• Bronius Grigelionis
Part of the SpringerBriefs in Statistics book series (BRIEFSSTATIST)


Considering a sample of independent observations \(X_1,\ldots ,X_n\) from the normal population with mean \(\alpha \) and variance \(\sigma ^2\) for testing the null hypothesis \(H_0:\alpha =\alpha _0\) against the alternative \(H_1:\alpha =\alpha _1\), Gosset (“Student”) in 1908 [1] suggested the test statistic


Multivariate Gaussian Distribution Inverse Gamma Variance Mixture Univariate Student Infinite Divisibility 
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  1. 1.
    Student: On the probable error of mean. Biometrika. 6, 1–25 (1908)Google Scholar
  2. 2.
    Fisher, R.A.: Applications of Student’s distribution. Metron 5, 109–112 (1925)Google Scholar
  3. 3.
    Fisher, R.A.: Introduction to Table of Hh functions. In: British Association Mathematical Tables, vol. 1, pp. 26–35. British Association, London, (1931)Google Scholar
  4. 4.
    Kotz, S., Nadarajah, S.: Multivariate \(t\)-Distributions and Their Applications. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  5. 5.
    Johnson, N.L., Kotz, S.: Distributions in Statistics: Continuous Univariate Distributions. Wiley, New York (1972)Google Scholar
  6. 6.
    Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  7. 7.
    Grosswald, E.: The Student \(t\)-distribution of any degree of freedom is infinitely divisible. Z. Wahrscheinlichkeitstheor. verw. Geb. 36, 103–109 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Halgreen, C.: Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions. Z. Wahrscheinlichkeitstheor. verw. Geb. 47, 13–17 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Ismail, M.E.H., Kelker, D.H.: The Bessesl polynomials and the Student \(t\)-distribution. SIAM J. Math. Anal. 7, 82–91 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kelker, D.: Infinite divisibility and variance mixtures of the normal distribution. Ann. Math. Statist. 42, 802–808 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Thorin, O.: On the infinite divisibility of lognormal distribution. Scand. Actuar. J. 3, 121–148 (1977)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bondesson, L.: Generalized gamma convolutions and related classes of distributions and densities. Lecture Notes in Statistics, vol. 76. Springer-Verlag, Berlin (1992)Google Scholar
  13. 13.
    Grigelionis, B.: Extending the Thorin class. Lith. Math. J. 51(2), 194–206 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    James, L.F., Roynette, B., Yor, M.: Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples. Probab. Surv. 5, 346–415 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Barndorff-Nielsen, O.E.: Exponentially decreasing distributions for the logarithm of particle size. Proc. Royal Soc. Lond. A 353, 401–419 (1977)CrossRefGoogle Scholar
  16. 16.
    Barndorff-Niesen, O.E., Shephard, N.: Non-Gaussian Ornstein Uhlenbeck-based models and some of their use in financial econometrics. J. R. Statist. Soc. B. 63, 167–241 (2001)Google Scholar
  17. 17.
    Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman& Hall/CRC, Boca Raton (2004)zbMATHGoogle Scholar
  18. 18.
    Heyde, C.C., Leonenko, N.N: Student processes, Adv. Appl. Prob. 37, 342–365 (2005)Google Scholar
  19. 19.
    Hurst, S.R., Platen, E., Rachev, S.R.: Subordinated Markov models: a comparison. Finan. Eng. Jpn. Markets 4, 97–124 (1997)zbMATHCrossRefGoogle Scholar
  20. 20.
    McNeil, A.J., Frey, R., Embrechts, P.: Quantitive Risk Management: Concepts, Techniques, and Tools. Princeton series in, finance (2005)Google Scholar
  21. 21.
    Schoutens, W.: Lévy Processes in Finance: Pricing Financial Derivatives. Wiley, New York (2003)Google Scholar
  22. 22.
    Woyczynski, W.A: Burgers-KPZ turbulence. Lecture Notes Mathematics, vol. 1700, Springer, Berlin (1998)Google Scholar
  23. 23.
    Kallsen, J., Tankov, P.: Characterization of dependence of multidimensional Lévy processes using Lévy copulas. J. Multivar. Anal. 97, 1551–1572 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Abramowitz, M., Stegun, I. (eds.): Handbook of Mathematical Functions. Dover, New York (1968)Google Scholar
  25. 25.
    Watson, G.N.: Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1958)Google Scholar

Copyright information

© The Author(s) 2013

Authors and Affiliations

  1. 1.University of VilniusVilniusLithuania

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