Abstract
Let % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaacI% cacaWGybWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiabeI7aXnaaBaaa% leaacaWGPbaabeaakiaacMcacaGG9baaaa!3ED1!\[\{ (X_i ,\theta _i )\} \] be a sequence of independent random vectors where X i , conditional on % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS% baaSqaaiaadMgaaeqaaaaa!38BD!\[\theta _i \], has the probability density of the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI% cacaWG4bGaaiiFaiabeI7aXnaaBaaaleaacaWGPbaabeaakiaacMca% cqGH9aqpcaWG1bGaaiikaiaadIhacaGGPaGaam4qaiaacIcacqaH4o% qCdaWgaaWcbaGaamyAaaqabaGccaGGPaGaaeyzaiaabIhacaqGWbGa% aiikaiabgkHiTiaadIhacaGGVaGaeqiUde3aaSbaaSqaaiaadMgaae% qaaOGaaiykaaaa!4FFF!\[f(x|\theta _i ) = u(x)C(\theta _i ){\text{exp}}( - x/\theta _i )\] and the unobservable % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS% baaSqaaiaadMgaaeqaaaaa!38BD!\[\theta _i \] are i.i.d. according to an unknown G in some class G of prior distributions on Θ, a subset of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiabeI% 7aXjabg6da+iaaicdacaGG8bGaam4qaiaacIcacqaH4oqCcaGGPaGa% eyypa0JaaiikaiaadAgacaWG1bGaaiikaiaadIhacaGGPaGaaeyzai% aabIhacaqGWbGaaeikaiabgkHiTiaadIhacaGGVaGaeqiUdeNaaiyk% aiaadsgacaWG4bGaaiykamaaCaaaleqabaGaeyOeI0IaaGymaaaaki% abg6da+iaaicdacaGG9baaaa!54DE!\[\{ \theta > 0|C(\theta ) = (fu(x){\text{exp(}} - x/\theta )dx)^{ - 1} > 0\} \]. For a S(X 1 , ..., Xn, Xn+1)-measurable function % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdy2aaS% baaSqaaiaad6gaaeqaaOGaaiilaaaa!397F!\[\phi _n ,\] let % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa% aaleaacaWGUbaabeaakiabg2da9iaadweacaGGOaGaeqOXdy2aaSba% aSqaaiaad6gaaeqaaOGaeyOeI0IaeqiUde3aaSbaaSqaaiaad6gacq% GHRaWkcaaIXaaabeaakiaacMcadaahaaWcbeqaaiaaikdaaaaaaa!444A!\[R_n = E(\phi _n - \theta _{n + 1} )^2 \] denote the Bayes risk of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdy2aaS% baaSqaaiaad6gaaeqaaaaa!38C5!\[\phi _n \] and let R(G) denote the infimum Bayes risk with respect to G. For each integer s>1 we exhibit a class of S(X 1 , ..., Xn, Xn+1)-measurable functions % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdy2aaS% baaSqaaiaad6gaaeqaaaaa!38C5!\[\phi _n \] such that for δ in [s −1, 1], % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa% aaleaacaaIWaaabeaakiaad6gadaahaaWcbeqaaiabgkHiTiaaikda% caWGZbGaai4laiaacIcacaaIXaGaey4kaSIaaGOmaiaadohacaGGPa% aaaOGaeyizImQaamOuamaaBaaaleaacaWGUbaabeaakiaacIcacqaH% gpGzdaWgaaWcbaGaamOBaaqabaGccaGGSaGaam4raiaacMcacqGHsi% slcaWGsbGaaiikaiaadEeacaGGPaGaeyizImQaam4yamaaBaaaleaa% caaIXaaabeaakiaad6gadaahaaWcbeqaaiabgkHiTiaaikdacaGGOa% Gaam4Caiabes7aKjabgkHiTiaaigdacaGGPaGaai4laiaacIcacaaI% XaGaey4kaSIaaGOmaiaadohacaGGPaaaaaaa!5F94!\[c_0 n^{ - 2s/(1 + 2s)} \leqslant R_n (\phi _n ,G) - R(G) \leqslant c_1 n^{ - 2(s\delta - 1)/(1 + 2s)} \] under certain conditions on u and G. No assumptions on the form or smoothness of u is made, however. Examples of functions u, including one with infinitely many discontinuities, are given for which our conditions reduce to some moment conditions on G. When Θ is bounded, for each integer s>1 S(X 1 , ..., Xn, Xn+1)-measurable functions % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdy2aaS% baaSqaaiaad6gaaeqaaaaa!38C5!\[\phi _n \] are exhibited such that for δ in % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaaik% dacaGGVaGaam4CaiaacYcacaaIXaGaaiyxaiaadogadaqhaaWcbaGa% aGimaaqaaiaacEcaaaGccaWGUbWaaWbaaSqabeaacqGHsislcaaIYa% Gaam4Caiaac+cacaGGOaGaaGymaiabgUcaRiaaikdacaWGZbGaaiyk% aaaakiabgsMiJkaadkfadaWgaaWcbaGaamOBaaqabaGccaGGOaGaeq% OXdy2aaSbaaSqaaiaad6gaaeqaaOGaaiilaiaadEeacaGGPaGaeyOe% I0IaamOuaiaacIcacaWGhbGaaiykaiabgsMiJkaadogadaqhaaWcba% GaaGymaaqaaiaacEcaaaGccaWGUbWaaWbaaSqabeaacqGHsislcaaI% YaGaam4Caiabes7aKjaac+cacaGGOaGaaGymaiabgUcaRiaaikdaca% WGZbGaaiykaaaaaaa!637D!\[[2/s,1]c_0^' n^{ - 2s/(1 + 2s)} \leqslant R_n (\phi _n ,G) - R(G) \leqslant c_1^' n^{ - 2s\delta /(1 + 2s)} \]. Examples of functions u and class g are given where the above lower and upper bounds are achieved.
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Part of the research was carried out during R. S. Singh's visit to the University of Science and Technology of China.
Research supported in part by a Natural Sciences and Engineering Research Council of Canada Grant No. #A4631.
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Singh, R.S., Wei, L. Empirical Bayes with rates and best rates of convergence in u(x)C(θ) exp(-x/θ)-family: Estimation case. Ann Inst Stat Math 44, 435–449 (1992). https://doi.org/10.1007/BF00050697
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DOI: https://doi.org/10.1007/BF00050697