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Linnik distributions and processes

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Dale N. Anderson  and
Barry C. Arnold
Dale N. Anderson
Affiliation:
University of California, Riverside
Barry C. Arnold*
Affiliation:
University of California, Riverside
*
∗Postal address: Department of Statistics, University of California, Riverside, CA 92521, USA.
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Abstract

Using a simple characterization of the Linnik distribution, discrete-time processes having a stationary Linnik distribution are constructed. The processes are structurally related to exponential processes introduced by Arnold (1989), Lawrance and Lewis (1981) and Gaver and Lewis (1980). Multivariate versions of the processes are also described. These Linnik models appear to be viable alternatives to stable processes as models for temporal changes in stock prices.

Keywords

GEOMETRIC COMPOUNDINGTIME SERIESSTABLE DISTRIBUTIONSTOCK PRICES

MSC classification

Primary: 60G10: Stationary processes
Secondary: 62M99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 2 , June 1993 , pp. 330 - 340
Copyright
Copyright © Applied Probability Trust 1993 

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References

Abramowitz, M. and Stegun, I. (1968) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, 55.Google Scholar
Anderson, D. N. (1990) Some Time Series Models with Non-Additive Structure. , University of California, Riverside.Google Scholar
Anderson, D. N. (1991) A multivariate Linnik distribution. Statist. Prob. Lett. To appear.Google Scholar
Arnold, B. C. (1989) A logistic process constructed using geometric minimization. Statist. Prob. Lett. 7, 253–257.Google Scholar
Box, G. ?. P. and Jenkins, G. M. (1976) Time Series Analysis: Forecasting and Control. Holden-Day, Oakland.Google Scholar
Brockwell, P. J. and Davis, R. A. (1987) Time Series: Theory and Methods. Holden-Day, Oakland.Google Scholar
Chambers, J. M., Mallows, C. L. and Stuck, B. W. (1976) A method for simulating stable random variables. J. Amer. Statist. Assoc. 71, 340–344.Google Scholar
Devroye, L. (1990) A note on Linnik's distribution. Statist. Prob. Lett. 9, 305–306.Google Scholar
Dewald, L. S. and Lewis, P. A. W. (1985) A new Laplace second-order autoregressive time series model-NLAR (2). IEEE Trans. Inf. Theory 31, 645–652.Google Scholar
Dewald, L. S., Lewis, P. A. W. and Mckenzie, E. (1988) l-Laplace processes. Technical Report #NPS55-88-0011, Naval Postgraduate School, Monterey, California.Google Scholar
Fama, E. F. (1965) The behavior of stock market prices. J. Business 38, 34–105.Google Scholar
Gaver, D. P. and Lewis, P. A. W. (1980) First-order autoregressive gamma sequences and point processes. Adv. Appl. Prob. 12, 727–745.Google Scholar
Lau, A. H., Lau, H. and Wigender, J. (1990) The distribution of stock return: new evidence against the stable model. J. Business Econom. Statist. 8, 225–234.Google Scholar
Lawrance, A. J. and Lewis, P. A. W. (1981) A new autoregressive time series model in exponential variables (NEAR(1)). Adv. Appl. Prob. 13, 826–845.Google Scholar
Leitch, R. A. and Paulson, A. S. (1975) Estimation of stable law parameters: stock price behavior application. J. Amer. Statist. Assoc. 70, 690–697.Google Scholar
Linnik, Ju. V. (1963) Linear forms and statistical criteria I, II. Selected Translations in Mathematical Statistics and Probability, Volume 3, American Mathematical Society, 1–90.Google Scholar
Paulson, A. S., Holcomb, E. W. and Leitch, R. A. (1975) The estimation of the parameters of the stable laws. Biometrika 62, 163–170.Google Scholar
Pitman, E. J. G. (1968) On the behavior of the characteristic function of a probability distribution in the neighbourhood of the origin. J. Austral. Math. Soc. 8, 423–443.Google Scholar
Press, S. J. (1972a) Estimation in univariate and multivariate stable distributions. J. Amer. Statist. Assoc. 67, 842–846.Google Scholar
Press, S. J. (1972b) Multivariate stable distributions. J. Multivariate Anal. 2, 444–462.Google Scholar
Press, S. J. (1982) Applied Multivariate Analysis: Using Bayesian and Frequentist Methods of Inference. Robert E. Krieger, Malabar, FL.Google Scholar
Taylor, S. J. (1989) Modelling Financial Time Series. Wiley, Chichester.Google Scholar