The stability test for symmetric alpha-stable distributions

Symmetric alpha-stable distributions are a popular statistical model for heavy-tailed phenomena encountered in communications, radar, biomedicine, and econometrics. The use of the symmetric alpha stable model is often supported by empirical evidence, where qualitative criteria are used to judge the fit, leading to subjective decisions. Objective decisions can only be made through quantitative statistical tests. Here, a goodness-of-fit hypothesis test for symmetric alpha-stable distributions is developed based on their unique stability property. Critical values for the test are found using both asymptotic theory and from bootstrap estimates. Experiments show that the stability test, using bootstrap estimates of the critical values, is better able to discriminate between symmetric alpha stable distributions and other heavy-tailed distributions than classical tests such as the Kolmogorov-Smirnov test.

[1]  Abdelhak M. Zoubir,et al.  Testing gaussianity with the characteristic function: The i.i.d. case , 1996, Signal Process..

[2]  P. Hall,et al.  The Effect of Simulation Order on Level Accuracy and Power of Monte Carlo Tests , 1989 .

[3]  H. Keselman,et al.  Multiple Comparison Procedures , 2005 .

[4]  J. How,et al.  No evidence of stable distributions in radar clutter , 1997, Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics.

[5]  S. James Press,et al.  Estimation in Univariate and Multivariate Stable Distributions , 1972 .

[6]  Abdelhak M. Zoubir,et al.  Testing for Impulsive Behavior: A Bootstrap Approach , 2001, Digit. Signal Process..

[7]  B. Mandelbrot.  The Variation of Some Other Speculative Prices , 1967 .

[8]  Ing Rj Ser.  Approximation Theorems of Mathematical Statistics , 1980 .

[9]  Boualem Boashash,et al.  The bootstrap and its application in signal processing , 1998, IEEE Signal Process. Mag..

[10]  S. Csőrgő.  Limit Behaviour of the Empirical Characteristic Function , 1981 .

[11]  David Middleton,et al.  Non-Gaussian Noise Models in Signal Processing for Telecommunications: New Methods and Results for Class A and Class B Noise Models , 1999, IEEE Trans. Inf. Theory.

[12]  E. Fama.  The Behavior of Stock-Market Prices , 1965 .

[13]  O. Barndorff-Nielsen,et al.  Lévy processes : theory and applications , 2001 .

[14]  J. Nolan,et al.  Multivariate stable distributions: approximation, estimation, simulation and identification , 1998 .

[15]  E. Fama,et al.  Parameter Estimates for Symmetric Stable Distributions , 1971 .

[16]  Dimitrios Hatzinakos,et al.  Analytic alpha-stable noise modeling in a Poisson field of interferers or scatterers , 1998, IEEE Trans. Signal Process..

[17]  A. S. Paulson,et al.  Modified weighted squared error estimation procedures with special emphasis on the stable laws , 1985 .

[18]  Gutti Jogesh Babu.  Bootstrap Techniques for Signal Processing , 2005, Technometrics.

[19]  Ralph B. D'Agostino,et al.  Goodness-of-Fit-Techniques , 2020 .

[20]  A. Feuerverger,et al.  The Empirical Characteristic Function and Its Applications , 1977 .

[21]  M. Kenward,et al.  An Introduction to the Bootstrap , 2007 .

[22]  Rama Chellappa,et al.  Adaptive target detection in foliage-penetrating SAR images using alpha-stable models , 1999, IEEE Trans. Image Process..

[23]  竹中 茂夫.  G.Samorodnitsky,M.S.Taqqu:Stable non-Gaussian Random Processes--Stochastic Models with Infinite Variance , 1996 .

[24]  Ioannis A. Koutrouvelis,et al.  Regression-Type Estimation of the Parameters of Stable Laws , 1980 .

[25]  T. W. Epps.  Characteristic Functions and Their Empirical Counterparts: Geometrical Interpretations and Applications to Statistical Inference , 1993 .

[26]  C. Mallows,et al.  A Method for Simulating Stable Random Variables , 1976 .

[27]  V. Zolotarev.  One-dimensional stable distributions , 1986 .

[28]  J. Nolan,et al.  Fitting Data and Assessing Goodness � of � t with Stable Distributions , 1999 .

[29]  John Teichmoeller.  A Note on the Distribution of Stock Price Changes , 1971 .

[30]  Ramon Brcic.  Some aspects of signal processing in heavy tailed noise , 2002 .

[31]  Athina P. Petropulu,et al.  Power-Law Shot Noise and Its Relationship To Long-Memory �-Stable Processes , 2000 .

[32]  Dimitris N. Politis,et al.  Computer-intensive methods in statistical analysis , 1998, IEEE Signal Process. Mag..

[33]  R. Officer.  The Distribution of Stock Returns , 1972 .

[34]  C. L. Nikias,et al.  Signal processing with alpha-stable distributions and applications , 1995 .

[35]  C. E. Heathcote.  A TEST OF GOODNESS OF FIT FOR SYMMETRIC RANDOM VARIABLES1 , 1972 .

[36]  David E. Booth,et al.  Applied Multivariate Analysis , 2003, Technometrics.

[37]  J. Huston McCulloch,et al.  13 Financial applications of stable distributions , 1996 .

[38]  Chrysostomos L. Nikias,et al.  Parameter estimation and blind channel identification in impulsive signal environments , 1995, IEEE Trans. Signal Process..

[39]  John P. Nolan,et al.  Tail Behavior, Modes and other Characteristics of Stable Distributions , 1999 .

[40]  M. Taqqu,et al.  Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance , 1995 .

[41]  A. Tamhane,et al.  Multiple Comparison Procedures , 1989 .

[42]  B. Stuck,et al.  A statistical analysis of telephone noise , 1974 .

[43]  R. Adler,et al.  A practical guide to heavy tails: statistical techniques and applications , 1998 .

[44]  Calyampudi Radhakrishna Rao,et al.  Statistical methods in finance , 1996 .

[45]  Clive W. J. Granger,et al.  “Infinite Variance” and Research Strategy in Time Series Analysis , 1972 .

[46]  D. Buckle.  Bayesian Inference for Stable Distributions , 1995 .