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The Limiting Log-Likelihood Process for Discontinuous Multiparameter Density Families

In: Probability and Statistical Inference

Author

Listed:
  • Georg Ch. Pflug

    (University of Vienna, Institute of Statistics)

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.

Suggested Citation

  • Georg Ch. Pflug, 1982. "The Limiting Log-Likelihood Process for Discontinuous Multiparameter Density Families," Springer Books, in: Wilfried Grossmann & Georg Ch. Pflug & Wolfgang Wertz (ed.), Probability and Statistical Inference, pages 287-295, Springer.
    • Handle: RePEc:spr:sprchp:978-94-009-7840-9_27
    DOI: 10.1007/978-94-009-7840-9_27
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