Variance Estimation in a Model With Gaussian Submodels
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具有高斯子模型结构的方差估计
- 研究问题
- 在均值参数受到有限子模型结构约束的高斯模型中，使用同一数据先选模型再估计方差，会如何影响风险及估计效率？
- 方法与思路
- 作者区分忽略结构的全局估计、适应性选择的两步估计及贝叶斯启发的加权估计，并以尺度不变二次损失进行理论和模拟比较。
- 材料陈述的贡献
- 摘要称，利用子模型结构可降低风险；子模型较多或接近时，合适先验产生的加权估计更有优势，而高度可区分时两步估计更合适。
- 阅读关注点
- 应核对子模型之间距离的定义、选择规则与先验设定，明确风险比较适用的参数范围，以及同样本选择如何影响方差估计。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明；摘要未说明风险界、先验细节和比较条件，不能理解为加权或两步估计的普遍优越性。
原文摘要
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.