Reciprocal Inverse Gaussian Research Articles

A family of asymmetric kernels based on log-symmetric distributions

This paper proposes a class of asymmetric kernels based on log-symmetric (LS) distributions for probability density function estimation in the context of strictly positive skewed data. Some asymptotic properties (bias, variance and mean integrated squared error) of the LS kernel density estimators are established. The choice of bandwidth is investigated by adapting the rule-of-thumb and cross-validation methods. A simulation study investigates the performance of the proposed LS kernel density estimators and compare their performances with the Kakizawa’s LS kernel estimators [Nonparametric density estimation for nonnegative data, using symmetrical-based inverse and reciprocal inverse Gaussian kernels through dual transformation. Journal of Statistical Planning and Inference. 2018; 193:117–135]. Finally, an application on real data is analyzed.

A Study of Seven Asymmetric Kernels for the Estimation of Cumulative Distribution Functions

In this paper, we complement a study recently conducted in a paper of H.A. Mombeni, B. Masouri and M.R. Akhoond by introducing five new asymmetric kernel c.d.f. estimators on the half-line [0,∞), namely the Gamma, inverse Gamma, LogNormal, inverse Gaussian and reciprocal inverse Gaussian kernel c.d.f. estimators. For these five new estimators, we prove the asymptotic normality and we find asymptotic expressions for the following quantities: bias, variance, mean squared error and mean integrated squared error. A numerical study then compares the performance of the five new c.d.f. estimators against traditional methods and the Birnbaum–Saunders and Weibull kernel c.d.f. estimators from Mombeni, Masouri and Akhoond. By using the same experimental design, we show that the LogNormal and Birnbaum–Saunders kernel c.d.f. estimators perform the best overall, while the other asymmetric kernel estimators are sometimes better but always at least competitive against the boundary kernel method from C. Tenreiro.

Stein's method in two limit theorems involving the generalized inverse Gaussian distribution

The generalized hyperbolic (GH) distribution converges in law to the generalized inverse Gaussian (GIG) distribution under certain conditions on the parameters. When the edges of an infinite rooted tree are equipped with independent resistances that are inverse Gaussian or reciprocal inverse Gaussian distributions, the total resistance converges almost surely to some random variable which follows the reciprocal inverse Gaussian (RIG) distribution. In this paper we provide explicit upper bounds for the distributional distance between GH (resp. infinite tree) distribution and their limiting GIG (resp. RIG) distribution applying Stein's method.

Multivariate reciprocal inverse Gaussian distributions from the Sabot–Tarrès–Zeng integral

In Sabot and Tarrès (2015), the authors have explicitly computed the integral STZn=∫exp(−〈x,y〉)(detMx)−1∕2dxwhere Mx is a symmetric matrix of order n with fixed non-positive off-diagonal coefficients and with diagonal (2x1,…,2xn). The domain of integration is the part of Rn for which Mx is positive definite. We calculate more generally for b1≥0,…bn≥0 the integral ∫exp−〈x,y〉−12b⊤Mx−1b(detMx)−1∕2dx,we show that it leads to a natural family of distributions in Rn, called the MRIGn probability laws. This family is stable by marginalization and by conditioning, and it has number of properties which are multivariate versions of familiar properties of univariate reciprocal inverse Gaussian distribution. In general, if the power of detMx under the integral in STZn is distinct from −1∕2 it is not known how to compute the integral. However, introducing the graph G having V={1,…,n} for set of vertices and the set E of {i,j}′ s of non-zero entries of Mx as set of edges, we show also that in the particular case where G is a tree, the integral ∫exp(−〈x,y〉)(detMx)q−1dxwhere q>0, is computable in terms of the MacDonald function Kq.

Local linear regression with reciprocal inverse Gaussian kernel

In this paper, we propose a local linear estimator for the regression model $$Y=m(X)+\varepsilon $$ based on the reciprocal inverse Gaussian kernel when the design variable is supported on $$(0,\infty )$$ . The conditional mean-squared error of the proposed estimator is derived, and its asymptotic properties are thoroughly investigated, including the asymptotic normality and the uniform almost sure convergence. The finite sample performance of the proposed estimator is evaluated via simulation studies and a real data application. A comparison study with other existing estimation methods is also made, and the pros and cons of the proposed estimator are discussed.

