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TheBerry-Esseen bound for minimum contrast estimates

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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:

$$\left| {P_\vartheta ^n \left\{ {x \in X^n :\frac{{\vartheta _n (x) - \vartheta }}{{\beta (\vartheta )}}n^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} < t} \right\} - \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^t {\exp [{{ - r^2 } \mathord{\left/ {\vphantom {{ - r^2 } 2}} \right. \kern-\nulldelimiterspace} 2}]dr} } \right| \leqq c_K n^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}} .$$

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.

  • Kelley, J. L.: General topology. New York 1969.

  • Michel, R. andJ. Pfanzagl: The accuracy of the normal approximation for minimum contrast estimates. Z. Wahrscheinlichkeitstheorie18, 73–84, 1971.

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  • 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

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