TheBerry-Esseen bound for minimum contrast estimates
TheBerry-Esseen bound for minimum contrast estimates
复制标题
最小对比度估计的 Berry-Esseen 界限
作者：
J. Pfanzagl
SummaryLet (X,A) be a measurable space,Θ ⊂ ℝ an open interval andPϑ|A, ϑ ∈Θ, a family of probability measures fulfilling certain regularity conditions. Letϑn be a minimum contrast estimate for the sample sizen. It is shown that for every compact setK ⊂ Θ there exists a constantcK such that for allϑ ∈ K, n ∈ ℕ, t ∈ ℝ:
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$$\left| {P_\vartheta ^n \left\{ {x \in X^n :\frac{{\vartheta _n (x) - \vartheta }}{{\beta (\vartheta )}}n^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} < t} \right\} - \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^t {\exp [{{ - r^2 } \mathord{\left/ {\vphantom {{ - r^2 } 2}} \right. \kern-\nulldelimiterspace} 2}]dr} } \right| \leqq c_K n^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}} .$$
This theorem improves an earlier result ofMichel andPfanzagl where the boundcKn−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 estimatingcK will not be taken up in this paper.
SummaryLet (X,A) be a measurable space,Θ ⊂ ℝ an open interval andPϑ|A, ϑ ∈Θ, a family of probability measures fulfilling certain regularity conditions. Letϑn be a minimum contrast estimate for the sample sizen. It is shown that for every compact setK ⊂ Θ there exists a constantcK such that for allϑ ∈ K, n ∈ ℕ, t ∈ ℝ:
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$$\left| {P_\vartheta ^n \left\{ {x \in X^n :\frac{{\vartheta _n (x) - \vartheta }}{{\beta (\vartheta )}}n^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} < t} \right\} - \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^t {\exp [{{ - r^2 } \mathord{\left/ {\vphantom {{ - r^2 } 2}} \right. \kern-\nulldelimiterspace} 2}]dr} } \right| \leqq c_K n^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}} .$$
This theorem improves an earlier result ofMichel andPfanzagl where the boundcKn−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 estimatingcK will not be taken up in this paper.