Normal Inverse Gaussian Distribution Research Articles

Parametric description of intermittent probability distribution functions in solar wind and magnetohydrodynamic turbulence

ABSTRACT In this work, we find empirical evidence that the scale-dependent statistical properties of solar wind and magnetohydrodynamic (MHD) turbulence can be described in terms of a family of parametric probability distribution functions (PDFs) known as Normal Inverse Gaussian (NIG). Understanding these PDFs is one of the most important goals in turbulence theory, as they are inherently connected to the intermittent properties of solar wind turbulence. We investigate the properties of PDFs of Elsasser increments based on a large statistical sample from solar wind observations and high-resolution numerical simulations of MHD turbulence. In order to measure the PDFs and their corresponding properties, three experiments are presented: fast and slow solar wind for experimental data and a simulation of reduced MHD (RMHD) turbulence. Conditional statistics on a 23-yr-long sample of WIND data near 1 au and high-resolution pseudo-spectral simulation of steadily driven RMHD turbulence on a $2048^3$ mesh are used to construct scale-dependent PDFs. The empirical PDFs are fitted to NIG distributions, which depend on four free parameters. Our analysis shows that NIG distributions accurately capture the evolution of the PDFs, with scale-dependent parameters, from large scales characterized by a Gaussian distribution, turning to exponential tails within the inertial range and stretched exponentials at dissipative scales. We also show that empirically-measured NIG parameters exhibit well-defined scaling properties that are similar across the three empirical data sets, which may be indicative of universal behaviour.

Value-at-Risk Effectiveness: A High-Frequency Data Approach with Semi-Heavy Tails

In the broader landscape of cryptocurrency risk management, this study delves into the nuanced estimation of Value-at-Risk (VaR) for a uniformly weighted portfolio of cryptocurrencies, employing the bivariate Normal Inverse Gaussian distribution renowned for its semi-heavy tails. Utilizing high-frequency data spanning between 1 January 2017 and 25 October 2022, with a primary focus on Bitcoin and Ethereum, our research seeks to accentuate the resilience of VaR methodology as a paramount risk assessment tool. The essence of our investigation lies in advancing the comprehension of VaR accuracy by quantitatively comparing the observed returns of both cryptocurrencies with their corresponding estimated values, with a central theme being the endorsement of the Normal Inverse Gaussian distribution as a potent model for risk measurement, particularly in the domain of high-frequency data. To bolster the statistical reliability of our results, we adopt a forward test methodology, showcasing not only a contribution to the evolution of risk assessment techniques in Finance but also underscoring the practicality of sophisticated distributional models in econometrics. Our findings not only contribute to the refinement of risk assessment methods but also highlight the applicability of such models in precisely modeling and forecasting financial risk within the dynamic realm of cryptocurrencies, epitomized by the case study of Bitcoin and Ethereum.

Skewness and kurtosis of solar wind proton distribution functions: The normal inverse-Gaussian model and its implications

Context. In the solar wind (SW), the particle distribution functions are generally not Gaussian. They present nonthermal features that are related to underlying acceleration and heating processes. These processes are critical in the overall dynamics of this expanding astrophysical fluid. Aims. The Proton Alpha Sensor (PAS) on board Solar Orbiter commonly observes skewed proton distributions, with a more populated high-energy side in the magnetic field direction than the Gaussian distribution. Our objectives are: (1) to identify a theoretical statistical function that adequately models the observed distributions and (2) to use its statistical interpretation to constrain the acceleration and heating processes. Methods. We analyzed the 3D velocity distribution functions (VDFs) measured by PAS and compared them to model statistical functions. Results. We show that the normal inverse Gaussian (NIG), a type of hyperbolic statistical distribution, provides excellent fits of skewed and leptokurtic proton distributions. NIG can model both the core distribution and the beam, if present. We propose an interpretation that is inspired by the mathematical formulation of the NIG. It assumes that the acceleration or heating mechanism can be modeled as a drifting diffusion process in velocity space, controlled (or subordinated) by the time of interaction of the particles with “accelerating structures”. The probability function of the interaction time is an inverse Gaussian (IG), obtained by considering a random drift across structures of a given size. The control of the diffusion by interaction times that follow an IG probability function formally defines the NIG distribution. Following this model, we show that skewness and kurtosis can be used to estimate the kinetic and thermal energy gains provided by the interaction with structures. For example, in the case studies presented here, the analyzed populations would have gained kinetic energy representing approximately two to four times their thermal energy, with an increase in velocity – due to acceleration – of from one-tenth to one-third of the observed flow velocity. We also show that the model constrains the initial temperature of the populations. Conclusions. Overall, the NIG model offers excellent fits of the observed proton distributions. Combining the skewness and the kurtosis, it also leads to constraints in the part of acceleration and heating due to the interactions with structures in the formation of the proton populations. We suggest that these effects add to the classical thermal evolution of the bulk velocity and temperature resulting from SW expansion.

