Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-15T20:43:59.070Z Has data issue: false hasContentIssue false
  • English
  • Français

Simple Goodness-of-Fit Tests for Symmetric Stable Distributions

Published online by Cambridge University Press:  19 October 2009

Erwin M. Saniga  and
Jack C. Hayya
Get access Rights & Permissions [Opens in a new window]

Extract

Stable distributions are becoming increasingly popular as appropriate models for stock price changes and other economic phenomena. As a result, there is an expanding body of literature on inferential procedures for this family of distributions. Computationally simple estimators for the parameters of symmetric stable distributions have been provided by Fama and Roll. Little attention, though, has been given to goodness-of-fit tests for members of this family other than the normal.

It is the purpose of this paper to discuss simple goodness-of-fit hypothesis tests using kurtosis, b2, to distinguish among members of the stable family. The b2 tests of hypothesis comprise: 1) a null normal versus a nonnormal symmetric stable alternative; 2) a null nonnormal symmetric stable versus a normal alternative; and 3) a null nonnormal stable versus another nonnormal stable alternative. Tables that give the percentage points of b2 and that are necessary for these tests of hypothesis are given. Apart from providing critical values for the tests, the tables allow the researcher to calculate the power. It will be seen that the b2 test exhibits excellent power.

It is then hoped that computational convenience will make b2 an important tool for researchers and practitioners in finance. It is also hoped that the procedures we provide will aid these researchers and practitioners in the construction of appropriate financial models.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 12 , Issue 2 , June 1977 , pp. 276 - 289
Copyright
Copyright © School of Business Administration, University of Washington 1977

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bergstrom, Harold. “On Some Expansions of Stable Distributions.” Arkiv for Matematic, Vol. 11 (1952), pp. 375–378.Google Scholar
[2]D'Agostino, R., and Pearson, E. S.. “Tests for Departure from Normality. Empirical Results for the Distributions of b2 and .” Biometrika, Vol. 60 (1973), pp. 613–622.Google Scholar
[3]D'Agostino, R., and Tietjen, G.. “Simulation Probability Points of b2 for Small Samples.” Biometrika, Vol 58 (1971), pp. 669–672.Google Scholar
[4]DuMonceaux, R.; Antle, C. E.; and Hass, G.. “Likelihood Ratio Test for Discrimination between Two Models with Unknown Location and Scale Parameters.” Technometrics, Vol. 15 (1973), pp. 19–31.Google Scholar
[5]DuMouchel, W. H. “Stable Distributions in Statistical Inference.” Ph.D. dissertation, Yale University (1971).Google Scholar
[6]DuMouchel, W. H. “On the Asymptotic Normality of the Maximum Likelihood Estimate When Sampling from a Stable Distribution.” The Annals of Statistics, Vol. 1, No. 5 (1973), pp. 948–957.Google Scholar
[7]DuMouchel, W. H. “Stable Distributions in Statistical Inference: 1. Symmetric Stable Distributions Compared to Other Symmetric Long Tailed Distributions.” Journal of the American Statistical Association, Vol. 68 (1973), pp. 469–477.Google Scholar
[8]Dyer, A. R. “Hypothesis Testing Procedures for Separate Families of Hypotheses.” Journal of the American Statistical Association, Vol. 69 (1974), pp. 140–145.Google Scholar
[9]Fama, E. F. “The Behavior of Stock Market Prices.” Journal of Business, Vol. 38 (1965), pp. 34–105.Google Scholar
[10]Fama, E., and Roll, R.. “Some Properties of Symmetric Stable Distributions.” Journal of the American Statistical Association, Vol. 63 (1968), pp. 817–836.Google Scholar
[11]Fama, E. “Parameter Estimates for Symmetric Stable Distributions.” Journal of the American Statistical Association, Vol. 66 (1971), pp. 331–338.CrossRefGoogle Scholar
[12]Grubbs, F. E. “Procedures for Detecting Outlying Observations in Samples.” Technometrics, Vol. 44 (1969), pp. 1–21.Google Scholar
[13]Pearson, E. S. “A Further Development of Tests for Normality.” Biometrika, Vol. 22 (1930), pp. 239–249.CrossRefGoogle Scholar
[14]Pearson, E.S. “Tables of Percentages Points of and b2 in Normal Samples; A Rounding Off.” Biometrika, Vol. 52 (1965), pp. 282–285.Google Scholar
[15]Saniga, E. M. “Parameter Estimation and Goodness-of-Fit Tests for Symmetric Stable Distributions.” Unpublished Ph.D. dissertation, The Pennsylvania State University (August 1975).Google Scholar
[16]Saniga, E. M.; Pfaffenberger, R.; and Hayya, J.. “Estimation and Goodness-of-Fit Tests for Symmetric Stable Distributions.”1975 Proceedings of Business and Economics Statistics Section, American Statistical Association, pp. 530–534.Google Scholar
[17]Shapiro, S. S.; Wilk, M. B.; and Chen, J.. “A Comparative Study of Various Tests for Normality.” Journal of the American Statistical Association, Vol. 63 (1968), pp. 1343–1372.CrossRefGoogle Scholar
[18]Smith, K. V. “A Simulation Analysis of the Power of Several Tests for Detecting Heavy-Tailed Distributions.” Journal of the American Statistical Association, Vol. 70 (1975), pp. 662–665.Google Scholar
[19]Snedecor, G., and Cochran, W.. Statistical Methods. Ames, Iowa: Iowa State Press (1968).Google Scholar
[20]Stephens, M. A. “EDF Statistics for Goodness-of-Fit and Some Comparisons.” Journal of the American Statistical Association, Vol. 69 (1974), pp. 730–737.Google Scholar