normal mean–variance mixtures from the point of view of various applications. We have found closed form expressions for the cumulative distribution and partial-moment-generating functions of the hyperbolic distribution. The obtained formulas use the values of the Humbert confluent hypergeometric and Whittaker special functions. The results are applied to the problem of European option pricing in the related Lévy model of financial market. The research demonstrates that the discussed normal mean–variance mixture is analytically tractable.

Time series clustering based on latent volatility mixture modeling with applications in finance

Modeling financial time series data poses a significant challenge in the realm of time series analysis. The Autoregressive Conditional Heteroskedasticity (ARCH) model stands out as a potent tool for capturing time-varying volatility and heteroskedasticity in financial data. However, conventional ARCH models display sensitivity to departures from normality, leading to the development of extensions employing more flexible distributions. In this context, we propose a robust enhancement to the mixture of ARCH (MoARCH) model by integrating normal mean–variance mixture (NMVM) distributions to model component errors. The stochastic representation of the proposed model allows for a straightforward implementation of an Expectation Conditional Maximization Either (ECME) algorithm for obtaining Maximum Penalized Likelihood estimates (MPL). To thoroughly evaluate the model, we conduct four simulation studies to explore finite-sample properties, assess MPL estimators, scrutinize model robustness, and evaluate the accuracy of our proposal in fitting, clustering, and forecasting. Practical applications further highlight the effectiveness of our methodology, showcasing successful implementations across diverse real datasets.

Generalised shot-noise representations of stochastic systems driven by non-Gaussian Lévy processes

Abstract We consider the problem of obtaining effective representations for the solutions of linear, vector-valued stochastic differential equations (SDEs) driven by non-Gaussian pure-jump Lévy processes, and we show how such representations lead to efficient simulation methods. The processes considered constitute a broad class of models that find application across the physical and biological sciences, mathematics, finance, and engineering. Motivated by important relevant problems in statistical inference, we derive new, generalised shot-noise simulation methods whenever a normal variance-mean (NVM) mixture representation exists for the driving Lévy process, including the generalised hyperbolic, normal-gamma, and normal tempered stable cases. Simple, explicit conditions are identified for the convergence of the residual of a truncated shot-noise representation to a Brownian motion in the case of the pure Lévy process, and to a Brownian-driven SDE in the case of the Lévy-driven SDE. These results provide Gaussian approximations to the small jumps of the process under the NVM representation. The resulting representations are of particular importance in state inference and parameter estimation for Lévy-driven SDE models, since the resulting conditionally Gaussian structures can be readily incorporated into latent variable inference methods such as Markov chain Monte Carlo, expectation-maximisation, and sequential Monte Carlo.

Introducing a Family of Distributions by Using the Class of Normal Mean–Variance Mixture

Introducing a Family of Distributions by Using the Class of Normal Mean–Variance Mixture

Random games under normal mean–variance mixture distributed independent linear joint chance constraints

In this paper, we study an n player game where the payoffs as well as the strategy sets are defined using random variables. The payoff function of each player is defined using expected value function and his/her strategy set is defined using a linear joint chance constraint. The random constraint vectors defining the joint chance constraint are independent and follow normal mean–variance mixture distributions. For each player, we reformulate the joint chance constraint in order to prove the existence of a Nash equilibrium using the Kakutani fixed-point theorem under mild assumptions. We illustrate our theoretical results by considering a game between two competing firms in financial market.

Estimation for multivariate normal rapidly decreasing tempered stable distributions

In this paper we describe a methodology for parameter estimation of multivariate distributions defined as normal mean-variance mixture where the mixing random variable is rapidly decreasing tempered stable distributed. We address some numerical issues resulting from the use of the characteristic function for density approximation. We focus our attention on the practical implementation of numerical methods involving the use of these multivariate distributions in the field of finance and we empirical assess the proposed algorithm through an analysis on a five-dimensional series of stock index log-returns.

