具有正态逆伽玛先验的正态分布中的方差参数在Stein损失下的贝叶斯后验估计量（英文）
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摘要
对于先验分布为正态逆伽玛分布的正态分布的方差参数,我们解析地计算了具有共轭的正态逆伽玛先验分布的在Stein损失函数下的贝叶斯后验估计量.这个估计量最小化后验期望Stein损失.我们还解析地计算了在平方误差损失函数下的贝叶斯后验估计量和后验期望Stein损失.数值模拟的结果例证了我们的如下理论研究:后验期望Stein损失不依赖于样本;在平方误差损失函数下的贝叶斯后验估计量和后验期望Stein损失要一致地大于在Stein损失函数下的对应的量.最后,我们计算了上证综指的月度的简单回报的贝叶斯后验估计量和后验期望Stein损失.
For the variance parameter of the normal distribution with a normal-inverse-gamma prior,we analytically calculate the Bayes posterior estimator with respect to a conjugate normalinverse-gamma prior distribution under Stein's loss function.This estimator minimizes the Posterior Expected Stein's Loss(PESL).We also analytically calculate the Bayes posterior estimator and the PESL under the squared error loss function.The numerical simulations exemplify our theoretical studies that the PESLs do not depend on the sample,and that the Bayes posterior estimator and the PESL under the squared error loss function are unanimously larger than those under Stein's loss function.Finally,we calculate the Bayes posterior estimators and the PESLs of the monthly simple returns of the SSE Composite Index.
For the variance parameter of the normal distribution with a normal-inverse-gamma prior,we analytically calculate the Bayes posterior estimator with respect to a conjugate normalinverse-gamma prior distribution under Stein's loss function.This estimator minimizes the Posterior Expected Stein's Loss(PESL).We also analytically calculate the Bayes posterior estimator and the PESL under the squared error loss function.The numerical simulations exemplify our theoretical studies that the PESLs do not depend on the sample,and that the Bayes posterior estimator and the PESL under the squared error loss function are unanimously larger than those under Stein's loss function.Finally,we calculate the Bayes posterior estimators and the PESLs of the monthly simple returns of the SSE Composite Index.
引文
[1] BROWN L D,PURVES R.Measurable selections of extrema[J].Ann Statist,1973,1(5):902-912.
[2] CASELLA G,BERGER R L.Statistical Inference[M].2nd ed.Pacific Grove:Duxbury Press,2001.
[3] LIU Y J,SUN D C.Should we use asis?[J].Chinese J Appl Probab Statist,2014,30(1):1-11.
[4] ANDERSON T W.An Introduction to Multivariate Statistical Analysis[M].2nd ed.New York:Wiley,1984.
[5] BOBOTAS P,KOUROUKLIS S.On the estimation of a normal precision and a normal variance ratio[J].Stat Methodol,2010,7(4):445-463.
[6] BROWN L D.Inadmissibility of the usual estimators of scale parameters in problems with unknown location and scale parameters[J].Ann Math Statist,1968,39(1):29-48.
[7] BROWN L D.[Developments in decision-theoretic variance estimation]:comment[J].Statist Sci,1990,5(1):103-106.
[8] JAMES W,STEIN C.Estimation with quadratic loss[C]//NEYMAN J.Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability,Volume 1:Contributions to the Theory of Statistics,Berkeley,CA:University of California Press,1961:361-379.
[9] OONO Y,SHINOZAKI N.On a class of improved estimators of variance and estimation under order restriction[J].J Statist Plann Inference,2006,136(8):2584-2605.
[10] PARSIAN A,NEMATOLLAHI N.Estimation of scale parameter under entropy loss function[J].J Statist Plann Inference,1996,52(1):77-91.
[11] PETROPOULOS C,KOUROUKLIS S.Estimation of a scale parameter in mixture models with unknown location[J].J Statist Plann Inference,2005,128(1):191-218.
[12] ZHANG Y Y.The Bayes rule of the variance parameter of the hierarchical normal and inverse gamma model under Stein's loss[J].Comm Statist Theory Methods,2017,46(14):7125-7133.
[13] DEY D K,SRINIVASAN C.Estimation of a covariance matrix under Stein's loss[J].Ann Statist,1985,13(4):1581-1591.
[14] KONNO Y.Estimation of a normal covariance matrix with incomplete data under Stein's loss[J].J Multivariate Anal,1995,52(2):308-324.
[15] KONNO Y.Estimation of normal covariance matrices parametrized by irreducible symmetric cones under Stein's loss[J].J Multivariate Anal,2007,98(2):295-316.
[16] MA T F,JIA L J,SU Y S.A new estimator of covariance matrix[J].J Statist Plann Inference,2012,142(2):529-536.
[17] SHEENA Y,TAKEMURA A.Inadmissibility of non-order-preserving orthogonally invariant estimators of the covariance matrix in the case of Stein's loss[J].J Multivariate Anal,1992,41(1):117-131.
[18] SUN X Q,SUN D C,HE Z Q.Bayesian inference on multivariate normal covariance and precision matrices in a star-shaped model with missing data[J].Comm Statist Theory Methods,2010,39(4):642-666.
[19] TSUKUMA H.Estimation of a high-dimensional covariance matrix with the Stein loss[J].J Multivariate Anal,2016,148:1-17.
