Skip to main content
SpringerLink
Log in

Generalized Mittag-Leffler Distributions and Processes for Applications in Astrophysics and Time Series Modeling

  • Conference paper
  • First Online:
Proceedings of the Third UN/ESA/NASA Workshop on the International Heliophysical Year 2007 and Basic Space Science

Part of the book series: Astrophysics and Space Science Proceedings ((ASSSP))

Abstract

Geometric generalized Mittag-Leffler distributions having the Laplace transform \(\frac{1} {1+\beta \log (1+{t}^{\alpha })},0 < \alpha \leq 2,\beta > 0\) is introduced and its properties are discussed. Autoregressive processes with Mittag-Leffler and geometric generalized Mittag-Leffler marginal distributions are developed. Haubold and Mathai (Astrophysics and Space Science 273 53–63, 2000) derived a closed form representation of the fractional kinetic equation and thermonuclear function in terms of Mittag-Leffler function. Saxena et al. (2002; Astrophysics and Space Science 209 299–310 2004a; Physica A 344 657–664 2004b) extended the result and derived the solutions of a number of fractional kinetic equations in terms of generalized Mittag-Leffler functions. These results are useful in explaining various fundamental laws of physics. Here we develop first-order autoregressive time series models and the properties are explored. The results have applications in various areas like astrophysics, space sciences, meteorology, financial modeling and reliability modeling.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Anderson, D.N., Arnold, B.C.: Linnik distributions and processes. Journal of Applied Probability 30 330–340 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  • Dewald, D.N., Lewis, P.A.W.: A new Laplace second order auto regressive time series model-NLAR (2). IEEE Transactions in Information Theory 31 645–651 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  • Devroye, L.: A note on Linnik’s distribution. Statistical Probability Letters 9 305–306 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  • Erdélyi, A.: Higher Transcendental Functions. Vol. 3, McGraw Hill, New York (1955)

    MATH  Google Scholar 

  • Fujita, Y.: A generalization of the results of Pillai. Annals of the Institute of Statistical Mathematics 45 361–365 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  • Galambos, J., Kotz, S.: Characterizations of probability distributions. Lecture Notes in Mathematics, Vol. 675, Springer-Verlag, New York (1978)

    Google Scholar 

  • Gaver, D.P., Lewis, P.A.W.: First-order autoregressive gamma sequences and point processes. Advances in Applied Probability 12 727–745 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  • Haubold, H.J., Mathai, A.M.: The fractional kinetic equation and thermonuclear functions. Astrophysics and Space Science 273 53–63 (2000)

    Article  MATH  ADS  Google Scholar 

  • Jacques, C., Remillard, B., Theodorescu, R.: Estimation of Linnik parameters. Statist. Decisions 17 213–236 (1999)

    MATH  MathSciNet  Google Scholar 

  • Jayakumar, K., Pillai, R.N.: On class L distributions. Journal of the Indian Statistical Association 30 103–108 (1992)

    MathSciNet  Google Scholar 

  • Jayakumar, K., Pillai, R.N.: The first-order autoregressive Mittag-Leffler process. Journal of Applied Probability 30 462–466 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  • Jayakumar, K., Kalyanaraman, K., Pillai, R.N.: α-Laplace Processes. Mathematics of Computational Modelling 22 109–116 (1995)

    Article  MATH  Google Scholar 

  • Jayakumar, K., Ajitha, B.K.: On the geometric Mittag-Leffler distributions. Calcutta Statistical Association Bulletin 54 Nos. 215–216, 195–208 (2003)

    Google Scholar 

  • Jose, K.K., Seetha Lekshmi, V.: On geometric exponential distribution and its applications. Journal of the Indian Statistical Association 37 51–58 (1997)

    Google Scholar 

  • Klebanov, L.B., Maniya, G.M., Melamed, I.A.: A problem of Zolotarev and analogs of infinitely divisible and stable distributions in a scheme for summing a random number of random variables. Theory of Probability Applications 24 791–794 (1984)

    Google Scholar 

  • Kotz, S., Kozubowski, T.J., Podgorski, K.: The Laplace Distributions and Generalizations. Birkhaeuser, Boston (2001)

    Google Scholar 

  • Kotz, S., Ostrovskii, I.V.: A mixture representation of the Linnik distribution. Statistical Probability Letters 26 61–64 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Kozubowski, T.J., Rachev, S.T.: Univariate geometric stable laws. Journal of Computational Analysis and Applications, preprint (1999)

    Google Scholar 

  • Kozubowski, T.J.: Fractional moment estimation of Linnik and Mittag-Leffler parameters. Mathematics of Computational Modelling. Special issue: Stable Non-Gaussian Models in Finance and Econometrics 34 1023–1035 (2000)

    MathSciNet  Google Scholar 

  • Kozubowski, T.J.: Mixture representation of Linnik distribution revisited. Statistical Probability Letters 38 157–160 (1998)

    Article  MathSciNet  Google Scholar 

  • Lawrance, A.J.: Some autoregressive models for point processes. Colloquia Mathematica Societatis Janos Bolyai 24 Point Processes and Queuing Problems, Hungary, 257–275 (1978)

