小池, 健一

橋本, 真太郎

Kyoto University Research Information Repository

Titleベイズリスクに対するバッタチャリャ型情報不等式 (種々のモデルのための漸近展開とそれらに関連する話題)Author(s)橋本, 真太郎; 小池, 健一Citation数理解析研究所講究録 (2013), 1860: 1-8Issue Date2013-11URL http://hdl.handle.net/2433/195298RightType Departmental Bulletin PaperTextversionpublisherKyoto University*Shintaro HashimotoGraduate School of Pure and Applied SciencesUniversity of Tsukuba\daggerKen-ichi KoikeFaculty of Pure and Applied SciencesUniversity of Tsukuba1Borovkov and Sakhanenko (1980)Koike(2006)Borovkov and Sakhanenko (1980)Brown and Gajek (1990) Sato and Akahira(1996) ( )Brown and Gajek (1990)305-0006 1-1-11860 2013 1-8 12$X_{1},$$\ldots,$$X_{n}$ ( $\sigma$ $\mu$ ) $f_{0}(x, \theta)(\theta\in$$\Theta)$$\Theta$$a,$ $b(-\infty\leq a<b\leq\infty)$$X:=(X_{1}, \ldots, X_{n})$ $f(x, \theta):=\Pi_{i=1}^{n}f_{0}(x_{i}, \theta)$$f(x, \theta)$ , $\theta$ $\theta$ $g(\theta)$$L(\theta, d)=(d-g(\theta))^{2}$ $\theta$ () $q(\theta)$ $\delta$ $R(\theta, \delta)=E_{\theta}\{L(\theta, \delta)\}$$\delta$ ( ), $B(q, \delta)=\int R(\theta, \delta)q(\theta)d\theta$ $B(q)= \inf_{\delta}B(q, \delta)$$q$$\theta$$\partial f/\partial\theta=f’,$ $\partial^{2}f/\partial\theta^{2}=f"$(Al) (A2)(Al) $g(\theta)$ $\delta$ $e(\theta)=E_{\theta}(\delta)$ ($)$ :$Var_{\theta}(\delta)\geq(e’(\theta), e"(\theta))V_{n}^{-1}(e’(\theta), e"(\theta))^{T}$ (2.1)$V_{n}=(\begin{array}{ll}I_{n} K_{n}K_{n} J_{n}\end{array})$$I_{n}(\theta)=E_{\theta}\{(f’/f)^{2}\},$ $J_{n}(\theta)=E_{\theta}\{(f"/f)^{2}\},$ $K_{n}(\theta)=E_{\theta}(f’f"/f^{2})$$(n\geq 1)$ . $I_{1}(\theta)$ (2.1) Zacks(1971)(A2) $\theta$ $q(\theta)$ $q(a)=q(b)=0$2.1. (Koike (2006)) (Al) (A2)$I_{n}=nI_{1}, J_{n}=nJ_{1}+n(n-1)I_{1}^{2}, K_{n}=nK_{1}$ . (2.2)3Borovkov and Sakhanenko (1980)23.1. (Borovkov and Sakhanenko (1980)) $n=1$ (Al) (A2)$B(q, \delta)\geq\frac{C^{2}}{C+D}$ , (3.1)$C:= \int_{\Theta}\frac{g^{\prime 2}q}{I_{1}}d\theta, D:=\int_{\Theta}\frac{1}{q}(\frac{g’q}{I_{1}})^{\prime 2}d\theta.$$n\geq 1$ (2.2) (3.1)$B(q, \delta)\geq Cn^{-1}+(-D)n^{-2}+O(n^{-3}) (narrow\infty)$ (3.2)Brown and Gajek (1990)( )3.2. (Brown and Gajek (1990)) $n=1$ (Al) (A2)$B(g, \delta)\geq\frac{C^{2}}{C+D}+Q_{1}(b)+Q_{2}(b)+Q_{3}(b)$ , (3.3)$b(\theta)=E_{\theta}\{\delta(X)\}-g(\theta)$ ,$\gamma(b)=\int_{\Theta}b(\frac{g’q}{I_{1}})’d\theta=-\int_{\Theta}\frac{g’b’q}{I_{1}}d\theta$, (3.4)$Q_{1}(b):= \frac{C+D}{CD}(\gamma-\frac{CD}{C+D})^{2}\geq 0,$$Q_{2}(b):= \int_{\Theta}(b’+\frac{g’\gamma}{C})^{2}\frac{q}{I_{1}}d\theta\geq 0,$$Q_{3}(b):= \int_{\Theta}\{b-\frac{\gamma}{D}\frac{1}{q}(\frac{g’q}{I_{1}})’\}^{2}qd\theta\geq 0.