TheBerry-Esseen bound for minimum contrast estimates

Abstract

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: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaaca% WGqbWaa0baaSqaaiabeg9akbqaaiaad6gaaaGcdaGadaqaaiaadIha% cqGHiiIZcaWGybWaaWbaaSqabeaacaWGUbaaaOGaaiOoamaalaaaba% Gaeqy0dO0aaSbaaSqaaiaad6gaaeqaaOGaaiikaiaadIhacaGGPaGa% eyOeI0Iaeqy0dOeabaGaeqOSdiMaaiikaiabeg9akjaacMcaaaGaam% OBamaaCaaaleqabaWaaSGbaeaacaaIXaaabaGaaGOmaaaaaaGccqGH% 8aapcaWG0baacaGL7bGaayzFaaGaeyOeI0YaaSaaaeaacaaIXaaaba% WaaOaaaeaacaaIYaGaeqiWdahaleqaaaaakmaapehabaGaciyzaiaa% cIhacaGGWbGaai4wamaalyaabaGaeyOeI0IaamOCamaaCaaaleqaba% GaaGOmaaaaaOqaaiaaikdaaaGaaiyxaGqaaiaa-rgacaWGYbaaleaa% cqGHsislcqGHEisPaeaacaWG0baaniabgUIiYdaakiaawEa7caGLiW% oatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab+zMi% gkaadogadaWgaaWcbaGaam4saaqabaGccaWGUbWaaWbaaSqabeaada% WcgaqaaiabgkHiTiaaigdaaeaacaaIYaaaaaaakiaac6caaaa!7AF1! $$\left| {P_\vartheta ^n \left\{ {x \in X^n :\frac{{\vartheta _n (x) - \vartheta }}{{\beta (\vartheta )}}n^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} < t} \right\} - \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^t {\exp [{{ - r^2 } \mathord{\left/ {\vphantom {{ - r^2 } 2}} \right. \kern-\nulldelimiterspace} 2}]dr} } \right| \leqq c_K n^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}} .$$ 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.