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The likelihood ratio for densities with singularities

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Abstract

Let tn be the Bayesian estimator of the parameter θ constructed from independent observations in the case of infinite information and probability distribution density with some singularities. It is shown that under certain conditions on the behavior of the densities near their singularities, the normalizing factor ϕ(n) ensuring that Δn=ϕ(n) (tn-θ) has a nontriviallimiting distribution for n →∞ is regularly varying in Karamata's sense. The limiting distribution of Δn is determined.

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Literature cited

  1. I. A. Ibragimov and R. Z. Khas'minskii, “On moments of generalized Bayesian estimators and maximum likelihood estimators,” Teor. Veroyatn. Primen.,18, No. 3 (1973).

  2. I. A. Ibragimov and R. Z. Khas'minskii, “Asymptotic behavior of location parameter estimators for samples with a continuous density with singularities,” J. Sov. Math.,9, No. 1 (1978).

  3. W. Feller, Introduction to Probability Theory and Its Applications, Wiley (1968).

  4. I. A. Ibragimov and R. Z. Khas'minskii, “Asymptotic behavior of estimators for samples with discontinuous density,” Mat. Sb.,87, No. 4 (1972).

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Authors
  1. N. K. Bakirov
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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 74, pp. 66–82, 1977.

I would like to acknowledge the guidance of I. A. Ibragimov in the course of this research.

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Bakirov, N.K. The likelihood ratio for densities with singularities. J Math Sci 34, 1403–1413 (1986). https://doi.org/10.1007/BF01085008

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  • DOI: https://doi.org/10.1007/BF01085008

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