Heavy-tailed distribution, GARCH models and the silver returns

After serving as a medium of exchange for the human society, silver is still widely used in our daily life. From the jewellery, electronic and electrical industries as well as medicine, optics, the power industry, automotive industry and many other industries, silver is still playing a very active role. In addition to the industrial usage, silver also serves as an investment tool for many financial institutions. Thus, it is crucial to develop effective quantitative risk management tool for those financial institutions. In this paper, we investigate the conditional heavy tails of daily silver spot returns under the GARCH framework. Our results indicate that that it is important to introduce heavy-tailed distributions to the GARCH framework and the normal reciprocal inverse Gaussian (NRIG) distribution, a newly-developed distribution, has the best empirical performance in capture the daily silver spot returns dynamics.

A Flexible Generalized Hyperbolic Option Pricing Model and Its Special Cases*

We study a generalized hyperbolic (GH) time-changed Levy process for option pricing and examine six three-parameter special cases: the variance gamma (VG) model of Madan, Carr, and Chang (1998), t, hyperbolic (H), normal inverse Gaussian (NIG), reciprocal hyperbolic (RH), and normal reciprocal inverse Gaussian (NRIG) option pricing models. We study the GH model’s moment properties of the associated risk-neutral distribution of logarithmic spot returns, and obtain an explicit pricing formula for European options facilitated by the time-change Levy process construction. Using S&P 500 Index European options during low and high volatility sample periods, we compare the GH model empirically with existing benchmark models such as the finite-moment log-stable model and the Black–Scholes model. The GH model offers the best in- and out-of-sample performance overall, and a proposed t model special case generally outperforms the existing VG special case. We also present a stochastic volatility extension of the GH model.

Properties and applications of the Poisson-reciprocal inverse Gaussian distribution

ABSTRACTThe barely known continuous reciprocal inverse Gaussian distribution is used in this paper to introduce the Poisson-reciprocal inverse Gaussian discrete distribution. Several of its most relevant statistical properties are examined, some of them directly inherited from the reciprocal of the inverse Gaussian distribution. Furthermore, a mixed Poisson regression model that uses the reciprocal inverse Gaussian as mixing distribution is presented. Parameters estimation in this regression model is performed via an EM type algorithm. In light of the numerical results displayed in the paper, the distributions introduced in this work are competitive with the classical negative binomial and Poisson-inverse Gaussian distributions.

Nonparametric density estimation for nonnegative data, using symmetrical-based inverse and reciprocal inverse Gaussian kernels through dual transformation

The classical Birnbaum–Saunders (BS) distribution has recently been generalized in various ways to introduce flexible parametric models for nonnegative data, focusing on the parametric fitting. In this paper, a new symmetrical-based inverse/reciprocal inverse Gaussian density, through dual transformation, is applied to the context of nonparametric density estimation for nonnegative data. The beauty and importance of new density estimator lies in its general formulation via the density generator, including a log-symmetrical kernel density estimator. We provide sufficient conditions under which the proposed estimator has desirable asymptotic properties, and discuss the asymptotic comparison between the proposed estimator and the previous (normal-based) estimator.

Conditional Heavy Tails, Volatility Clustering and Asset Prices of the Precious Metal

The world palladium demand has increased steadily and dramatically in the years of 21th century. However, its spot price still has not reached the peak level observed in January 17, 2001. In 2008, a single-day increase of the palladium spot price has exceeded 37%, which witnesses significant risk for investments in the world palladium market. In this paper, we apply the GARCH model with heavy-tailed distributions into the palladium spot returns series for risk management purpose. We compare empirical performance of the Student’s t distribution and the normal reciprocal inverse Gaussian (NRIG) distribution. Our results show the newlydeveloped distribution, the NRIG, cannot outperform the older fashion one, the Student’s t distribution. Nevertheless, our results do demonstrate that it is important to incorporate conditional heavy tails for precious metals’ spot return modelling.