$3Borovkov and Sakhanenko (1980) Koike (2006)3.3. (Koike (2006)) $n=1$ (Al) (A2)$B(q, \delta)\geq(\int_{\Theta}\frac{g^{\prime 2}q}{I_{1}}d\theta, -\int_{\Theta}\frac{g’g"q}{I_{1}}d\theta)W^{-1}(\int_{\Theta}\frac{g^{\prime 2}q}{I_{1}}d\theta, -\int_{\Theta}\frac{g’g"q}{I_{1}}d\theta)^{T}$ (3.5)$W=\{E(S_{i}S_{j})\}_{i,j=1,2}$ $2\cross 2$$S_{i}= \frac{1}{f_{0}q}\frac{\partial^{i}}{\partial\theta^{i}}(\frac{g’qf_{0}}{I_{1}}) (i,j=1,2)$ .3.1. (Koike (2006)) $n\geq 1$ (Al) (A2) (3.5)$B(q, \delta)\geq Cn^{-1}+(-D+E)n^{-2}+O(n^{-3}) (narrow\infty)$ (3.6)$E:= \frac{[\int_{\Theta}\{_{I_{1}}L_{2^{-K_{1}-g_{I_{1}}}}^{\prime 2}\acute{s}_{-\}qd\theta]^{2}}"}{2\int_{\Theta}g^{\prime 2}qd\theta}\geq 0.$(3.6) (3.2) $n^{-2}$4(A3) $\delta$$b( \theta)=E_{\theta}\{\delta(X)\}-g(\theta)=\frac{b_{1}(\theta)}{n}+O(n^{-2}) (narrow\infty)$ .(A3) Brown and Gajek (1990)$B(q, \delta)\geq Cn^{-1}+(R_{1}+R_{2})n^{-2}+O(n^{-3}) (narrow\infty)$ (4.1)$R_{1}:=2 \int_{\Theta}\frac{b_{1}’g’q}{I_{1}}d\theta, R_{2}:=\int_{\Theta}b_{1}^{2}qd\theta.$Brown and Gajek (1990)44.1. $n=1$ (Al) (A2)$B(q, \delta)\geq\int_{\Theta}(g’+b’, g"+b")V_{1}^{-1}(g’+b’, g"+b")^{T}qd\theta+\int_{\Theta}b^{2}qd\theta$ . (4.2)4.1. $n\geq 1$ (Al) (A3) (4.2)$B(q, \delta)\geq Cn^{-1}+(R_{1}+R_{2}+R_{3})n^{-2}+O(n^{-3}) (narrow\infty)$ (4.3)$R_{3};= \int_{\Theta}\frac{(g’K_{1}-g"I_{1})^{2}}{I_{1}^{4}}qd\theta\geq 0.$(3.3) (4.2) (4.1) (4.3)$R_{3}n^{-2}+O(n^{-3})\geq 0 (narrow\infty)$ .$4.1$ . Brown and Gajek (1990) 2.7 (4.3) $b_{1}$$B(q, \delta)\geq Cn^{-1}+(-D+R_{3})n^{-2}+O(n^{-3}) (narrow\infty)$ .$K_{1}=0$4.2. $n=1$ (Al) (A2) $K_{1}=0$$B(g, \delta)\geq\frac{C^{2}}{C+D}+Q_{1}(b)+Q_{2}(b)+Q_{3}(b)+Q_{4}(b)$ , (4.4)$Q_{4}(b);= \int_{\Theta}\frac{(g"+b")^{2}}{J_{1}}qd\theta\geq 0.$4.1. $X$ $N(\theta, 1)$ # $\theta$ $N(\mu, \sigma^{2})$$g(\theta)=\theta^{2}$ $X$ $\theta$$N( \frac{X+(\mu/\sigma^{2})}{1+(1/\sigma^{2})}, \frac{1}{1+(1/\sigma^{2})})$ $\theta^{2}$$\delta(X)=E(\theta^{2}|X)=\frac{1}{1+(1/\sigma^{2})}+\{\frac{X+(\mu/\sigma^{2})}{1+(1/\sigma^{2})}\}^{2}$5$b( \theta)=\frac{\mu^{2}+2\sigma^{4}+2\sigma^{2}(\mu-\theta)\theta-\theta^{2}}{(1+\sigma^{2})^{2}},$$B(q, \delta)=\frac{2\sigma^{2}(2\sigma^{4}+2\mu^{2}\sigma^{2}+2\mu^{2}+\sigma^{2})}{(1+\sigma^{2})^{2}}.