Bernoulli

  • Bernoulli
  • Volume 5, Number 4 (1999), 721-760.

The exponential statistical manifold: mean parameters, orthogonality and space transformations

Giovanni Pistone and Maria Piera Rogantin

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Abstract

Let ( X,cal X,μ) be a measure space, and let cal M (X,cal X,μ) denote the set of the μ -almost surely strictly positive probability densities. It was shown by Pistone and Sempi in 1995 that the global geometry on cal M (X,cal X,μ) can be realized by an affine atlas whose charts are defined locally by the mappings cal M (X,cal X,μ)cal U pqlog(q/p)+K(p,q)B p , where cal U p is a suitable open set containing p , K (p,q) is the Kullback--Leibler relative information and B p is the vector space of centred and exponentially ( pμ) -integrable random variables. In the present paper we study the transformation of such an atlas and the related manifold structure under basic transformations, i.e. measurable transformation of the sample space. A generalization of the mixed parametrization method for exponential models is also presented.

Article information

Source
Bernoulli, Volume 5, Number 4 (1999), 721-760.

Dates
First available in Project Euclid: 19 February 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1171899326

Mathematical Reviews number (MathSciNet)
MR1704564

Zentralblatt MATH identifier
0947.62003

Keywords
exponential families exponential statistical manifolds information mean parameters Orlicz spaces orthogonality

Citation

Pistone, Giovanni; Piera Rogantin, Maria. The exponential statistical manifold: mean parameters, orthogonality and space transformations. Bernoulli 5 (1999), no. 4, 721--760. https://projecteuclid.org/euclid.bj/1171899326


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References

  • [1] Amari, S.-I. (1982) Differential geometry of curved exponential families-curvature and information loss. Ann. Statist., 18, 357-385.
  • [2] Amari, S.-I. (1985) Differential-Geometrical Methods in Statistics. Lecture Notes Statist., 28. Berlin: Springer-Verlag.
  • [3] Amari, S.-I., Barndorff-Nielsen, O.E., Kass, R., Lauritzen, S.L. and Rao, C.R. (1987) Differential Geometry and Statistical Inference. Hayward, CA: Institute of Mathematical Statistics.
  • [4] Barndorff-Nielsen, O.E. (1978a) Information and Exponential Families in Statistical Theory. New York: Wiley.
  • [5] Barndorff-Nielsen, O.E. (1978b) Parametric Statistical Models and Likelihood. Lecture Notes Statist., 50. Berlin: Springer-Verlag.
  • [6] Barndorff-Nielsen, O.E. and Cox, D.R. (1989) Asymptotic Techniques for use in Statistics. London: Chapman & Hall.
  • [7] Barndorff-Nielsen, O.E. and Cox, D.R. (1994) Inference and Asymptotics. London: Chapman & Hall.
  • [8] Barndorff-Nielsen, O.E. and Jupp, P.E. (1989) Approximating exponential models. Ann. Inst. Statist. Math., 41, 247-267.
  • [9] Brigo, D. and Pistone, G. (1996) Projecting the Fokker-Planck equation onto a finite dimensional exponential family. Preprint 4, Dipartimento di Matematica Pura e Applicata Università di Padova.
  • [10] Cox, D.R. and Reid, N. (1987) Parameter orthogonality and approximate conditional inference (with discussion). J. Roy. Statist. Soc. B, 49, 1-39.
  • [11] Dawid, A.P. (1975) Discussion of a paper by B. Efron. Ann. Statist., 3, 1231-1234.
  • [12] Dawid, A.P. (1977) Further comments on a paper by Bradley Efron. Ann. Statist., 5, 1249-1249.
  • [13] Donsker, M.D. and Varadhan, S.R.S. (1975) Asymptotic evaluation of certain Markov processes expectations for large time I. Commun. Pure Appl. Math., 28, 1-47.
  • [14] Efron, B. (1975) Defining the curvature of a statistical problem (with applications to second-order efficiency) (with discussion). Ann. Statist., 3, 1189-1242.
  • [15] Efron, B. (1978) The geometry of exponential families. Ann. Statist., 6, 362-376.
  • [16] Ekeland, I. and Temam, R. (1974) Analyse Convexe et Problèmes Variationnels. Paris: Dunod, Gauthier-Villars.
  • [17] Gibilisco, P. and Pistone, G. (1998) Connections on non-parametric statistical manifolds by Orlicz space geometry, infinite dimensional analysis. Quantum Probab. Related Topics, 1, 325-347.
  • [18] Hardy, G.H., Littlewood, J.E. and Pólya, G. (1952) Inequalities, 2nd edn. London: Cambridge University Press.
  • [19] Jeffreys, H. (1946) An invariant form of the prior probability in estimation problems. Proc. Roy. Soc. London A, 196, 453-461.
  • [20] Kass, R.E. (1989) The geometry of asymptotic inference (with discussion). Statist. Sci., 4, 188-234.
  • [21] Krasnosel'skii, M.A. and Rutickii, Ya.B. (1961) Convex Functions and Orlicz Spaces. Groningen: Noordhoff. (Russian original (1958) Moscow: Fizmatgiz.)
  • [22] Kullback, S. (1997) Information Theory and Statistics, reprint of 2nd (1968) edn., Mineola, NY: Dover.
  • [23] Kullback, S. and Leibler, R.A. (1951) On information and sufficiency. Ann. Math. Statist., 22, 79-86.
  • [24] Lang, S. (1995) Differentiable Manifolds and Riemannian Manifolds. New York: Springer-Verlag.
  • [25] Letac, G. (1992) Lectures on Natural Exponential Families and their Variance Functions, Monogr. Mat., 50. Rio de Janeiro: Istituto de Matemaática Pura e Aplicada.
  • [26] Madsen, L.T. (1979) The geometry of statistical model-a generalization of curvature. Research Report 79-1, Statistics Research Unit, Danish Medical Research Council.
  • [27] Murray, M.K. and Rice, J.W. (1993) Differential Geometry and Statistics, Monogr. Statist. Appl. Probab., 48. London: Chapman & Hall.
  • [28] Neveu, J. (1972) Martingales à Temps Discrets. Paris: Masson.
  • [29] Pistone, G. and Rogantin, M.P. (1990) Gli strumenti della geometria differenziale nell'inferenza statistica. In P. Nastasi (ed.) Memorial Beniamino Gulotta, Giornate di Lavoro di Probabilità e Statistica, pp. 85-99. Palermo: Istituto Gramsci.
  • [30] Pistone, G. and Rogantin, M.P. (1994) The Transformation of the Non-Parametric Statistical Manifold under Conditioning and Sampling. Proceedings of the 57th IMS Annual Meeting and Third World Congress of the Bernoulli Society, Chapel Hill, NC, 20-25 June 1994.