    Google Scholar 

  • Lawrance, A.J., Lewis, P.A.W.: A mixed exponential time series model. Management Science 28 9 1045–1053 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  • Lin, G.D.: A note on the characterization of positive Linnik laws. Australian New Zealand Journal of Statistics 43 17–20 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Lin, G.D.: A note on the Linnik distributions. Journal of Mathematical Analysis and Applications 217 701–706 (1998a)

    Article  MATH  MathSciNet  Google Scholar 

  • Lin, G.D.: On the Mittag-Leffler distributions. Journal of Statistical Planning Inference 74 1–9 (1998b)

    Article  MATH  Google Scholar 

  • Lin, G.D.: Characterizations of the Laplace and related distributions via geometric compound. Sankhya 56 1–9 (1994)

    MATH  Google Scholar 

  • Linnik, Yu.V.: Linear forms and statistical criteria, I, II. Ukrainian Mathematical Zhurnal 5 207–243 (1962) (English Translations in Mathematical Statistics and Probability 3 1–40, 41–90, American Mathematical Society, Providence, R.I.)

    Google Scholar 

  • Pakes, A.G.: Mixture representations for symmetric generalized Linnik laws. Statistical Probability Letters 37 213–221 (1998)

    Article  MathSciNet  Google Scholar 

  • Pillai, R.N.: Semi-α-Laplace distributions. Communications in Statistical Theoretical Methods 14 991–1000 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  • Pillai, R.N.: On Mittag-Leffler and related distributions. Annals of the Institute of Statistical Mathematics 42 157–161 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  • Pillai, R.N., Sandhya, E.: Distributions with complete monotone derivative and geometric infinite divisibility. Advances in Applied Probability 22 751–754 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  • Pillai, R.N., Jayakumar, K.: Specialized class L property and stationary autoregressive process. Statistical Probability Letters 19 51–56 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  • Pillai, R.N., Jayakumar, K.: Discrete Mittag-Leffler distributions. Statistical Probability Letters 23 271–274 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  • Rao, C.R., Rubin, H.: On characterization of the Poisson distribution, Sankhya, Ser. A, 26 294–298 (1964)

    MathSciNet  Google Scholar 

  • Rényi, A.: A characterization of the Poisson process. Magyar Tud. Akad. Mat. Kutato Int. Kozl. 1 519–527 (1956) (in Hungarian). (Translated into English in Selected Papers of Alfred Rényi, Vol.1, Akademiai Kiadó, Budapest, 1976)

    Google Scholar 

  • Saxena, R.K., Mathai, A.M., Haubold, H.J.: On fractional kinetic equations. Astrophysics and Space Science 282 281–287 (2002)

    Article  ADS  Google Scholar 

  • Saxena, R.K., Mathai, A.M., Haubold, H.J.: Unified fractional kinetic equation and a fractional diffusion equation. Astrophysics and Space Science 209 299–310 (2004a)

    Article  ADS  Google Scholar 

  • Saxena, R.K., Mathai, A.M., Haubold, H.J.: On generalized fractional kinetic equations. Physica A 344 657–664 (2004b)

    Article  MathSciNet  ADS  Google Scholar 

  • Seetha Lekshmi, V., Jose, K.K.: Geometric Mittag-Leffler tailed autoregressive processes. Far East Journal of Theoretical Statistics 6 147–153 (2002)

    MATH  MathSciNet  Google Scholar 

  • Seetha Lekshmi, V., Jose, K.K.: Geometric Mittag-Leffler distributions and processes. Journal of Applied Statistical Sciences (accepted for publication) (2003)

    Google Scholar 

  • Seetha Lekshmi, V., Jose, K.K.: An autoregressive process with geometric α-Laplace marginals. Statistical Papers 45 337–350 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics 52 479–487 (1988)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Weron, K., Kotulski, M.: On the Cole–Cole relaxation function and related Mittag-Leffler distribution. Physica A 232 180–188 (1996)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, St. Thomas College, Pala, M.G. University, Kerala, 686 574, India

    Kanichukattu Korakutty Jose, Padmini Uma, Vanaja Seetha Lekshmi & Hans Joachim Haubold

Authors
  1. Kanichukattu Korakutty Jose
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Padmini Uma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vanaja Seetha Lekshmi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hans Joachim Haubold
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kanichukattu Korakutty Jose .

Editor information

Editors and Affiliations

  1. Space Affairs (UNOOSA), United Nations Office for Outer, Wagramer Str. 5, Wien, 1400, Austria

    Hans J. Haubold

  2. Dept. Mathematics & Statistics, McGill University, Sherbrooke St. W. 805, Montreal, H3A 2K6, Canada

    A.M. Mathai

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Jose, K.K., Uma, P., Lekshmi, V.S., Haubold, H.J. (2010). Generalized Mittag-Leffler Distributions and Processes for Applications in Astrophysics and Time Series Modeling. In: Haubold, H., Mathai, A. (eds) Proceedings of the Third UN/ESA/NASA Workshop on the International Heliophysical Year 2007 and Basic Space Science. Astrophysics and Space Science Proceedings. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03325-4_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-03325-4_9

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03388-9

  • Online ISBN: 978-3-642-03325-4

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation