Skip to main content
Log in

Continuous l n,p -symmetric distributions

  • Published:
Lithuanian Mathematical Journal Aims and scope Submit manuscript

Abstract

For p > 0, the l n,p -generalized surface measure on the l n,p -unit sphere is studied and used for deriving a geometric measure representation for l n,p -symmetric distributions having a density.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. T.W. Anderson, An Introduction to Multivariate Statistical Analysis. 2nd ed., Wiley, N.Y., 1984.

    MATH  Google Scholar 

  2. T.W. Anderson and K.T. Fang, On the theory of multivariate elliptically contoured distributions and their applications, in K.T. Fang and T.W. Anderson (Eds.), Statistical Inference in Elliptically Contoured and Related Distributions, Allerton Press Inc., N.Y., 1990, pp. 1–23.

    Google Scholar 

  3. K. Breitung and W.-D. Richter, A geometric approach to an asymptotic expansion for large deviation probabilities of Gaussian random vectors, J. Multivariate Anal., 58(1):1–20, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  4. K.T. Fang and T.W. Anderson (Eds.), Statsitical Inference in Elliptically Contoured and Related Distributions, Allerton Press Inc., N.Y., 1990.

    Google Scholar 

  5. K.T. Fang, S. Kotz, and K.W. Ng, Symmetric Multivariate and Related Distributions, Chapman and Hall, London, New York, 1990.

    MATH  Google Scholar 

  6. K.T. Fang and Y.T. Zang, Generalized Multivariate Analysis, Springer Verlag, N.Y., 1990.

    MATH  Google Scholar 

  7. V. Fatalov and W.-D. Richter, Gaussian probabilities of large deviations for fixed and increasing dimensions, J. Contemporary Math. Anal., 27(1):1–16, 1992.

    MathSciNet  Google Scholar 

  8. H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin–Heidelberg–New York, 1969.

    MATH  Google Scholar 

  9. R. Fisher, T. Lewis, and B. Embleton, Statistical Analysis of Spherical Data, University Press, Cambridge, 1987.

    MATH  Google Scholar 

  10. I.R. Goodman and S. Kotz, Multivariate θ-generalized normal distributions, J. Multivariate Anal., 3:204–219, 1973.

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Gupta and D. Song, l p -norm spherical distributions, J. Statist. Plann. Inference, 60:241–260, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Gupta and T. Varga, Elliptically Contoured Models in Statistics, Kluwer Academic publishers, Dordrecht, Boston, London, 1993.

    MATH  Google Scholar 

  13. V. Henschel and W.-D. Richter, Geometric generalization of the exponential law, J. Multivariate Anal., 81:189–204, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  14. C. Ittrich, D. Krause, and W.-D. Richter, Probabilities and large quantiles of noncentral generalized chi-square distributions, Statistics, 34:53–101, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  15. C. Ittrich and W.-D. Richter, Exact tests and confidence regions in nonlinear regression, Statistics, 39(1):13–42, 2005.

    MATH  MathSciNet  Google Scholar 

  16. D.R. Jensen and D.E. Ramirez, Some exact properties of Cook’s d i statistic, in N. Balakrishnan and C. Rao (Eds.), Handbook of Statistics, vol. 16, Elsevier Science Publishers, Amsterdam, 1998, pp. 387–402.

    Google Scholar 

  17. M. Johnson, Multivariate Statistical Simulation, J. Wiley Sons, N.Y., 1987.

    MATH  Google Scholar 

  18. N.L. Johnson and S. Kotz, Distributions in Statistics-Continuous Univariate Distributions-2, Houghton Mifflin Co, Boston, 1970.

    Google Scholar 

  19. J.D. Kalbfleisch and R.L. Prentice, The Statistical Analysis of Failure Time Data, J. Wiley Sons, N.Y., 1980.

    MATH  Google Scholar 

  20. D. Kelker, Distribution theory of spherical distributions and a location-scale parameter generalization, Sankhya Ser. A, 32:419–430, 1970.