Heavy-Tailed Distributions, Volatility Clustering and Asset Prices of the Precious Metal

In the years of 21st century, the world palladium demand has increased steadily and dramatically. However, its spot price still has not reached the peak level observed in January 17, 2001. In 2008, a single-day increase of the palladium spot price has exceeded 37%, which witnesses significant risk for investments in the world palladium market. In this paper, we apply the GARCH model with heavy-tailed distributions into the palladium spot returns series for risk management purpose. We compare empirical performance of the Student’s t distribution and the normal reciprocal inverse Gaussian (NRIG) distribution. Our results show the newly-developed distribution, the NRIG, cannot outperform the older fashion one, the Student’s t distribution. Nevertheless, our results do demonstrate that it is important to incorporate conditional heavy tails for precious metals’ spot return modelling.

GARCH Model With Fat-Tailed Distributions and Bitcoin Exchange Rate Returns

In the era of diminishing power from US dollar and increasing competition among world currencies, Bitcoin, as a completely new concept as a medium of exchange, has received increasing attentions over the world. Nowadays, Bitcoin also becomes an investment vehicle, which carries attractive opportunities but also significant risks for the investment community. In this paper, we have compared the empirical performance of a newly-developed heavy-tailed distribution, the normal reciprocal inverse Gaussian (NRIG), with the most popular heavy-tailed distribution, the Student’s t distribution, under the GARCH framework in fitting the daily Bitcoin exchange rate returns. Our results indicate the heavy-tailed distribution has better performance in capture the daily Bitcoin exchange rate returns dynamics than the standard normal distribution. Our results also show the older fashioned Student’s t distribution still performs better than the new heavy-tailed distribution.

Volatility Clustering, Fat Tails and Bitcoin Exchange Rate Returns

In the era of diminishing power from US dollar and increasing competition among world currencies, Bitcoin, as a completely new concept as a medium of exchange, has received increasing attentions over the world. Nowadays, Bitcoin also becomes an investment vehicle, which carries attractive opportunities but also significant risks for the investment community. In this paper, we have compared the empirical performance of a newly-developed heavy-tailed distribution, the normal reciprocal inverse Gaussian (NRIG), with the most popular heavy-tailed distribution, the Student’s t distribution, under the GARCH framework in fitting the daily Bitcoin exchange rate returns. Our results indicate the heavy-tailed distribution has better performance in capture the daily Bitcoin exchange rate returns dynamics than the standard normal distribution. Our results also show the older fashioned Student’s t distribution still performs better than the new heavy-tailed distribution.

GARCH Models with a New Class of Heavy-Tailed Distribution for the Silver Returns

After serving as a medium of exchange for the human society, silver is still widely used in our daily life. From the jewelry, electronic and electrical industries as well as medicine, optics, the power industry, automotive industry and many other industries, silver is still playing a very active role. In addition to the industrial usage, silver also serves as an investment tool for many financial institutions. Thus, it is crucial to develop effective quantitative risk management tool for those financial institutions. In this paper, we investigate the conditional heavy tails of daily silver spot returns under the GARCH framework. Our results indicate that that it is important to introduce heavy-tailed distributions to the GARCH framework and the normal reciprocal inverse Gaussian (NRIG) distribution, a newly-developed distribution, has the best empirical performance in capture the daily silver spot returns dynamics.

A comparison study between three different Kernel estimators for the Hazard rate function

Kernel estimation is one of the most important data analytical tool, if we consider the non parametric approach in the estimation of the probability density function. In parallel, its used to estimate the hazard rate function, which is one of the most important ways for representing the life time distribution in the survival analysis. As the support of the hazard rate function is in the non negative part of the real line , its will be under the boundary effect near zero when the estimation is done using symmetric kernels such as the Gaussian kernel. Two kernel estimators for the hazard rate function were proposed using asymmetric kernels are the Reciprocal Inverse Gaussian and Inverse Gaussian kernel estimators to avoid the high bias near zero. In this paper, we propose a theoretical comparison between those estimators by looking at their asymptotic bias, variance and the mean squared error. Also, a comparison of the practical performance of the two estimators based on simulated and real data will be present.