Properties of the entropic risk measure EVaR in relation to selected distributions

Entropic Value-at-Risk (EVaR) measure is a convenient coherent risk measure. Due to certain difficulties in finding its analytical representation, it was previously calculated explicitly only for the normal distribution. We succeeded to overcome these difficulties and to calculate Entropic Value-at-Risk (EVaR) measure for Poisson, compound Poisson, Gamma, Laplace, exponential, chi-squared, inverse Gaussian distribution and normal inverse Gaussian distribution with the help of Lambert function that is a special function, generally speaking, with two branches.

Volatility Forecasting Using Quasi-Score-Driven Models with an Application to the Coronavirus Pandemic Period

Abstract We study the statistical and volatility forecasting performances of the recent quasi-score-driven EGARCH (exponential generalized autoregressive conditional heteroscedasticity) models. We compare the quasi-score-driven EGARCH models with GARCH, asymmetric power ARCH (A-PARCH), and all relevant score-driven EGARCH models of the literature. For score-driven and quasi-score-driven EGARCH, we use the following seven score-driven probability distributions: Student’s t-distribution; general error distribution (GED); generalized t-distribution (Gen-t); skewed generalized t-distribution (Skew-Gen-t); exponential generalized beta distribution of the second kind (EGB2); normal-inverse Gaussian distribution (NIG); Meixner distribution (MXN). We use all combinations of those distributions for (i) the probability distribution of the dependent variable, and (ii) the probability distribution which defines the quasi-score function updating term of the quasi-score-driven filters. We use daily data for the Standard & Poor’s 500 (S&P 500) index. We find that both in-sample and out-of-sample, quasi-score-driven EGARCH is superior to GARCH, A-PARCH, and score-driven EGARCH. We report in-sample results for the period of January 2000 to December 2020, providing evidence in favor of the quasi-score-driven EGARCH model for the last two decades. We report out-of-sample volatility forecasting results for a period within the coronavirus disease 2019 (COVID-19) pandemic, providing evidence in favor of the quasi-score-driven EGARCH model for a crisis period.

Product of bi-dimensional VAR(1) model components. An application to the cost of electricity load prediction errors

Abstract The multi-dimensional vector autoregressive (VAR) time series is often used to model the impulse-response functions of macroeconomics variables. However, in some economical applications, the variable of main interest is the product of time series describing market variables, like e.g. the cost, being the product of price and volume. In this paper, we analyze the product of the bi-dimensional VAR(1) model components. For the introduced time series, we derive general formulas for the autocovariance function and study its properties for different cases of cross-dependence between the VAR(1) model components. The theoretical results are then illustrated in the simulation study for two types of bivariate distributions of the residual series, namely the Gaussian and Student’s t. The obtained results are applied for the electricity market case study, in which we show that the additional cost of balancing load prediction errors prior to delivery can be well described by time series being the product of the VAR(1) model components with the bivariate normal inverse Gaussian distribution.