[20] XU K,HE D J.Further results on estimation of covariance matrix[J].Statist Probab Lett,2015,101:11-20.
[21] YE R D,WANG S G.Improved estimation of the covariance matrix under Stein's loss[J].Statisi Probab Lett,2009,79(6):715-721.
[22] ROBERT C P.The Bayesian Choice:From Decision-Theoretic Foundations to Computational Implementation[M].2nd ed.New York:Springer,2007.
[23] RAIFFA H,SCHLAIFER R.Applied Statistical Decision Theory[M].Cambridge,MA:Harvard University Press,1961.
[24] MAO S S,TANG Y C.Bayesian Statistics[M].2nd ed.Beijing:China Statistics Press,2012.(in Chinese)
[25] CHEN M H.Bayesian statistics lecture[R].Statistics Graduate Summer School,School of Mathematics and Statistics,Northeast Normal University,Changchun,China,July 2014.
[26] R Core Team.R:a language and environment for statistical computing[CP].Vienna,Austria:R Foundation for Statistical Computing,2017.
[27] RYAN J A.quantmod:quantitative financial modelling framework[CP].R package version 0.4-7,2016.
[2] CASELLA G,BERGER R L.Statistical Inference[M].2nd ed.Pacific Grove:Duxbury Press,2001.
[3] LIU Y J,SUN D C.Should we use asis?[J].Chinese J Appl Probab Statist,2014,30(1):1-11.
[4] ANDERSON T W.An Introduction to Multivariate Statistical Analysis[M].2nd ed.New York:Wiley,1984.
[5] BOBOTAS P,KOUROUKLIS S.On the estimation of a normal precision and a normal variance ratio[J].Stat Methodol,2010,7(4):445-463.
[6] BROWN L D.Inadmissibility of the usual estimators of scale parameters in problems with unknown location and scale parameters[J].Ann Math Statist,1968,39(1):29-48.
[7] BROWN L D.[Developments in decision-theoretic variance estimation]:comment[J].Statist Sci,1990,5(1):103-106.
[8] JAMES W,STEIN C.Estimation with quadratic loss[C]//NEYMAN J.Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability,Volume 1:Contributions to the Theory of Statistics,Berkeley,CA:University of California Press,1961:361-379.
[9] OONO Y,SHINOZAKI N.On a class of improved estimators of variance and estimation under order restriction[J].J Statist Plann Inference,2006,136(8):2584-2605.
[10] PARSIAN A,NEMATOLLAHI N.Estimation of scale parameter under entropy loss function[J].J Statist Plann Inference,1996,52(1):77-91.
[11] PETROPOULOS C,KOUROUKLIS S.Estimation of a scale parameter in mixture models with unknown location[J].J Statist Plann Inference,2005,128(1):191-218.
[12] ZHANG Y Y.The Bayes rule of the variance parameter of the hierarchical normal and inverse gamma model under Stein's loss[J].Comm Statist Theory Methods,2017,46(14):7125-7133.
[13] DEY D K,SRINIVASAN C.Estimation of a covariance matrix under Stein's loss[J].Ann Statist,1985,13(4):1581-1591.
[14] KONNO Y.Estimation of a normal covariance matrix with incomplete data under Stein's loss[J].J Multivariate Anal,1995,52(2):308-324.
[15] KONNO Y.Estimation of normal covariance matrices parametrized by irreducible symmetric cones under Stein's loss[J].J Multivariate Anal,2007,98(2):295-316.
[16] MA T F,JIA L J,SU Y S.A new estimator of covariance matrix[J].J Statist Plann Inference,2012,142(2):529-536.
[17] SHEENA Y,TAKEMURA A.Inadmissibility of non-order-preserving orthogonally invariant estimators of the covariance matrix in the case of Stein's loss[J].J Multivariate Anal,1992,41(1):117-131.
[18] SUN X Q,SUN D C,HE Z Q.Bayesian inference on multivariate normal covariance and precision matrices in a star-shaped model with missing data[J].Comm Statist Theory Methods,2010,39(4):642-666.
[19] TSUKUMA H.Estimation of a high-dimensional covariance matrix with the Stein loss[J].J Multivariate Anal,2016,148:1-17.
[20] XU K,HE D J.Further results on estimation of covariance matrix[J].Statist Probab Lett,2015,101:11-20.
[21] YE R D,WANG S G.Improved estimation of the covariance matrix under Stein's loss[J].Statisi Probab Lett,2009,79(6):715-721.
[22] ROBERT C P.The Bayesian Choice:From Decision-Theoretic Foundations to Computational Implementation[M].2nd ed.New York:Springer,2007.
[23] RAIFFA H,SCHLAIFER R.Applied Statistical Decision Theory[M].Cambridge,MA:Harvard University Press,1961.
[24] MAO S S,TANG Y C.Bayesian Statistics[M].2nd ed.Beijing:China Statistics Press,2012.(in Chinese)
[25] CHEN M H.Bayesian statistics lecture[R].Statistics Graduate Summer School,School of Mathematics and Statistics,Northeast Normal University,Changchun,China,July 2014.
[26] R Core Team.R:a language and environment for statistical computing[CP].Vienna,Austria:R Foundation for Statistical Computing,2017.
[27] RYAN J A.quantmod:quantitative financial modelling framework[CP].R package version 0.4-7,2016.