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2007 | 5 | 1 | 50-83
Tytuł artykułu

Neighbourhoods of independence and associated geometry in manifolds of bivariate Gaussian and Freund distributions

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We provide explicit information geometric tubular neighbourhoods containing all bivariate distributions sufficiently close to the cases of independent Poisson or Gaussian processes. This is achieved via affine immersions of the 4-manifold of Freund bivariate distributions and of the 5-manifold of bivariate Gaussians. We provide also the α-geometry for both manifolds. The Central Limit Theorem makes our neighbourhoods of independence limiting cases for a wide range of bivariate distributions; the topological character of the results makes them stable under small perturbations, which is important for applications in models of stochastic processes.
Wydawca
Czasopismo
Rocznik
Tom
5
Numer
1
Strony
50-83
Opis fizyczny
Daty
wydano
2007-03-01
online
2007-03-01
Twórcy
  • University of Manchester
Bibliografia
  • [1] S.-I. Amari, O.E. Barndorff-Neilsen, R.E. Kass, S.L. Lauritzen and C.R. Rao: Differential Geometrical Methods in Statistics, Springer Lecture Notes in Statistics 28, Springer-Verlag, Berlin, 1985.
  • [2] S-I. Amari and H. Nagaoka: Methods of Information Geometry, American Mathematical Society, Oxford University Press, 2000.
  • [3] Khadiga Arwini: Differential geometry in neighbourhoods of randomness and independence, PhD thesis, UMIST, 2004.
  • [4] Khadiga Arwini and C.T.J. Dodson: “Information geometric neighbourhoods of randomness and geometry of the McKay bivariate gamma 3-manifold”, Sankhya: Indian Journal of Statistics, Vol. 66(2), (2004), pp. 211–231.
  • [5] Khadiga Arwini and C.T.J. Dodson: “Neighbourhoods of independence for random processes via information geometry”, Math. J., Vol. 9(4), (2005).
  • [6] Khadiga Arwini, L. Del Riego and C.T.J. Dodson: “Universal connection and curvature for statistical manifold geometry”, Houston J. Math., in press, (2006).
  • [7] Y. Cai, C.T.J. Dodson, O. Wolkenhauer and A.J. Doig: “Gamma Distribution Analysis of Protein Sequences shows that Amino Acids Self Cluster”, J. Theoretical Biology, Vol. 218(4), (2002), pp. 409–418.
  • [8] C.T.J. Dodson: “Spatial statistics and information geometry for parametric statistical models of galaxy clustering”, Int. J. Theor. Phys., Vol. 38(10), (1999), pp. 2585–2597. http://dx.doi.org/10.1023/A:1026609310371
  • [9] C.T.J. Dodson: “Geometry for stochastically inhomogeneous spacetimes”, Nonlinear Analysis, Vol. 47, (2001), pp. 2951–2958. http://dx.doi.org/10.1016/S0362-546X(01)00416-3
  • [10] C.T.J. Dodson and Hiroshi Matsuzoe: “An affine embedding of the gamma manifold”, Appl. Sci., Vol. 5(1), (2003), pp. 1–6.
  • [11] R.J. Freund: “A bivariate extension of the exponential distribution”, J. Am. Stat. Assoc., Vol. 56, (1961), pp. 971–977. http://dx.doi.org/10.2307/2282007
  • [12] T.P. Hutchinson and C.D. Lai: Continuous Multivariate Distributions, Emphasising Applications, Rumsby Scientific Publishing, Adelaide 1990.
  • [13] S. Kotz, N. Balakrishnan and N. Johnson: Continuous Multivariate Distributions, Volume 1, 2nd ed., John Wiley, New York, 2000.
  • [14] S. Leurgans, T.W.-Y. Tsai and J. Crowley: “Freund’s bivariate exponential distribution and censoring”, In: R.A. Johnson (Ed.): Survival Analysis, IMS Lecture Notes, Hayward, California, Institute of Mathematical Statistics, 1982.
  • [15] A.F.S. Mitchell: “The information matrix, skewness tensor and α-connections for the general multivariate elliptic distribution”, Ann. Ins. Stat. Math., Vol. 41, (1989), pp. 289–304. http://dx.doi.org/10.1007/BF00049397
  • [16] Y. Sato, K. Sugawa and M. Kawaguchi: The geometrical structure of the parameter space of the two-dimensional normal distribution, Division of information engineering, Hokkaido University, Sapporo, Japan, 1977.
  • [17] L.T. Skovgaard: “A Riemannian geometry of the multivariate normal model”, Scandinavian J. Stat., Vol. 11, (1984), pp. 211–223.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-006-0034-5
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