Risk Premium of Bitcoin and Ethereum during the COVID-19 and Non-COVID-19 Periods: A High-Frequency Approach

This paper reports our findings on the return dynamics of Bitcoin and Ethereum using high-frequency data (minute-by-minute observations) from 2015 to 2022 for Bitcoin and from 2016 to 2022 for Ethereum. The main objective of modeling these two series was to obtain a dynamic estimation of risk premium with the intention of characterizing its behavior. To this end, we estimated the Generalized Autoregressive Conditional Heteroskedasticity in Mean with Normal-Inverse Gaussian distribution (GARCH-M-NIG) model for the residuals. We also estimated the other parameters of the model and discussed their evolution over time, including the skewness and kurtosis of the Normal-Inverse Gaussian distribution. Similarly, we determined the parameters that define the evolution of the estimated variance, i.e., the parameters related to the fitted past variance, square error and long-term average value. We found that, despite the market uncertainty during the COVID-19 emergency period (2020 and 2021), the selected cryptocurrencies’ return volatility and kurtosis were even greater for several other subperiods within our sample’s time frame. Our model represents an analytical tool that estimates the risk premium that should be delivered by Bitcoin and Ethereum and is therefore of interest to risk managers, traders and investors.

New highly efficient high‐breakdown estimator of multivariate scatter and location for elliptical distributions

AbstractHigh‐breakdown‐point estimators of multivariate location and shape matrices, such as the ‐estimator with smoothed hard rejection and the Rocke ‐estimator, are generally designed to have high efficiency for Gaussian data. However, many phenomena are non‐Gaussian, and these estimators can therefore have poor efficiency. This article proposes a new tunable ‐estimator, termed the ‐estimator, for the general class of symmetric elliptical distributions, a class containing many common families such as the multivariate Gaussian, ‐, Cauchy, Laplace, hyperbolic, and normal inverse Gaussian distributions. Across this class, the ‐estimator is shown to generally provide higher maximum efficiency than other leading high‐breakdown estimators while maintaining the maximum breakdown point. Furthermore, the ‐estimator is demonstrated to be distributionally robust, and its robustness to outliers is demonstrated to be on par with these leading estimators while also being more stable with respect to initial conditions. From a practical viewpoint, these properties make the ‐estimator broadly applicable for practitioners. These advantages are demonstrated with an example application—the minimum‐variance optimal allocation of financial portfolio investments.

The Distribution of Commodity Futures: A Test of the Generalized Hyperbolic Process

ABSTRACT Using daily closing price data spreading over 3 April 1990, to 5 May 2020, this study explores the skewness and excess kurtosis behaviour across energy, metals, and agricultural commodity futures. Subsequently, it compares the fitting of an empirical distribution under normal distribution assumption with those under the Generalized Hyperbolic distribution. The generalized hyperbolic distribution includes Hyperbolic distribution, Variance-Gamma distribution, and Normal Inverse Gaussian distribution. The results show that the Normal Inverse Gaussian distribution for natural gas, gold, platinum, copper, sugar, and feeder cattle futures captures skewness as well as excess kurtosis of the daily logarithmic returns. The findings are robust to the sub-sample analysis.

The expectation-maximization algorithm for autoregressive models with normal inverse Gaussian innovations

In this paper, we study the autoregressive (AR) model with normal inverse Gaussian (NIG) innovations. The NIG distribution is semi heavy-tailed and is helpful in capturing the extreme observations present in the data. The expectation-maximization (EM) algorithm is used to estimate the parameters of the considered AR(p) model. The efficacy of the estimation procedure is shown on the simulated data for AR(2) and AR(1) models. A comparative study is presented, where the classical estimation algorithms are also incorporated, namely, Yule-Walker and conditional least squares methods along with EM method for model parameter estimation. In simulation study, the maximum likelihood estimation (MLE) of NIG distribution by EM algorithm and iterative Newton-Raphson method are also compared. The real-life applications of the introduced model are demonstrated on the NASDAQ stock market index data and US gasoline price data. The studies show that AR(1) model with NIG residuals is good fit for financial data with extreme values as well as for gasoline price data.

Correlated-Weighted Statistically Modeled Contourlet and Curvelet Coefficient Image-Based Breast Tumor Classification Using Deep Learning

Deep learning-based automatic classification of breast tumors using parametric imaging techniques from ultrasound (US) B-mode images is still an exciting research area. The Rician inverse Gaussian (RiIG) distribution is currently emerging as an appropriate example of statistical modeling. This study presents a new approach of correlated-weighted contourlet-transformed RiIG (CWCtr-RiIG) and curvelet-transformed RiIG (CWCrv-RiIG) image-based deep convolutional neural network (CNN) architecture for breast tumor classification from B-mode ultrasound images. A comparative study with other statistical models, such as Nakagami and normal inverse Gaussian (NIG) distributions, is also experienced here. The weighted entitled here is for weighting the contourlet and curvelet sub-band coefficient images by correlation with their corresponding RiIG statistically modeled images. By taking into account three freely accessible datasets (Mendeley, UDIAT, and BUSI), it is demonstrated that the proposed approach can provide more than 98 percent accuracy, sensitivity, specificity, NPV, and PPV values using the CWCtr-RiIG images. On the same datasets, the suggested method offers superior classification performance to several other existing strategies.

