Summary
Let (X,A) be a measurable space,Θ ⊂ ℝ an open interval andP ϑ|A, ϑ ∈Θ, a family of probability measures fulfilling certain regularity conditions. Letϑ n be a minimum contrast estimate for the sample sizen. It is shown that for every compact setK ⊂ Θ there exists a constantc K such that for allϑ ∈ K, n ∈ ℕ, t ∈ ℝ:
This theorem improves an earlier result ofMichel andPfanzagl where the boundc Kn−1/2 (logn)1/2 was obtained. The bound obtained now cannot be improved any more as far as the order ofn is concerned. The problem of estimatingc K will not be taken up in this paper.
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References
Feller, W.: An introduction to probability theory and its applications. Vol. I, New York 1957.
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Michel, R. andJ. Pfanzagl: The accuracy of the normal approximation for minimum contrast estimates. Z. Wahrscheinlichkeitstheorie18, 73–84, 1971.
Pfanzagl, J.: On the measurability and consistency of minimum contrast estimates. Metrika14, 249–272, 1969.
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Pfanzagl, J. TheBerry-Esseen bound for minimum contrast estimates. Metrika 17, 82–91 (1971). https://doi.org/10.1007/BF02613813
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DOI: https://doi.org/10.1007/BF02613813