Multivariate measurement error models with normal mean‐variance mixture distributions

The class of normal mean‐variance mixture (NMVM) distributions is a rich family of asymmetric and heavy‐tailed distributions and has been widely considered in parametric modeling of the data for robust statistical inference. This paper proposes an extension of measurement error models by assuming the NMVM distributions for the unobserved covariates and error terms in the model, referred to as the NMVM‐MEM. An expectation conditional maximization either (ECME) algorithm is developed to compute the maximum likelihood (ML) estimates of model parameters. Furthermore, an information‐based approach is performed to derive the asymptotic covariance matrix of ML estimators. The analysis of a blood pressure dataset illustrates the superiority of NMVM‐MEM to accommodate asymmetry and outliers over the normal counterpart. Two simulation studies are undertaken to validate our proposed techniques.

Robust mixture regression modeling based on the normal mean-variance mixture distributions

Mixture regression models (MRMs) are widely used to capture the heterogeneity of relationships between the response variable and one or more predictors coming from several non-homogeneous groups. Since the conventional MRMs are quite sensitive to departures from normality caused by extra skewness and possible heavy tails, various extensions built on more flexible distributions have been put forward in the last decade. The class of normal mean-variance mixture (NMVM) distributions that arise from scaling both the mean and variance of a normal random variable with a common mixing distribution encompasses many prominent (symmetric or asymmetrical) distributions as special cases. A unified approach to robustifying MRMs is proposed by considering the class of NMVM distributions for component errors. An expectation conditional maximization either (ECME) algorithm, which incorporates membership indicators and the latent scaling variables as the missing data, is developed for carrying out maximum likelihood (ML) estimation of model parameters. Four simulation studies are conducted to examine the finite-sample property of ML estimators and the robustness of the proposed model against outliers for contaminated and noisy data. The usefulness and superiority of our methodology are demonstrated through applications to two real datasets.

Modeling spatial extremes using normal mean-variance mixtures

Classical models for multivariate or spatial extremes are mainly based upon the asymptotically justified max-stable or generalized Pareto processes. These models are suitable when asymptotic dependence is present, i.e., the joint tail decays at the same rate as the marginal tail. However, recent environmental data applications suggest that asymptotic independence is equally important and, unfortunately, existing spatial models in this setting that are both flexible and can be fitted efficiently are scarce. Here, we propose a new spatial copula model based on the generalized hyperbolic distribution, which is a specific normal mean-variance mixture and is very popular in financial modeling. The tail properties of this distribution have been studied in the literature, but with contradictory results. It turns out that the proofs from the literature contain mistakes. We here give a corrected theoretical description of its tail dependence structure and then exploit the model to analyze a simulated dataset from the inverted Brown–Resnick process, hindcast significant wave height data in the North Sea, and wind gust data in the state of Oklahoma, USA. We demonstrate that our proposed model is flexible enough to capture the dependence structure not only in the tail but also in the bulk.

Modelling Skewed and Heavy-tailed Data Using a Normal Weighted Inverse Gaussian Distribution

The normal distribution is inadequate in capturing skewed and heavy-tailed behaviour of data taken over short time intervals. In addition the data can be leptokurtic. For this reason a normal weighted inverse Gaussian distribution is proposed as an alternative to the normal inverse Gaussian distribution to handle such data. The mixing distribution used in the normal variance mean mixture is a finite mixture of two special cases of Generalized Inverse Gaussian \((\textit{GIG})\) distribution. The two special cases and the finite mixture are weighted inverse Gaussian distribution. The motivation for this work is that a finite mixture is more flexible than a single/standard distribution. The \(textit{EM}\)-algorithm has been used for parameter estimation.

Value at Risk and Expected Shortfall for Normal Variance Mean Mixtures of Finite Weighted Inverse Gaussian Distributions

The Normal Inverse Gaussian (NIG) distribution, a special case of the Generalized Hyperbolic Distribution (GHD) has been frequently used for financial modelling and risk measures. In this work, we consider other normal Variance mean mixtures based on finite mixtures of special cases for Generalised Inverse Gaussian as mixing distributions. The Expectation-Maximization (EM) algorithm has been used to obtain the Maximum Likelihood (ML) estimates of the proposed models for some financial data. We estimate Value at risk (VaR) and Expected Shortfall (ES) for the fitted models. The Kupiec likelihood ratio (LR) has been applied for backtesting of VaR. Akaike Information Creterion (AIC), Bayesian Information Creterion (BIC) and Log-likelihood have been used for model selection. The results clearly show that the proposed models are good alternatives to NIG for determining VaR and ES.