A robust multiplicative watermarking technique for digital images in curvelet domain using normal inverse Gaussian distribution

A robust multiplicative watermarking technique for digital images in curvelet domain using normal inverse Gaussian distribution

Infinite Mixtures of Multivariate Normal-Inverse Gaussian Distributions for Clustering of Skewed Data

Mixtures of multivariate normal inverse Gaussian (MNIG) distributions can be used to cluster data that exhibit features such as skewness and heavy tails. For cluster analysis, using a traditional finite mixture model framework, the number of components either needs to be known a priori or needs to be estimated a posteriori using some model selection criterion after deriving results for a range of possible number of components. However, different model selection criteria can sometimes result in different numbers of components yielding uncertainty. Here, an infinite mixture model framework, also known as Dirichlet process mixture model, is proposed for the mixtures of MNIG distributions. This Dirichlet process mixture model approach allows the number of components to grow or decay freely from 1 to \(\infty\) (in practice from 1 to N) and the number of components is inferred along with the parameter estimates in a Bayesian framework, thus alleviating the need for model selection criteria. We run our algorithm on simulated as well as real benchmark datasets and compare with other clustering approaches. The proposed method provides competitive results for both simulations and real data.

Score-driven multi-regime Markov-switching EGARCH: empirical evidence using the Meixner distribution

Abstract In this paper, statistical and volatility forecasting performances of the non-path-dependent score-driven multi-regime Markov-switching (MS) exponential generalized autoregressive conditional heteroskedasticity (EGARCH) models are explored. Three contributions to the existing literature are provided. First, we use all relevant score-driven distributions from the literature - namely, the Student’s t-distribution, general error distribution (GED), skewed generalized t-distribution (Skew-Gen-t), exponential generalized beta distribution of the second kind (EGB2), and normal-inverse Gaussian (NIG) distribution. We then introduce the score-driven Meixner (MXN) distribution-based EGARCH model to the literature on score-driven models. Second, proving the sufficient conditions of the asymptotic properties of the maximum likelihood (ML) estimator for non-path-dependent score-driven MS-EGARCH models is an unsolved problem. We provide a partial solution to that problem by proving necessary conditions for the asymptotic theory of the ML estimator. Third, to the best of our knowledge, this work includes the largest number of international stock indices from the G20 countries in the literature, covering the period of 2000–2022. We provide a discussion on the major events which caused common or non-common switching to the high-volatility regime for the G20 countries. The statistical performance and volatility forecasting results support the adoption of score-driven MS-EGARCH for the G20 countries.

Multivariate tempered stable additive subordination for financial models

We study a class of multivariate tempered stable distributions and introduce the associated class of tempered stable Sato subordinators. These Sato subordinators are used to build additive inhomogeneous processes by subordination of a multiparameter Brownian motion. The resulting process is additive and time inhomogeneous and it is a generalization of multivariate Lévy processes with good fit properties on financial data. We specify the model to have unit time normal inverse Gaussian distribution and we discuss the ability of the model to fit time inhomogeneous correlations on real data.

