Symmetric Stable Distribution Research Articles

A simple estimator for the characteristic exponent of the stable Paretian distribution

A new estimator for the index α of symmetric stable Paretian distributions is proposed. In addition to being numerically simple, fast, and reliable, simulations suggest that even in small samples, the estimator is unbiased for α ∈ [1, 2], and almost exactly normally distributed. Standard errors are also given, the magnitudes of which are shown to depend only on sample size and not on the true value of α.

Comparison of estimators in stable models

In this paper, several estimators of the index and scale parameter of a symmetric univariate stable distribution are analyzed and compared. It turns out that minimum distance estimators with a suitable choice of the weight function have good theoretical and empirical performance. Also some modifications of the moment method are reliable in large parts of the parameter space. The classical Hill estimator, however, works well only for a small stability index and for an extremely large number of observations. One can construct tail-estimators with a considerably improved behaviour compared to the Hill estimator if the scale is known.

Superdiffusion and stable laws

The superdiffusion equation with a fractional Laplacian Δα/2 in N-dimensional space describes the asymptotic (t→∞) behavior of a generalized Poisson process with the range (discontinuity) distribution density ∼|x|−α−1. The solutions of this equation belong to a class of spherically symmetric stable distributions. The main properties of these solutions are given together with their representations in the form of integrals and series and the results of numerical calculations. It is shown that allowance for the finite velocity of free particle motion for α>1 merely amounts to a reduction in the diffusion coefficient with the form of the distribution remaining stable. For α<1 the situation changes radically: the expansion velocity of the diffusion packet exceeds the velocity of free particle motion and the superdiffusion equation becomes physically meaningless.

Simulation of random vectors from three-dimensional spherically symmetric stable distributions

A simple approximate algorithm of sampling random vectors from three-dimensional symmetric stable distributions based on analytical approximation of inverse distribution functions is proposed. Explicit expressions for simulating the stable distributions with indices α=0.2(0.1) 1.8 are presented. The errors of the algorithm do not exceed 1–2% in the probability density function.

On Robust Analysis of a Normal Location Parameter

Summary Pericchi and Smith considered a normal location parameter problem with double-exponential and Student t prior distributions. These two prior distributions both belong to the class of scale mixtures of normal distributions and are useful in providing a robust analysis of the normal location parameter problem. In this paper we extend the analysis to other scale mixtures of normal distributions, such as the exponential power and the symmetric stable distributions.

Reply [to “Comment on ‘Evidence for non‐Gaussian scaling behavior in heterogeneous sedimentary formations’ by Scott Painter”

Water Resources ResearchVolume 33, Issue 4 p. 909-910 CommetariesFree Access Reply [to “Comment on ‘Evidence for non-Gaussian scaling behavior in heterogeneous sedimentary formations’ by Scott Painter”] Scott Painter, Scott PainterSearch for more papers by this author Scott Painter, Scott PainterSearch for more papers by this author First published: 01 April 1997 https://doi.org/10.1029/96WR03789Citations: 6AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL No abstract is available for this article. References Fama, E., R. Roll, Parameter estimates for symmetric stable distributions, J. Am. Stat. Assoc., 66, 331, 1972. Liu, H. H., andF. J. Molz, Comment on “Evidence for non-Gaussian scaling behavior in heterogeneous sedimentary formations” by Scott Painter, 4. Painter, S., Evidence for non-Gaussian scaling behavior in heterogeneous sedimentary formations, Water Resources Research, 325, 1183– 1195, 1996a. Painter, S., Stochastic interpolation of aquifer properties using fractional Lévy motion, Water Resources Research, 325, 1323– 1332, 1996b. Citing Literature Volume33, Issue4April 1997Pages 909-910 ReferencesRelatedInformation

On the pointwise central limit theorem and mixtures of stable distributions

Let X 1, X 2, … be i.i.d. r.v.'s with EX 1 = 0. EX 2 1 − 1 and put S n = X 1 + … + X n . We investigate the a.s. limiting behavior of L N= 1 log N ∑ k⩽N 1 k I S k a k <χ for general norming sequences ( a k ). The pointwise central limit theorem shows that L N converges a.s. to the normal distribution if a k = √ k; in our paper we prove the surprising result that for suitably chosen ( a k ) the expression L N can converge also to non-Gaussian limits, in particular, any symmetric stable distribution is a possible limit of L N . We shall determine the class of limit distributions of L N and extend the result to the case when X n belong to the domain of attraction of a stable law.

