Family Of Stable Distributions Research Articles

Circular-Statistics-Based Estimators and Tests for the Index Parameter α of Distributions for High-Volatility Financial Markets

The distributions for highly volatile financial time-series data are playing an increasingly important role in current financial scenarios and signal analyses. An important characteristic of such a probability distribution is its tail behaviour, determined through its tail thickness. This can be achieved by estimating the index parameter of the corresponding distribution. The normal and Cauchy distributions, and, sometimes, a mixture of the normal and Cauchy distributions, are suitable for modelling such financial data. The family of stable distributions can provide better modelling for such financial data sets. Financial data in high-volatility markets may be better modelled, in many cases, by the Linnik distribution in comparison to the stable distribution. This highly flexible family of distributions is better capable of modelling the inflection points and tail behaviour compared to the other existing models. The estimation of the tail thickness of heavy-tailed financial data is important in the context of modelling. However, the new probability distributions do not admit any closed analytical form of representation. Thus, novel methods need to be developed, as only a few can be found in the literature. Here, we recall a recent novel method, developed by the authors, based on a trigonometric moment estimator using circular distributions. The linear data may be transformed to yield circular data. This transformation is solely for yielding a suitable estimator. Our aim in this paper is to provide a review of the few existing methods, discuss some of their drawbacks, and also provide a universal (∀α∈(0,2]), efficient, and easily implementable estimator of α based on the transformation mentioned above. Novel, circular-statistics-based tests for the index parameter α of the stable and Linnik distributions are introduced and also exemplified with real-life financial data. Two real-life data sets are analysed to exemplify the methods recommended and enhanced by the authors.

Exact and asymptotic identification-robust inference for dynamic structural equations with an application to New Keynesian Phillips Curves

Many models in econometrics involve endogeneity and lagged dependent variables. We start by observing that usual identification-robust (IR) tests are unreliable when model variables are nonstationary or nearly nonstationary. We propose IR methods which are also robust to nonstationarity: one Anderson-Rubin type procedure and two split-sample procedures. Our procedures are also robust to missing instruments. For distributional theory, three different sets of assumptions are considered. First, on assuming Gaussian structural errors, we show that two of the proposed statistics follow the standard F distribution. Second, for more general cases, we assume that the distribution of errors is completely specified up to an unknown scale factor, allowing the Monte Carlo test method to be applied. This assumption enables one to deal with non-Gaussian error distributions. For example, even when errors follow heavy-tailed distribution, such as the Cauchy distribution or more generally the family of stable distributions—which may not have moments and thus make inference difficult—our procedures provide simple and exact solutions. Third, we establish the asymptotic validity of our procedures under quite general distributional assumptions. We present simulation results showing that our procedures control their level correctly and have good power properties. The methods are applied to an empirical example, the New Keynesian Phillips curve, in which both weak identification and nonstationarity present challenges. The results of this empirical study suggest forward-looking behavior of U.S. inflation.

An Universal, Simple, Circular Statistics-Based Estimator of α for Symmetric Stable Family

The aim of this article is to obtain a simple and efficient estimator of the index parameter of symmetric stable distribution that holds universally, i.e., over the entire range of the parameter. We appeal to directional statistics on the classical result on wrapping of a distribution in obtaining the wrapped stable family of distributions. The performance of the estimator obtained is better than the existing estimators in the literature in terms of both consistency and efficiency. The estimator is applied to model some real life financial datasets. A mixture of normal and Cauchy distributions is compared with the stable family of distributions when the estimate of the parameter α lies between 1 and 2. A similar approach can be adopted when α (or its estimate) belongs to (0.5,1). In this case, one may compare with a mixture of Laplace and Cauchy distributions. A new measure of goodness of fit is proposed for the above family of distributions.

Visual Enhancement of Touch and Body Ownership: an ERP study

The choice of kernel in an integro-difference equation (IDE) approach to model spatio-temporal data is studied. By using approximations to stochastic partial differential equations, it is shown that higher order cumulants and tail behavior of the kernel affect how an IDE process evolves over time. The asymmetric Laplace and the family of stable distributions are presented as alternatives to the Gaussian kernel. The asymmetric Laplace has an extra parameter controlling skewness, whereas the class of stable distributions includes parameters controlling both tail behavior and skewness. Simulations show that failing to account for kernel shape may lead to poor predictions from the model. For an illustration with real data, the IDE model with flexible kernels is applied to ozone pressure measurements collected biweekly by radiosonde at varying altitudes. The results obtained with the different kernel families are compared and it is confirmed that better model prediction may be achieved by electing to use a more flexible kernel.

