Abstract
Let P N and Q N , N≥1, be two possible probability distributions of a random vector X N =(XN1,...,XNN), whose components are independent. Suppose P N and Q N have respective densities % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamiCamaaBaaaleaacaWGobaabeaakiabg2da9maaxadabaGaeuiO% dafaleaacaWGPbGaeyypa0JaaGymaaqaaiaad6eaaaGccaWGMbGaai% ikaiaadIhadaWgaaWcbaGaamOtaiaadMgaaeqaaOGaeyOeI0YaaCbi% aeaacqaH4oqCaSqabeaacaGGFbaaaOWaaSbaaSqaaiaad6eaaeqaaO% Gaaiykaaaa!4DEC!\[p_N = \mathop \Pi \limits_{i = 1}^N f(x_{Ni} - \mathop \theta \limits^\_ _N )\] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamyCamaaBaaaleaacaWGobaabeaakiabg2da9maaxadabaGaeuiO% dafaleaacaWGPbGaeyypa0JaaGymaaqaaiaad6eaaaGccaWGMbGaai% ikaiaadIhadaWgaaWcbaGaamOtaiaadMgaaeqaaOGaeyOeI0IaeqiU% de3aaSbaaSqaaiaad6eacaWGPbaabeaakiaacMcaaaa!4DA5!\[q_N = \mathop \Pi \limits_{i = 1}^N f(x_{Ni} - \theta _{Ni} )\], where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaCbiaeaacqaH4oqCaSqabeaacaGGFbaaaOWaaSbaaSqaaiaad6ea% aeqaaOGaeyypa0JaamOtamaaCaaaleqabaGaeyOeI0IaaGymaaaakm% aaqahabaGaeqiUde3aaSbaaSqaaiaad6eacaWGPbaabeaaaeaacaWG% PbGaeyypa0JaaGymaaqaaiaad6eaa0GaeyyeIuoaaaa!4C75!\[\mathop \theta \limits^\_ _N = N^{ - 1} \sum\limits_{i = 1}^N {\theta _{Ni} } \], such that % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaCbeaeaacaqGTbGaaeyyaiaabIhaaSqaaiaaigdacqGHKjYOcaWG% PbGaeyizImQaamOtaaqabaGccaGG8bGaeqiUde3aaSbaaSqaaiaad6% eacaWGPbaabeaakiabgkHiTmaaxacabaGaeqiUdehaleqabaGaai4x% aaaakmaaBaaaleaacaWGobaabeaakiaacYhacqGH9aqpcaWGpbGaai% ikaiaad6eadaahaaWcbeqaaiabgkHiTiaaigdacaGGVaGaaGOmaaaa% kiaacMcaaaa!5647!\[\mathop {{\rm{max}}}\limits_{1 \le i \le N} |\theta _{Ni} - \mathop \theta \limits^\_ _N | = O(N^{ - 1/2} )\], f(x)>0 for almost every real x, f is absolutely continuous, and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaCbeaeaaciGGZbGaaiyDaiaacchaaSqaaiabeI7aXjaad+gacqGH% KjYOcqaH4oqCcqGHKjYOcqaH4oqCcaWGVbaabeaakmaapedabaGaai% 4waiaadAgaaSqaaiabg6HiLcqaaiabg6HiLcqdcqGHRiI8aOGaai4j% aiaacIcacaWG4bGaeyOeI0IaeqiUdeNaaiykamaaCaaaleqabaGaaG% Omaaaakiaac+cacaWGMbGaaiikaiaadIhacaGGPaGaamizaiaadIha% cqGH8aapcqGHEisPaaa!5ECE!\[\mathop {\sup }\limits_{\theta o \le \theta \le \theta o} \int_\infty ^\infty {[f} '(x - \theta )^2 /f(x)dx < \infty \] for some % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaeqiUde3aaSbaaSqaaiaaicdaaeqaaOGaeyOpa4JaaGimaaaa!3FD4!\[\theta _0 > 0\]. The contiguity of {q N } to {p N } is well known. In this paper it is proven that under these conditions {Q N } preserves C.-T.L.D. (Cramér-type large deviation) from {P N } for a general class of statistics ℱ which includes R-, U- and L-statistics as members. That means, for any {S N =SN(XN)} from ℱ, a C.-T.L.D. theorem with range C≤x≤o(Nδ) (any C≤0), 0<δ≤4-1, holds for {S N } under {P N }, implying that the same theorem holds for {S N } under {Q N }. It also provides a quick and simple way to establish C.-T.L.D. results for statistics under {Q N }.
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Research supported in part by grant VE87080 from the National Science Council, Republic of China.
Part of the research was done while the author was visiting the Institute of Statistical Science, Academia Sinica, Taipei, Taiwan.
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Wu, TJ. Contiguous alternatives which preserve Cramér-type large deviations for a general class of statistics. Ann Inst Stat Math 41, 649–660 (1989). https://doi.org/10.1007/BF00057731
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DOI: https://doi.org/10.1007/BF00057731