Α-stable Model Research Articles

A Bootstrap Test for Circular Data

ABSTRACT This article presents a bootstrap test for simultaneous hypotheses on the mean direction and the dispersion of a sample of i.i.d. circular observations with unimodal distribution. This test is a nonparametric extension of the multivariate saddlepoint test for wrapped symmetric α-stable models proposed by Gatto (2000). Because the test statistic has an asymptotic chi-squared distribution with second-order accuracy, the typical sampling stage of most bootstrap procedures is not required. A simulation study illustrates the numerical accuracy and the effectiveness of this new test. In this simulation, samples are generated from circular distributions within the wrapped symmetric α-stable class.

An α-stable approach to the study of the P(D) distribution of unresolved point sources in CMB sky maps

We present a new approach to the statistical study and modelling of number counts of faint point sources in astro- nomical images, i.e. counts of sources whose flux falls below the detection limit of a survey. The approach is based on the theory of α-stable distributions. We show that the non-Gaussian distribution of the intensity fluctuations produced by a generic point source population - whose number counts follow a simple power law - belongs to the α-stable family of distributions. Even if source counts do not follow a simple power law, we show that the α-stable model is still useful in many astrophysical scenarios. With the α-stable model it is possible to totally describe the non-Gaussian distribution with a few parameters which are closely related to the parameters describing the source counts, instead of an infinite number of moments. Using statisti- cal tools available in the signal processing literature, we show how to estimate these parameters in an easy and fast way. We demonstrate that the model proves valid when applied to realistic point source number counts at microwave frequencies. In the case of point extragalactic sources observed at CMB frecuencies, our technique is able to successfully fit the P(D) distribution of deflections and to precisely determine the main parameters which describe the number counts. In the case of the Planck mission, the relative errors on these parameters are small either at low and at high frequencies. We provide a way to deal with the presence of Gaussian noise in the data using the empirical characteristic function of the P(D). The formalism and methods here presented can be very useful also for experiments in other frequency ranges, e.g. X-ray or radio Astronomy.

Synthesis of bidimensional α-stable models with long-range dependence

In the context of linear modeling, the main advantage of stable distributions is to allow the definition of non-Gaussian processes whose statistical properties are easy to characterize. In this work, we are interested in the design of a specific class of 2D discrete-space processes with stable distributions. A frequency domain method for the synthesis of these fields will be proposed which is similar to algorithms already used in the Gaussian case. The considered models represent 2D α-stable extensions of the 1D fractional Gaussian noise. They exhibit long-range dependence properties and, consequently, they could provide interesting alternatives to image modeling techniques based on the 2D fractional Brownian motion. However, the conditions for the existence of such processes deserve special attention and they are derived in this paper.

Testing α-stable processes in capturing the queuing behavior of broadband teletraffic

The paper analyzes the applicability of α-stable processes in traffic modelling. This study is suggested by the ability of α-stable processes in capturing not only the self-similarity of actual traffic, but also the heavy tails of its marginal distribution. The relevance of this property is proven by means of discrete event simulations carried out considering two different traffic data sets, related to a LAN-to-LAN interconnection and to an entertainment video service, respectively. The performance of the α-stable model is evaluated in terms of the ability in capturing the queuing behavior of actual traffic, in cases of realistic systems (i.e. with finite buffer) and ideal models (i.e. with infinite buffer). In both cases, the analysis is mainly carried out by means of discrete event simulations; in addition to this, in the infinite buffer scenario, a theoretical lower bound is also given and its tightness is discussed. The queuing simulations emphasize the improvements in the performance prediction introduced by the higher flexibility of the α-stable model with respect to the widely used fractional Brownian motion.

Financial Data Analysis with Two Symmetric Distributions

AbstractThe normal inverted gamma mixture or generalized Student t and the symmetric double Weibull, as well as their logarithmic counterparts, are proposed for modeling some loss distributions in non-life insurance and daily index return distributions in financial markets. For three specific data sets, the overall goodness-offit from these models, as measured simultaneously by the negative log-likelihood, chi-square and minimum distance statistics, is found to be superior to that of various “good” competitive models including the log-normal, the Burr, and the symmetric α-stable distribution. Furthermore, the study justifies on a statistical basis different important models of financial returns like the model of Black-Scholes (1973), the log-Laplace model of Hürlimann (1995), the normal mixture by Praetz (1972), the symmetric α-stable model by Mandelbrot (1963) and Fama (1965), and the recent double Weibull as limiting geometric-multiplication stable scheme in Mittnik and Rachev (1993). As an application, the prediction of one-year index returns from daily index returns is discussed.

Recent results in applications and processing of α-stable-distributed time series

Symmetric, α-stable random variables and processes have, recently, been receiving increasing attention from the signal processing and communication communities as statistical models for signals and noises that contain impulsive components. This paper is intended as a comprehensive review of the fundamental concepts and results of signal processing with α-stable processes with emphasis placed on acquainting engineers with this emerging discipline in signal processing and revealing its potential applications. In particular, we start with deriving α-stable models for impulsive noise. This derivation serves as an illustration of the Generalized Central Limit Theorem, which states that the first-order distributions in all observed time series follow, to a higher or lesser degree, a stable law. We proceed to present new, fast algorithms for estimation of the parameters of α-stable interference and address two signal detection problems. These problems also build intuition on the differences between Gaussian and non-Gaussian, α-stable signal processing, as well as indicate the performance gains that are to be expected if the signal processing algorithms are designed on the basis of a non-Gaussian, α-stable assumption rather than on a Gaussianity assumption. In the paper, we follow a presentation style that emphasizes only the highlights of the field and omit the fine mathematical details. However, we have included a large number of references to the literature for the interested reader to study further.