The Limiting Log-Likelihood Process for Discontinuous Multiparameter Density Families

  • Georg Ch. Pflug

Abstract

Let \({\{ f(\theta ,x)\} _{\theta \in \Theta }}\) be a family of probability densities on a measure space (X,A,μ) with multidimensional parameter \(\theta \in \Theta \subseteq {\mathbb{R}^k}\). Let \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X} _n} = ({X_1}, \ldots ,{X_n})\) be a i. i. d. sample in Xn, distributed according to f(θ,.). We study the asymptotic distribution of the log-likelihood process
$${Y_n}(t) = \sum\limits_{i = 1}^n {\log } \frac{{f(\theta + t.1/n,{X_i})}}{{f(\theta ,{X_i})}}{\text{ }}t \in {\mathbb{R}^k}$$
under the special assumption, that the densities have — as function of θ — discontinuities of the first kind.

Keywords

Manifold 

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References

  1. [1]
    Gihman, I. I., Skorohod, A. V.:1969, Introduction to the theory of random processes, Saunders, Philadelphia.Google Scholar
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    Ibragimov, I. A., Has’minskij, R. Z.:1972, Asymptotic behavior of statistical estimates for samples with a discontinuous density, Mat. Sbornik Tom 87 (129) No. 4.Google Scholar
  3. [3]
    Parthasarathy, K. R.:1967, Probability measures on metric spaces, Academic Press, New York and London.MATHGoogle Scholar
  4. [4]
    Pflug, G.: 1981, The limiting log-likelihood process for discontinuous density families, Preprint No. 40, Inst. f. Stat., Univ. Vienna, submitted to publication in Z. f. Wth.Google Scholar

Copyright information

© D. Reidel Publishing Company 1982

Authors and Affiliations

  • Georg Ch. Pflug
    • 1
  1. 1.Institute of StatisticsUniversity of ViennaAustria

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