在變異數未知情形及不軛先驗分配下有界常態平均數之貝氏估計

Bayesian Estimation of a Bounded Normal Mean under Unknown Variance and Conjugate Prior

羅琪(Chi Lo) ； 廖淑宛(Shu-Wan Liao)

《中國統計學報》 42卷3期 (2004/09) Pp. 245-257

https://doi.org/10.29973/JCSA.200409.0002

摘要

在統計文獻上，在損失函數為平方誤差損失（squared-errorloss）且變異數已知時，用最大概似估計量估計有界的常態平均數，此估計量既不滿足允許性（admissibility）不滿足大中取小性（minimaxity）。 Lo (1995)在其論文中曾探討用貝氏法在平方誤差損失且變異數已知的情形下，估計有界的常態平均數，文中假設未受限制的母體平均數的先驗分配為無訊息（noninformative）。另外 Lo ( 2000）曾探討相同的問題，但是先驗分配是取共軛分配。然而在實務上，通常平均數未知，變異數也未知，所以假設變異數為已知並不十分合理。因此廖（2000）將 Lo (1995)的研究，推廣到變異數未知的情形，本文則是將 Lo (2000)的研究，推廣到變異數未知的情形，兩研究皆集中於一維。本文考應在變異數未知及共軛先驗分配下之有界常態平均數的貝氏估計。研究的內容包含後驗分配與有限制的貝氏估計量的推導、與其他估計量（未受限制的及裁頭的貝氏估計量）的風險函數的比較及三個估計量平均損失的模擬研究。結論是貝氏法不但可以輕易地解決平均數有界的問題，而且有限制的貝氏估計量也優於另外兩個估計量。最後，並將本文結果與 Lo (1995, 2000）、廖（2000）的結果作成比較。

關鍵字

貝氏估計 ； 有界常態平均數 ； 有限制的貝氏估計量 ； 截頭的貝氏估計量

並列摘要

It is well known in the statistics literature that the MLE is neither admissible nor minimax when we use it to estimate the bounded normal mean vector under squared- error loss and known covariance matrix. The problem of Bayesian estimation of a bounded multivariate normal mean under squared-error loss and known covariance matrix has been considered by Lo (1995). In her research, the prior distribution of the unrestricted mean is assumed to be noninformative. For the case of using conjugate prior has been developed by Lo (2000). In practice, it is unrealistic to assume that the variance is known. Therefore, Liao (2000) has studied the problem when the variance is unknown and the prior distribution of the unrestricted mean is noninformative. This paper will consider the same problem under unknown variance and conjugate prior. The posterior distribution and the restricted Bayes estimator are obtained. Numerical results are used to compare the performance between the restricted Bayes, unrestricted Bayes, and truncated Bayes estimators. The overall conclusion is that the Bayes method can solve the bounded mean problem easily and the restricted Bayes estimator is generally superior than the other 2 estimators. Finally, the paper will relate/assimilate the results in current study with earlier results of Lo (1995, 2000) and Liao (2000).

並列關鍵字

Bayes estimation ； bounded normal mean ； restricted Bayes estimator ； truncated Bayes estimator

參考文獻

廖維琦(2000)。有限制的常態分配平均數的貝氏估計當變異數未知(碩士論文)。中華大學電機工程學系。

Berger, J. O.,J. M. Bernardo,M. H. Degroot,D. V. Lindley,A. F. M. Smith (Eds)(1985).Bayesian Statistics.North-Holland.

Berry, J. C.(1990).Minimax estimation of a bounded normal mean vector.Journal of Multivariate Analysis.35,130-139.

Bickel, P. J.(1981).Minimax estimation of the mean of a normal distribution when the parameter space is restricted.Annals of Statistics.9,1301-1309.

Casella, G.,Strawderman, W. E.(1981).Estimating a bounded normal mean.Annals of Statistics.9,870-878.