Modelling Skewed and Heavy-tailed Data Using a Normal Weighted Inverse Gaussian Distribution

The normal distribution is inadequate in capturing skewed and heavy-tailed behaviour of data taken over short time intervals. In addition the data can be leptokurtic. For this reason a normal weighted inverse Gaussian distribution is proposed as an alternative to the normal inverse Gaussian distribution to handle such data. The mixing distribution used in the normal variance mean mixture is a finite mixture of two special cases of Generalized Inverse Gaussian \((\textit{GIG})\) distribution. The two special cases and the finite mixture are weighted inverse Gaussian distribution. The motivation for this work is that a finite mixture is more flexible than a single/standard distribution. The \(textit{EM}\)-algorithm has been used for parameter estimation.

A Normal Weighted Inverse Gaussian Distribution for Skewed and Heavy-Tailed Data

High frequency financial data is characterized by non-normality: asymmetric, leptokurtic and fat-tailed behaviour. The normal distribution is therefore inadequate in capturing these characteristics. To this end, various flexible distributions have been proposed. It is well known that mixture distributions produce flexible models with good statistical and probabilistic properties. In this work, a finite mixture of two special cases of Generalized Inverse Gaussian distribution has been constructed. Using this finite mixture as a mixing distribution to the Normal Variance Mean Mixture we get a Normal Weighted Inverse Gaussian (NWIG) distribution. The second objective, therefore, is to construct and obtain properties of the NWIG distribution. The maximum likelihood parameter estimates of the proposed model are estimated via EM algorithm and three data sets are used for application. The result shows that the proposed model is flexible and fits the data well.

Hessian orderings of multivariate normal variance-mean mixture distributions and their applications in evaluating dependent multivariate risk portfolios

Hessian orderings of multivariate normal variance-mean mixture distributions and their applications in evaluating dependent multivariate risk portfolios

A Special Weight for Inverse Gaussian Mixing Distribution in Normal Variance Mean Mixture with Application

Normal Variance-Mean Mixture (NVMM) provides a general framework for deriving models with desirable properties for modelling financial market variables such as exchange rates, equity prices, and interest rates measured over short time intervals, i.e. daily or weekly. Such data sets are characterized by non-normality and are usually skewed, fat-tailed and exhibit excess kurtosis. The Generalised Hyperbolic distribution (GHD) introduced by Barndorff-Nielsen (1977) which act as Normal variance-mean mixtures with Generalised Inverse Gaussian (GIG) mixing distribution nest a number of special and limiting case distributions. The Normal Inverse Gaussian (NIG) distribution is obtained when the Inverse Gaussian is the mixing distribution, i.e., the index parameter of the GIG is . The NIG is very popular because of its analytical tractability. In the mixing mechanism, the mixing distribution characterizes the prior information of the random variable of the conditional distribution. Therefore, considering finite mixture models is one way of extending the work. The GIG is a three parameter distribution denoted by and nest several special and limiting cases. When , we have which is called an Inverse Gaussian (IG) distribution. When , , , we have , and distributions respectively. These distributions are related to and are called weighted inverse Gaussian distributions. In this work, we consider a finite mixture of and and show that the mixture is also a weighted Inverse Gaussian distribution and use it to construct a NVMM. Due to the complexity of the likelihood, direct maximization is difficult. An EM type algorithm is provided for the Maximum Likelihood estimation of the parameters of the proposed model. We adopt an iterative scheme which is not based on explicit solution to the normal equations. This subtle approach reduces the computational difficulty of solving the complicated quantities involved directly to designing an iterative scheme based on a representation of the normal equation. The algorithm is easily programmable and we obtained a monotonic convergence for the data sets used.

