The Rao–Blackwell theorem offers a procedure for converting a crude unbiased estimator of a parameter θ into a “better” one, in fact unique and optimal if the improvement is based on a minimal sufficient statistic that is complete. In contrast, behind every minimal sufficient statistic that is not complete, there is an improvable Rao–Blackwell improvement. This is illustrated via a simple example based on the uniform distribution, in which a rather natural Rao–Blackwell improvement is uniformly improvable. Furthermore, in this example the maximum likelihood estimator is inefficient, and an unbiased generalized Bayes estimator performs exceptionally well. Counterexamples of this sort can be useful didactic tools for explaining the true nature of a methodology and possible consequences when some of the assumptions are violated. [Received December 2014. Revised September 2015.]

中文翻译：

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可改进的 Rao-Blackwell 改进、低效最大似然估计器和无偏广义贝叶斯估计器的示例

Rao-Blackwell 定理提供了一种将参数 θ 的粗略无偏估计量转换为“更好”估计量的过程，如果改进基于完整的最小充分统计量，则实际上是唯一和最优的。相比之下，在每个不完整的最小充分统计量背后，都有一个可改进的 Rao-Blackwell 改进。这是通过一个基于均匀分布的简单示例来说明的，其中相当自然的 Rao-Blackwell 改进是一致可改进的。此外，在这个例子中，最大似然估计器效率低下，而无偏广义贝叶斯估计器表现得非常好。此类反例可能是有用的教学工具，可用于解释方法论的真实性质以及违反某些假设时可能产生的后果。