Well‐posedness of Bayesian inverse problems in quasi‐Banach spaces with stable priors

AbstractThe Bayesian perspective on inverse problems has attracted much mathematical attention in recent years. Particular attention has been paid to Bayesian inverse problems (BIPs) in which the parameter to be inferred lies in an infinite‐dimensional space, a typical example being a scalar or tensor field coupled to some observed data via an ODE or PDE. This article gives an introduction to the framework of well‐posed BIPs in infinite‐dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451–559, 2010) and others. This framework has the advantage of ensuring uniformly well‐posed inference problems independently of the finite‐dimensional discretisation used for numerical solution. Recently, this framework has been extended to the case of a heavy‐tailed prior measure in the family of stable distributions, such as an infinite‐dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen–Loève expansion for square‐integrable random variables can be used to sample such measures on quasi‐Banach spaces. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger and total variation metrics upon perturbations of the misfit function and observed data. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors

This article extends the framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451--559, 2010) and others, to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen--Lo\`eve expansion for square-integrable random variables can be used to sample such measures on quasi-Banach spaces. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger metric upon perturbations of the misfit function and observed data.

Flexible integro-difference equation modeling for spatio-temporal data

The choice of kernel in an integro-difference equation (IDE) approach to model spatio-temporal data is studied. By using approximations to stochastic partial differential equations, it is shown that higher order cumulants and tail behavior of the kernel affect how an IDE process evolves over time. The asymmetric Laplace and the family of stable distributions are presented as alternatives to the Gaussian kernel. The asymmetric Laplace has an extra parameter controlling skewness, whereas the class of stable distributions includes parameters controlling both tail behavior and skewness. Simulations show that failing to account for kernel shape may lead to poor predictions from the model. For an illustration with real data, the IDE model with flexible kernels is applied to ozone pressure measurements collected biweekly by radiosonde at varying altitudes. The results obtained with the different kernel families are compared and it is confirmed that better model prediction may be achieved by electing to use a more flexible kernel.

A Finite Sample Test for the Composite Hypothesis of Stable Law

The Family of stable distributions serves as an error model for a wide-and even widening -variety of applications. The importance of these distributions is due to the properties that sums of independent and identically distributed stable random variables are again stable and that these are the only possible limiting distributions of sums of independent and identically distributed random variables. There are many notable potential applications in economics, physics, insurance, risk management, and especially finance, to name just a few. Several types of parameter estimation procedures have been implemented and many empirical studies concerning financial time series have been conducted under the proposition that the stable distributions provide viable models of price behavior. The viability of stable models has remained in doubt because of the lack of quantitative test for fit - despite appealing stable-like characteristics exhibited by the data, e.g. estimates of shape parameter less than 2, pronounced skewness, etc. In this paper we propose and provide a (family of) finite sample test(s) for the composite hypothesis of stable distribution with all parameters unspecified. The test procedure is based on the property of stability under addition (sums of m random variables) as captured by the m th power of the empirical characteristic function, and a variant of the estimation procedure of Paulson and Delehanty. The test has attractive power properties against an array of alternatives including normal mixtures, the lognormal distributions, Student's t, and other competing model distributions. We provide percentage points for sample size n ≥ 50. Detailed studies of the behavior of the test under various conditions involving the skewness, scale and location parameters are provided. We briefly comment on several applications from financial markets.

One-step R-estimation in linear models with stable errors

Classical estimation techniques for linear models either are inconsistent, or perform rather poorly, under α-stable error densities; most of them are not even rate-optimal. In this paper, we propose an original one-step R-estimation method and investigate its asymptotic performances under stable densities. Contrary to traditional least squares, the proposed R-estimators remain root-n consistent (the optimal rate) under the whole family of stable distributions, irrespective of their asymmetry and tail index. While parametric stable-likelihood estimation, due to the absence of a closed form for stable densities, is quite cumbersome, our method allows us to construct estimators reaching the parametric efficiency bounds associated with any prescribed values (α0,b0) of the tail index α and skewness parameter b, while preserving root-n consistency under any (α,b) as well as under usual light-tailed densities. The method furthermore avoids all forms of multidimensional argmin computation. Simulations confirm its excellent finite-sample performances.

INFERENCES OF -STABLE DISTRIBUTION OF THE UNDERLYING NOISE COMPONENTS IN GEODETIC DATA

We investigate the use of distribution functions to characterise the geophysical signals and noise components embedded in the geodetic Very Long Baseline Interferometry (VLBI) data sets across some of the International VLBI Service (IVS) stations. The rationale of using α-stable distributions as a tool to model the noise components in geodetic observables is due to the existence of impulsive signals/noise bursts (which often take the form of excursions with intermittent occurrences) in the data sets suggesting deviations from Gaussian distribution. A deviation from Gaussian distribution type would therefore suggest that statistical techniques such as least squares analysis, often used for analyzing the geodetic data (which are often based on Gaussian assumptions) could not be robust. In this paper, the properties of a long-range α-stable distribution with long tails and infinite moments in geodetic data are investigated by way of statistically testing their distribution using a family of stable distributions. The choice of stable distributions is based on the ease with which the statistical properties of the non-Gaussian processes are defined. Results indicate that the independent geophysical noise components reconstructed from geodetic VLBI baseline data exhibit distributions that have asymptotic power-law decay (albeit variable power indices) whose underlying process can be modelled as a long-range dependent process with an α-stable distribution (i.e., the stable varieties have small characteristic exponents).