$$g’(\theta)=2\theta,$ $g”(\theta)=2,$ $I_{1}(\theta)=1,$ $K_{1}(\theta)=0$ 4.1$(4.2)= \frac{2\sigma^{2}(2\sigma^{4}+2\mu^{2}\sigma^{2}+2\mu^{2}+\sigma^{2})}{(1+\sigma^{2})^{2}}.$(4.2)Koike (2006) $n\geq 1$$X_{1},$$\ldots,$$X_{n}$ $\theta$ is $N( \frac{\Sigma_{i=1}^{n}X_{i}+(\mu/\sigma^{2})}{n+(1/\sigma^{2})}, \frac{1}{n+(1/\sigma^{2})})$$\theta^{2}$$\delta(X)=E(\theta^{2}|X)=\frac{1}{n+(1/\sigma^{2})}+\{\frac{\sum_{i--1}^{n}X_{i}+(\mu/\sigma^{2})}{n+(1/\sigma^{2})}\}^{2}$$b( \theta)=\frac{2(\mu\theta+\sigma^{2}-\theta^{2})}{\sigma^{2}}n^{-1}+o(n^{-1})$ ,$B(q, \delta)=4(\mu^{2}+\sigma^{2})n^{-1}+\frac{-2(2\mu^{2}+3\sigma^{2})}{\sigma^{2}}n^{-2}+O(n^{-3}) (narrow\infty)$ .4.1$(4.3)=4( \mu^{2}+\sigma^{2})n^{-1}+\{-4(2+\frac{\mu^{2}}{\sigma^{2}})+2\}n^{-2}+O(n^{-3})$as $narrow\infty$ . (4.3) $O(n^{-3})(narrow\infty)$(3.6)$4( \mu^{2}+\sigma^{2})n^{-1}+\{-4(2+\frac{\mu^{2}}{\sigma^{2}})+\frac{2\mu^{2}}{\mu^{2}+\sigma^{2}}\}n^{-2}+O(n^{-3})$ , (4.5)(4.3) (4.5)$(2- \frac{2\mu^{2}}{\mu^{2}+\sigma^{2}})n^{-2}+O(n^{-3}) (narrow\infty)$ , (4.6)$2- \frac{2\mu^{2}}{\mu^{2}+\sigma^{2}}\geq 0$ (4.3) (3.6) $n^{-2}$64.2. $X$ $Po(\lambda)$ $\lambda$$Ga(a, b)(a, b>0)$ $g(\theta)=e^{-\lambda}$ ($X=0$ )$X$ $\lambda$ $Ga(a+X, )$$e^{-\lambda}$$\delta(X)=E(e^{-\lambda}|X)=(1+\frac{b}{1+b})^{-a-X}$(4.2)(3.1)(3.3)(4.2)$Ga(1,1)$ $2.614\cross 10^{-2}$ $4.635\cross 10^{-2}$ 4. $753\cross 10^{-2}$ $4.761\cross 10^{-2}$$Ga(1,2)$ $1.592\cross 10^{-2}$ 3. $971\cross 10^{-2}$ $5.269\cross 10^{-2}$ $5.285\cross 10^{-2}$$Ga(2,2) 3.312\cross 10^{-3} 1.403\cross 10^{-2} 1.490\cross 10^{-2} 1.507\cross10^{-2}$$Ga(4,2)$ $2.258\cross 10^{-5}$ $9.054\cross 10^{-4}$ 9. $611\cross 10^{-4}$ $9.785\cross 10^{-4}$(4.2)54.1 $\theta$$(A1)-(A3)$5.1. $\epsilon>0$ $a>0$ $\theta\in(\theta_{0}-\epsilon, \theta_{0}+\epsilon)$$0<a\leq|K_{1}(\theta)|$ (5.1)$(\theta_{0}-\epsilon, \theta_{0}+\epsilon)\subset\Theta$ $(A1)-(A3)$ $\theta_{0}$$narrow\infty$$\sup_{\theta\in(\theta_{0}-\epsilon,\theta_{0}+\epsilon)}E_{\theta}\{(\hat{\theta}-\theta)^{2}\}\geq\frac{1}{I^{*}}n^{-1}-\frac{\pi^{2}}{\epsilon^{2}I_{*}^{2}}n^{-2}+\frac{a^{2}}{2I^{*}4}n^{-2}+O(n^{-3})$ (5.2)$\hat{\theta}$$\theta$ $I^{*}= \sup_{\theta}I_{1}(\theta),$ $I_{*}= \inf_{\theta}I_{1}(\theta)$75.1. Borovkov (1998)$\sup_{\theta\in(\theta_{0}-\epsilon,\theta_{0}+\epsilon)}E_{\theta}\{(\hat{\theta}-\theta)^{2}\}\geq\frac{1}{I^{*}}n^{-1}-\frac{\pi^{2}}{\epsilon^{2}I_{*}^{2}}n^{-2}+O(n^{-3})$ $(narrow\infty)$ . (5.3)(5.2) (5.3) $n^{-2}$[1] Borovkov, A. $A$ . (1998). Mathematical Statistics. Amsterdam: Gordon andBreach.