Asymptotic properties of symmetric stable distributions with small index

The family of symmetric stable distributions is considered. A local limit theorem is established which takes account of large deviations as the characteristic exponent α tends to zero. On the basis of this theorem the asymptotics of Fisher information with respect to α is obtained.

Detection of impulsive stochastic transients over background noise

We develop a receiver which detects impulsive stochastic transients superimposed on background noise and show that the new detector outperforms existing square- and vth-law detectors. The receiver is based on the assumption of a symmetric stable distribution for the transients. To derive the receiver, we first study the structure of the probability density function (pdf) of random variables which are formed as the sum of two or more independent, symmetric stable random variables of different characteristic exponents. For this pdf, we present two asymptotic series expansions, valid for small and for large argument, respectively.

Distribution of incremental static stress caused by earthquakes

Abstract. Theoretical calculations, simulations and measurements of rotation of earthquake focal mechanisms suggest that the stress in earthquake focal zones follows the Cauchy distribution which is one of the stable probability distributions (with the value of the exponent α equal to 1). We review the properties of the stable distributions and show that the Cauchy distribution is expected to approximate the stress caused by earthquakes occurring over geologically long intervals of a fault zone development. However, the stress caused by recent earthquakes recorded in instrumental catalogues, should follow symmetric stable distributions with the value of α significantly less than one. This is explained by a fractal distribution of earthquake hypocentres: the dimension of a hypocentre set, δ, is close to zero for short-term earthquake catalogues and asymptotically approaches 2¼ for long-time intervals. We use the Harvard catalogue of seismic moment tensor solutions to investigate the distribution of incremental static stress caused by earthquakes. The stress measured in the focal zone of each event is approximated by stable distributions. In agreement with theoretical considerations, the exponent value of the distribution approaches zero as the time span of an earthquake catalogue (ΔT) decreases. For large stress values α increases. We surmise that it is caused by the δ increase for small inter-earthquake distances due to location errors.

Optimal Linear Filtering and Smoothing for a Discrete Time Stable Linear Model

Properties of conditional expectation and metric projection for multivariate symmetric stable distributions are studied. Linear filtering, smoothing, and prediction problems for a discrete time stable linear model with constant coefficients are solved.

Pitman's measure of closeness for symmetric stable distributions

This paper considers symmetric stable distributions with different exponents γ (0 < γ ⩽ 2), and studies Pitman's measure of closeness of sample averages (both weighted and unweighted) based on different sample sizes. The behavior of measures of concentration of such averages around the point of symmetry is also studied. It is found that while Pitman's closeness criterion is compatible with the measure of concentration for 1 ⩽ γ ⩽ 2, this need not always be so for 0 < γ < 1. The relationship of our results with the ones given in Blyth and Pathak (1985) is also pointed out.

On Distributions Whose Component Ratios are Cauchy

Abstract This article considers a class of multivariate (dependent) variables that includes those that are obtained as scale mixtures of elliptically symmetric distributions. We provide a simple, intuitive proof that the ratio of two such variables has a general Cauchy distribution. This result extends the results of DeSilva concerning properties of a certain class of dependent multivariate symmetric stable distributions.

Adaptive robust estimation of location for stable distributions

An adaptive procedure based on using two selector statistics and ,where Xp is the pth sample quantile, is described and used to choose among some robust estimators of location with respect to a class V of symmetric stable distributions with a characteristic exponent a ranging from 0.5 to 2.0 with steps of 0.1 or 0.05. The estimators considered are special symmetrical linear combinations of order statistics. Asymptotic variances of these estimators are computed using the characteristic function. Optimum asymptotic efficiencies of these estimators for the entire class V as well as for several subclasses of it are given.

STOCHASTIC DOMINANCE AND TRUNCATED SAMPLE DATA

AbstractThe purpose of this paper is to examine whether the chance of making erroneous investment decisions can be reduced by applying stochastic dominance rules to truncated, rather than complete, sample data of rates of return. A simulation approach is used that assumes the rates of return follow the symmetric stable probability distribution. Using a variety of relationships between probability distributions of rates of return, it is demonstrated that sample truncation has the potential of significantly reducing sampling errors in the selection between alternative investments.

Tests for the Chacteristic Exponent and the Scale Parameter of Symmetric Stable Distributions

Let S (α, β, a, c) be a stable distribution, where α∈ (0, 2), −1 ≤ β ≤ 1, a ∈ IR and c > 0 are respectively, the characteristic exponent, the skewness, the location and the scale parameters. In this paper we shall develop tests for the parameters α and c of a symmetric stable random variable S (α, 0,0, c). The test statistics for the hypthesis H0: α = α0 vs. H1: α > α0 and H0:c = c0 vs. H1 : c > c0 are functions of the empirical characteristic function , where X1, …, Xn is a random sample of size n from a stable population S (α, 0,0, c). We shall study the asymptotic properties of the power functions of the tests above and their behaviour in the neighborhood of α0 and c0.

