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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

Asymptotic Distribution Special Assumption Discontinuous Density Multidimensional Parameter Density Family 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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
  2. [2]
    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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