One-Step R-Estimation in Linear Models with Stable Errors

Classical estimation techniques for linear models either are inconsistent, or perform somewhat poorly under stable error densities; most of them are not even rate-optimal. In this paper, we develop an original R-estimation method and investigate its asymptotic performances under stable densities. Contrary to traditional least squares, the proposed R-estimators, remain root-n consistent (the optimal rate) under the whole family of stable distributions, irrespective of their asymmetry and tail index. While stable-likelihood estimation, due to the absence of a closed form for stable densities, is generally considered unfeasible, our method allows us to construct estimators reaching the parametric efficiency bounds associated with any prescribed values of tail index alpha and the skewness parameter beta, while preserving root-n consistency under any alpha and beta. The method furthermore avoids all forms of multidimensional argmin computation. Simulations confirm its excellent finite-sample performances.

Intrinsic Bubbles and Fat Tails in Stock Prices: A Note

We study the constant discount rate present value model for stock pricing in a stochastic setting where the exogenous dividend stream is modeled as a random walk with innovations drawn from the family of stable distributions. We derive an exact analytical solution for the fundamental stock price. We evaluate the ability of the model fundamentals and the dividends-driven intrinsic bubbles to explain the observed variation in annual U.S. stock prices. We compare results obtained in this setting with those from the traditional model where all stochastic processes are driven by Gaussian shocks.

Intrinsic Bubbles and Fat Tails in Stock Prices

We study the constant discount rate present value model for stock pricing in a stochastic setting where the exogenous dividend stream is modeled as a random walk with innovations drawn from the family of stable distributions. We derive an exact analytical solution for the fundamental stock price. We evaluate the ability of the model fundamentals and the dividends-driven intrinsic bubbles to explain the observed variation in annual US stock prices. We compare results obtained in this setting with those from the traditional model where all stochastic processes are driven by Gaussian shocks.

Efficient posterior integration in stable paretian models

The paper proposes a Markov Chain Monte Carlo method for Bayesian analysis of general regression models with disturbances from the family of stable distributions with arbitrary characteristic exponent and skewness parameter. The method does not require data augmentation and is based on combining fast Fourier transforms of the characteristic function to get the likelihood function and a Metropolis random walk chain to perform posterior analysis. Both a validation nonlinear regression and a nonlinear model for the Standard and Poor’s composite price index illustrate the method.

Maturity Guarantees Revisited: Allowing for Extreme Stochastic Fluctuations using Stable Distributions

ABSTRACTThe paper examines the suitability of the stable family of distributions with the Maturity Guarantees Working Party's stochastic investment model (Ford et al, 1980). It then examines the effect of replacing the Gaussian assumption made by the working party with a more general stable distribution. It also explains how the appropriate stable distribution can be fitted.

INFRARED SCENE MODELING AND INTERPOLATION USING FRACTIONAL LÉVY STABLE MOTION

Many data arising from natural phenomena exhibit "1/f" behavior, indicating a long-range dependence structure in the increments. The data is said to be self-similar or fractal, which has been traditionally modeled by fractional Brownian motion (fBm). This stochastic fractal model assumes a Gaussian distribution of the increments which is at times too rigid, particularly for data emanating from a long-tailed distribution. Therefore, the fractional Lévy stable motion stochastic process is proposed as a means of modeling a wider range of data. For these processes the increments are assumed to be from the family of stable distributions which have been shown to be good models of long-tailed behavior. The model is applied to data from infrared scenes and used to perform fractal interpolation, preserving not only the self-similarity, but also the probability distribution of the increments over the newly generated scales. This offers a flexible new model for a broader class of data than the fBm model.

A note on the disappearance of day-of-the-week seasonals in the daily price changes of Treasury bond futures

Day-of-the-week seasonals in the Treasury bond futures market are investigated; parametric tests based on the assumption of normality are contrasted with the more general stable family of distributions and with non-parametric tests. The sample data corroborate the non-normality of the returns-generating process in the Treasury bond futures market found previously. Non-stationarity of the returns-generating process in the sample period, 1977-91, is investigated through parameter estimation in sub-periods of the sample. The disappearance of day-of-the-week seasonals in the Treasury bond futures market documented in prior studies is corroborated by the estimation methodology based on stable distributions.

Stable Distributions and the Mixtures of Distributions Hypotheses for Common Stock Returns

Abstract The form of the distribution underlying common stock returns has many implications for financial modeling. Among the frequently proposed models for stock return distributions is the family of stable distributions. Numerous studies of stock return distributions have also proposed various types of mixtures of normal distributions. This article examines the characteristics of several different combinations of mixtures of normal and stable distributions, and compares them with actual stock price distributions. The principal tests applied are based on the stability-under-addition property of stable distributions.

Stable Distributions and the Mixtures of Distributions Hypotheses for Common Stock Returns

Abstract The form of the distribution underlying common stock returns has many implications for financial modeling. Among the frequently proposed models for stock return distributions is the family of stable distributions. Numerous studies of stock return distributions have also proposed various types of mixtures of normal distributions. This article examines the characteristics of several different combinations of mixtures of normal and stable distributions, and compares them with actual stock price distributions. The principal tests applied are based on the stability-under-addition property of stable distributions.

Asymmetric Stable Distributions of Stock Price Changes

Abstract This note presents evidence that the asymmetric members of the stable family of distributions are appropriate for describing changes in stock prices.