    MATH  MathSciNet  Google Scholar 

  21. D. Krause and W.-D. Richter, Exact probabilities of correct classifications for uncorrelated repeated measurements from elliptically contoured distributions, J. Multivariate Anal., 89:36–69, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  22. F. Morgan, Geometric Measure Theory, Academic Press, San Diego, 1995.

    MATH  Google Scholar 

  23. A. Naor, The surface measure and cone measure on the sphere of l p n, Trans. Am. Math. Soc., 359(3):1045–1080, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  24. G. Pap and W.-D. Richter, Zum asymptotischen Verhalten der Dichten gewisser Funktionale Gaußscher Zufallsvektoren, Math. Nachr., 135:119–124, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  25. S.T. Rachev and L. Rüschendorf, Approximate independence of distributions on spheres and their stability properties, Ann. Probab., 19(3):1311–1337, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  26. W.-D. Richter, Laplace–Gauss integrals, Gaussian measure asymptotic behavior and probabilities of moderate deviations, Z. Anal. Anw., 4(3):257–267, 1985.

    MATH  Google Scholar 

  27. W.-D. Richter, Zur Restgliedabschätzung im mehrdimensionalen integralen Zentralen Grenzwertsatz der Wahrscheinlichkeitstheorie, Math. Nachr., 135:103–117, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  28. W.-D. Richter, Eine geometrische Methode in der Stochastik., Rostock. Math. Kolloq., 44:63–72, 1991.

    MATH  Google Scholar 

  29. W.-D. Richter, A geometric approach to the Gaussian law, in V Mammitzsch and Schneeweiß (Eds.), Symposia Gaussiana, Conf. B, W. de Gruyter and Co., Berlin, New York, 1995, pp. 25–45.

    Google Scholar 

  30. W.-D. Richter, Generalized spherical and simplicial coordinates, J. Math. Anal. Appl., 336:1187–1202, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  31. W.-D. Richter, On l 2,p -circle numbers, Lith. Math. J., 48(2):228–234, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  32. W.-D. Richter, On the π-function for non-convex l 2,p -circle discs, Lith. Math. J., 48(3):332–338, 2008.

    Article  MathSciNet  Google Scholar 

  33. W.-D. Richter and J. Schumacher, Asymptotic expansions for large deviation probabilities of noncentral generalized chi-square distributions, J. Multivariate Anal., 75:184–218, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  34. W.-D. Richter and J. Steinebach, A geometric approach to finite sample and large deviation properties in two-way ANOVA with spherically distributed error vectors, Metrika, 41:325–353, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  35. W.-D. Richter and V.V. Ulyanov, Two sides estimates for the Gaussian measure of the complements of balls in Hilbert space, Probab. Theory Appl., 36(4):805–806, 1991.

    Google Scholar 

  36. G. Schechtman and J. Zinn, On the volume of the intersection of two l n,p -balls, Proc. Am. Math. Soc., 110(1):217–224, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  37. G.E. Schilov, Mathematical Analysis. Functions of Several Real Variables, Nauka, Moscow, 1972 (in Russian).

    Google Scholar 

  38. I.J. Schoenberg, Metric spaces and completely monotone functions, Ann. Math., 39:811–841, 1938.

    Article  MathSciNet  Google Scholar 

  39. D. Song and A.K. Gupta, l p -norm uniform distributions, Proc. Am. Math. Soc., 125(2):595–601, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  40. M.T. Subbotin, On the law of frequency of errors, Mat. Sbornik, 31:296–301, 1923.

    Google Scholar 

  41. P.J. Szablowski, Uniform distribution on spheres in finite dimensional l α and their generalisations, J. Multivariate Anal., 64(2):103–117, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  42. T. Taguchi, On a generalization of Gaussian distribution, Ann. Inst. Stat. Math., 30:211–242, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  43. G. Watson, Statistics on Spheres, J. Wiley Sons, N.Y., 1983.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to W.-D. Richter.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Richter, WD. Continuous l n,p -symmetric distributions. Lith Math J 49, 93–108 (2009). https://doi.org/10.1007/s10986-009-9030-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10986-009-9030-3

Keywords

Navigation