GARCH Model With Fat-Tailed Distributions and Bitcoin Exchange Rate Returns

In the era of diminishing power from US dollar and increasing competition among world currencies, Bitcoin, as a completely new concept as a medium of exchange, has received increasing attentions over the world. Nowadays, Bitcoin also becomes an investment vehicle, which carries attractive opportunities but also significant risks for the investment community. In this paper, we have compared the empirical performance of a newly-developed heavy-tailed distribution, the normal reciprocal inverse Gaussian (NRIG), with the most popular heavy-tailed distribution, the Student’s t distribution, under the GARCH framework in fitting the daily Bitcoin exchange rate returns. Our results indicate the heavy-tailed distribution has better performance in capture the daily Bitcoin exchange rate returns dynamics than the standard normal distribution. Our results also show the older fashioned Student’s t distribution still performs better than the new heavy-tailed distribution.

Characterizations for the Power Inverse Gaussian and Generalized Inverse Gaussian Distributions Based on Conditional Moments

In this study, we characterize the power inverse Gaussian and generalized inverse Gaussian distributions based on recurrence relations for conditional moments in terms of hazard rate functions. Relations for conditional moments of the largest order statistic (upper record value) given the second largest order statistic (upper record value) are derived from these recurrence relations. Characterizations of the inverse Gaussian, reciprocal inverse Gaussian, gamma and reciprocal gamma distributions are obtained as special cases of both the power inverse Gaussian and the generalized inverse Gaussian distributions.

Re-formulation of inverse Gaussian, reciprocal inverse Gaussian, and Birnbaum–Saunders kernel estimators

We reveal the boundary bias problem of Birnbaum–Saunders, inverse Gaussian, and reciprocal inverse Gaussian kernel estimators (Jin and Kawczak, 2003, Scaillet, 2004) and re-formulate these estimators to solve the problem. We investigate asymptotic properties of a new class of asymmetric kernel estimators.

Generalization of the k-moment method using the maximum entropy principle. Application to the NBKM and full spectrum SLMB gas radiation models

The k-moment method is generalized by applying the maximum entropy principle to get several estimates of the k-distribution function on any kind of spectral interval as a function of the first two moments of the absorption coefficient. Corresponding formulations of the blackbody weighted band averaged transmission function of a gaseous uniform path are obtained. Different constraints involving the first and second order positive, first order negative and logarithmic moments are introduced together with a physical meaning whenever it is possible. Different sets of these constraints are considered to get maximum entropy estimates of the distributions functions: the Dirac, exponential, Gamma, inverse Gaussian and reciprocal inverse Gaussian k-distribution functions. Analytical formulas are provided for each of these distributions and for their associated transmission function, as a function of the mean and variance of the absorption coefficient. The methodology can be applied considering any spectral interval: narrow, wide, the full spectrum, continuous or not. Thus the resulting associated transmission and cumulative k-distribution functions can be utilized in the frame of a large variety of gas radiation models. Hence the k-moment method using the maximum entropy principle is assessed in the frame of the NBKM and full spectrum SLMB gas radiation models. A series of test cases implying comparisons with reference Line-by-Line results exhibits which maximum entropy k-distributions are likely to give the best estimations of narrow band or total emitted intensities, curves-of-growth of the total emission function and full spectrum cumulative k-distribution functions. In particular, the inverse Gaussian and Gamma k-distributions seem most of the time to perform very well.

A Non-Gaussian Kalman Filter With Application to the Estimation of Vehicular Speed

Single inductive loop detectors (ILDs), which provide online measurements of traffic volume and occupancy, are widely used devices in road systems. Because of the nature of traffic flow, fast estimation and forecasting of vehicular speed using the data collected by an ILD are crucial to online road traffic management. Here we formulate statistical inference for vehicular speed as a dynamic generalized linear model with a reciprocal inverse Gaussian observational distribution. The formulation motivates us to extend the Gaussian Kalman filter to this non-Gaussian scenario. This results in a set of simple recursive formulas in which the current estimate of the parameter of interest is updated as a weighted harmonic average of the previous estimate and the current observation. By applying the developed non-Gaussian Kalman filter to analyze traffic data collected by an ILD, we provide a competitive alternative for estimating vehicular speed at a minimum computational cost.