A Finite Mixture of Generalised Inverse Gaussian with Indexes -1/2 and -3/2 as Mixing Distribution for Normal Variance Mean Mixture with Application

Mixture models have become more popular in modelling compared to standard distributions. The mixing distributions play a role in capturing the variability of the random variable in the conditional distribution. Studies have lately focused on finite mixture models as mixing distributions in the mixing mechanism. In the present work, we consider a Normal Variance Mean mixture model. The mixing distribution is a finite mixture of two special cases of Generalised Inverse Gaussian distribution with indexes -1/2 and -3/2. The parameters of the mixed model are obtained via the Expectation-Maximization (EM) algorithm. The iterative scheme is based on a presentation of the normal equations. An application to some financial data has been done.

Pricing the correlation skew with normal mean-variance mixture copulas

In this paper we propose a new pricing methodology for European multi-asset options based on the family of normal mean-variance mixture copulas. The goal is to develop a copula-based method with the flexibility to reproduce the correlation skew, and at the same time efficient enough to be used for large baskets. Simplicity and ease of implementation are also properties of the model. After presenting the relevant pricing formulae, the methodology is applied to several market problems where the correlation skew is involved in different ways.

The effect of underlying distribution of asset returns on efficiency in DEA models

According to modern finance theory and increasing need for efficient investments, we evaluate the portfolio performance based on the data envelopment analysis method. By the fact that stock market’s return distributions usually exhibit skewness, kurtosis and heavy-tails, we consider some appropriate underlying distributions that affect the input and output of the model. In this regard, the multivariate skewed t and the multivariate generalized hyperbolic as the heavy-tailed distributions of Normal mean-variance mixture are applied. The models are inspired by the Range Directional Measure (RDM) model to deal with negative values. The value-at-risk (VaR) and conditional VaR (CVaR) as risk measures are used in these optimization problems. We estimate the parameters of such distributions by Expectation Maximization algorithm. Then we present an empirical investigation to measure the relative efficiency of two sets of seven groups of companies from different industries of Iran stock exchange market. By comparing the results of introduced models with previous RDM approach, we show that how well the distribution of assets affect the performance evaluation.

Inverse Gaussian quadrature and finite normal-mixture approximation of the generalized hyperbolic distribution

In this study, a numerical quadrature for the generalized inverse Gaussian distribution is derived from the Gauss–Hermite quadrature by exploiting its relationship with the normal distribution. The proposed quadrature is not Gaussian, but it exactly integrates the polynomials of both positive and negative orders. Using the quadrature, the generalized hyperbolic distribution is efficiently approximated as a finite normal variance–mean mixture. Therefore, the expectations under the distribution, such as cumulative distribution function and European option price, are accurately computed as weighted sums of those under normal distributions. The generalized hyperbolic random variates are also sampled in a straightforward manner. The accuracy of the methods is illustrated with numerical examples.

Linear Mixed Effects Models for Non-Gaussian Continuous Repeated Measurement Data

SummaryWe consider the analysis of continuous repeated measurement outcomes that are collected longitudinally. A standard framework for analysing data of this kind is a linear Gaussian mixed effects model within which the outcome variable can be decomposed into fixed effects, time invariant and time-varying random effects, and measurement noise. We develop methodology that, for the first time, allows any combination of these stochastic components to be non-Gaussian, using multivariate normal variance–mean mixtures. To meet the computational challenges that are presented by large data sets, i.e. in the current context, data sets with many subjects and/or many repeated measurements per subject, we propose a novel implementation of maximum likelihood estimation using a computationally efficient subsampling-based stochastic gradient algorithm. We obtain standard error estimates by inverting the observed Fisher information matrix and obtain the predictive distributions for the random effects in both filtering (conditioning on past and current data) and smoothing (conditioning on all data) contexts. To implement these procedures, we introduce an R package: ngme. We reanalyse two data sets, from cystic fibrosis and nephrology research, that were previously analysed by using Gaussian linear mixed effects models.