[2] Borovkov, A. $A$ ., Sakhanenko, A. $I$ . (1980). Estimates for averaged quadratic risk(in Russian). Probability and Mathematical Statistics 1: 185-195.[3] Blight, B. J. $N$ ., Rao, P. $V$ . (1974). The convergence of Bhattacharyya bounds.Biometrika 61: 137-142.[4] Brown, L. $D$ ., Gajek, L. (1990). Information inequalities for the Bayes risk. An-nals of Statistics 18: 1578-1594.[5] Ghosh, J. $K$ . (1994). Higher Order Asymptotics. NSF-CBMS Regional Confer-ence Series in Probability and Statistics, Vol.4, Inst. of Math. Statist., Hayward.[6] Ghosh, J. $K$ ., Sathe, Y. $S$ . (1987). Convergence of Bhattacharyya bounds–revisited. Sankhya Ser. $A$ 49: 37-42.[7] Khan, R. $A$ . (1984). On UMVU estimators and Bhattacharyya bounds in expo-nential distributions. Joumal of Statistical Planning and Inference 9: 199-206.[8] Koike, K. (2006). An integral Bhattacharyya type bound for the Bayes risk.Communications in $Statistics-$ Theory and Methods 35: 2185-2195.[9] Lehmann, E. $L$ ., Casella, G. (1998). Theory of Point Estimation, 2nd Ed. NewYork: Springer.[10] Sato, M., Akahira, M. (1996). An information inequality for the Bayes risk.Annals of Statistics 24: 2288-2295.[11] Takada, Y. (1999). Lower bounds on the Bayes risk for statistical predictionproblems. Communications in $Statistics-$ Theow and Methods 28: 693-703.[12] Zacks, S. (1971). The Theory of Statistical Inference. New York: Wiley.8

ベイズリスク ニ タイスル バッタチャリャガタ ジョウホウ フトウシキ シュジュ ノ モデル ノ タメ ノ ゼンキン テンカイ ト ソレラ ニ カンレン スル ワダイ

http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/195298/1/1860-01.pdf

ベイズリスクニタイスルバッタチャリャガタジョウホウフトウシキシュジュノモデルノタメノゼンキンテンカイトソレラニカンレンスルワダイ

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Kyoto University Research Information Repository

ベイズリスク ニ タイスル バッタチャリャガタ ジョウホウ フトウシキ シュジュ ノ モデル ノ タメ ノ ゼンキン テンカイ ト ソレラ ニ カンレン スル ワダイ

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Kyoto University Research Information Repository

ベイズリスクニタイスルバッタチャリャガタジョウホウフトウシキシュジュノモデルノタメノゼンキンテンカイトソレラニカンレンスルワダイ