Maximum Likelihood Estimates of Symmetric Stable Distribution Parameters

A method is developed to directly obtain maximum likelihood estimates of symmetric stable distribution parameters. This is a difficult estimation problem since the likelihood function is expressed as an integral. The estimation routine is tested on a Monte Carlo sample and produces reasonable estimates.

Some Properties of Symmetric Stable Distributions Close to the Normal Distribution

Previous article Next article Some Properties of Symmetric Stable Distributions Close to the Normal DistributionA. V. Nagaev and S. M. Shkol’nikA. V. Nagaev and S. M. Shkol’nikhttps://doi.org/10.1137/1133014PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] W. H. DuMouchel, Stable distributions in statistical inference. II. Information from stably distributed samples, J. Amer. Statist. Assoc., 70 (1975), 386–393 51:14360 0321.62017 CrossrefGoogle Scholar[2] William H. DuMouchel, Estimating the stable index $\alpha$ in order to measure tail thickness: a critique, Ann. Statist., 11 (1983), 1019–1031 87j:62053 0547.62022 CrossrefGoogle Scholar[3] V. M. Zolotarev, One-dimensional stable distributions, Translations of Mathematical Monographs, Vol. 65, American Mathematical Society, Providence, RI, 1986x+284 87k:60002 0589.60015 CrossrefGoogle Scholar[4] M. Abramowitz and , I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, Vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964xiv+1046 29:4914 0171.38503 Google Scholar[5] I. A. Ibragimov and , Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971, 443– 48:1287 0219.60027 Google Scholar[6] A. P. Prudnikov, , Yu. A. Brychkov and , O. I. Marichev, Integrals and Series, Nauka, Moscow, 1981, (In Russian.) 0511.00044 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Asymptotics of maximum likelihood estimation for stable law with continuous parameterizationCommunications in Statistics - Theory and Methods, Vol. 50, No. 15 | 23 January 2020 Cross Ref Some Improvements in Numerical Evaluation of Symmetric Stable Density and Its DerivativesCommunications in Statistics - Theory and Methods, Vol. 35, No. 1 | 1 Feb 2006 Cross Ref Robust decorrelating decision-feedback multiuser detection in non-Gaussian channelsSignal Processing, Vol. 81, No. 9 | 1 Sep 2001 Cross Ref Recent results in applications and processing of α-stable-distributed time seriesJournal of the Franklin Institute, Vol. 333, No. 4 | 1 Jul 1996 Cross Ref Asymptotic properties of symmetric stable distributions with small indexJournal of Mathematical Sciences, Vol. 76, No. 2 | 1 Aug 1995 Cross Ref The Suzdal International Seminar of 1991 on Stability Problems for Stochastic ModelsTheory of Probability & Its Applications, Vol. 36, No. 4 | 17 July 2006AbstractPDF (5581 KB)BibliographyChance and Stability Cross Ref Volume 33, Issue 1| 1989Theory of Probability & Its Applications1-205 History Submitted:27 October 1986Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1133014Article page range:pp. 139-144ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Spectral conditions for local nondeterminism

Let X( t) be a real Gaussian process with stationary increments and spectral distribution function F( x). Put φ(t)=F(∞) − F( 1 t ) . Sufficient conditions in terms of F are given for the process to be locally φ-nondeterministic. These are formulated for discrete and absolutely continuous functions F. The results in the discrete case are applied to the analysis of the local time of a random Fourier series with i.i.d. coefficients. The class of distributions of the coefficients includes not only the normal distribution but others such as the symmetric stable distribution.

Strong Approximations for Partial Sums of Random Variables Attracted to a Stable Law

Let Xi, i = 1, 2,…, be i.i.d. symmetric random variables in the domain of attraction of a symmetric stable distribution Gα with 0 < α < 2. Let Yi, i = 1, 2, …, be i.i.d. symmetric stable random variables with the common distribution Gα. It is known that under certain conditions the sequences {Xi} and {Yi} can be reconstructed on a new probability space without changing the distribution of each such that a.s. as n ∞, where α ≦ γ < 2 (see Stout [10]). We will give a second approximation by partial sums of i.i.d. stable (with characteristic exponent α*, α < α* ≦ 2) random variables Ui, i = 1, 2,…, n, and we will obtain strong upperbounds for the differences .