Risk Modelling Approaches for Student-like Models with Fractal Activity Time

The paper focuses on value at risk (V@R) measuring for Student-like models of markets with fractal activity time (FAT). The fractal activity time models were introduced by Heyde to try to encompass the empirically found characteristics of real data and elaborated on for Variance Gamma, normal inverse Gaussian and skewed Student distributions. But problem of evaluating an value at risk for this model was not researched. It is worth to mention that if we use normal or symmetric Student`s models than V@R can be computed using standard statistical packages. For calculating V@R for Student-like models we need Monte Carlo method and the iterative scheme for simulating N scenarios of stock prices. We model stock prices as a diffusion processes with the fractal activity time and for modeling increments of fractal activity time we use another diffusion process, which has a given marginal inverse gamma distribution.
 The aim of the paper is to perform and compare V@R Monte Carlo approach and Markowitz approach for Student-like models in terms of portfolio risk. For this purpose we propose procedure of calculating V@R for two types of investor portfolios. The first one is uniform portfolio, where d assets are equally distributed. The second is optimal Markowitz portfolio, for which variance of return is the smallest out of all other portfolios with the same mean return.
 The programmed model which was built using R-statistics can be used as to the simulations for any asset and for construct optimal portfolios for any given amount of assets and then can be used for understanding how this optimal portfolio behaves compared to other portfolios for Student-like models of markets with fractal activity time.
 Also we present numerical results for evaluating V@R for both types of investor portfolio. We show that optimal Markovitz portfolio demonstrates in the most of cases the smallest possible Value at Risk comparing with other portfolios. Thus, for making investor decisions under uncertainty we recommend to apply portfolio optimization and value at risk approach jointly.

Predicting VaR for China's stock market: A score-driven model based on normal inverse Gaussian distribution

Under the framework of dynamic conditional score, we propose a parametric forecasting model for Value-at-Risk based on the normal inverse Gaussian distribution (Hereinafter NIG-DCS-VaR), which creatively incorporates intraday information into daily VaR forecast. NIG specifies an appropriate distribution to return and the semi-additivity of the NIG parameters makes it feasible to improve the estimation of daily return in light of intraday return, and thus the VaR can be explicitly obtained by calculating the quantile of the re-estimated distribution of daily return. We conducted an empirical analysis using two main indexes of the Chinese stock market, and a variety of backtesting approaches as well as the model confidence set approach prove that the VaR forecasts of NIG-DCS model generally gain an advantage over those of realized GARCH (RGARCH) models. Especially when the risk level is relatively high, NIG-DCS-VaR beats RGARCH-VaR in terms of coverage ability and independence.

A Monte Carlo Approach to Bitcoin Price Prediction with Fractional Ornstein–Uhlenbeck Lévy Process

Since its inception in 2009, Bitcoin has increasingly gained main stream attention from the general population to institutional investors. Several models, from GARCH type to jump-diffusion type, have been developed to dynamically capture the price movement of this highly volatile asset. While fitting the Gaussian and the Generalized Hyperbolic and the Normal Inverse Gaussian (NIG) distributions to log-returns of Bitcoin, NIG distribution appears to provide the best fit. The time-varying Hurst parameter for Bitcoin price reveals periods of randomness and mean-reverting type of behaviour, motivating the study in this paper through fractional Ornstein–Uhlenbeck driven by a Normal Inverse Gaussian Lévy process. Features such as long-range memory are jump diffusion processes that are well captured with this model. The results present a 95% prediction for the price of Bitcoin for some specific dates. This study contributes to the literature of Bitcoin price forecasts that are useful for Bitcoin options traders.

An Adaptive Mixture of Normal-Inverse Gaussian Distributions for Structural Reliability Analysis

An Adaptive Mixture of Normal-Inverse Gaussian Distributions for Structural Reliability Analysis

Bühlmann credibility approach to systematic mortality risk modeling for sub-Saharan Africa populations (Kenya)

The classical mortality models such as the Cairns-Blake-Dowd (CBD), Lee-Carter (LC), Linear Regression (LR) models are used to model Systematic Mortality Risk (SMR) for many developed countries populations for actuarial product valuations. This research study aims at incorporating the Bühlmann credibility approach (BCA) to improve the SMR models to fit sub-Saharan African populations like Kenya. Since the Kenyan population does not exhibit the Gaussian properties used in modeling the classical error terms, we proposed using Normal Inverse Gaussian distribution to model these error terms instead of a Gaussian distribution. We model the error terms of the classical models (LC, CBD, and LR) as a Normal Inverse Gaussian (NIG) distribution through the Bühlmann credibility approach. This novel approach demonstrates an improved precision of the predicted SMR as shown by the values of MAPE and RMSE measures compared to those under classical mortality risk models. Ultimately, we have done actuarial valuations of annuities and assurances using our determined SMR, thus concluding that this BCA approach improves the accuracy of actuarial products sold in the Kenyan market.