Dukić Vanja M, Peña Edsel A
J Am Stat Assoc. 2005 Mar 1;100(469):296-309. doi: 10.1198/016214504000000818..
This paper considers the problem of estimating the dispersion parameter in a Gaussian model which is intermediate between a model where the mean parameter is fully known (fixed) and a model where the mean parameter is completely unknown. One of the goals is to understand the implications of the two-step process of first selecting a model among a finite number of sub-models, and then estimating a parameter of interest after the model selection, but using the same sample data. The estimators are classified into global, two-step, and weighted-type estimators. While the global-type estimators ignore the model space structure, the two-step estimators explore the structure adaptively and can be related to pre-test estimators, and the weighted estimators are motivated by the Bayesian approach. Their performances are compared theoretically and through simulations using their risk functions based on a scale invariant quadratic loss function. It is shown that in the variance estimation problem efficiency gains arise by exploiting the sub-model structure through the use of two-step and weighted estimators, especially when the number of competing sub-models is few; but that this advantage may deteriorate or be lost altogether for some two-step estimators as the number of sub-models increases or as the distance between them decreases. Furthermore, it is demonstrated that weighted estimators, arising from properly chosen priors, outperform two-step estimators when there are many competing sub-models or when the sub-models are close to each other, whereas two-step estimators are preferred when the sub-models are highly distinguishable. The results have implications regarding model averaging and model selection issues.
本文考虑了高斯模型中离散参数的估计问题，该模型介于均值参数完全已知（固定）的模型和均值参数完全未知的模型之间。目标之一是理解两步过程的影响，即首先在有限数量的子模型中选择一个模型，然后在模型选择之后使用相同的样本数据估计感兴趣的参数。估计量被分为全局估计量、两步估计量和加权型估计量。全局型估计量忽略模型空间结构，两步估计量自适应地探索该结构并且可以与预检验估计量相关，加权估计量则受贝叶斯方法的启发。基于尺度不变二次损失函数，通过理论分析和模拟比较了它们的风险函数来评估其性能。结果表明，在方差估计问题中，通过使用两步估计量和加权估计量利用子模型结构可以提高效率，特别是当竞争子模型的数量较少时；但随着子模型数量的增加或它们之间距离的减小，对于某些两步估计量，这种优势可能会恶化或完全丧失。此外，结果表明，当存在许多竞争子模型或子模型彼此接近时，由适当选择的先验产生的加权估计量优于两步估计量，而当子模型高度可区分时，两步估计量更受青睐。这些结果对模型平均和模型选择问题具有启示意义。
