Variance Estimation in a Model With Gaussian Submodels
This article considers the problem of estimating the dispersion parameter in a Gaussian model that 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 submodels, 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 estimators. Whereas the global-type estimators ignore the model space structure, the two-step estimators explore the structure adaptively and can be related to pretest 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 submodel structure through the use of two-step and weighted estimators, especially when the number of competing submodels is few, but that this advantage may deteriorate or be lost altogether for some two-step estimators as the number of submodels increases or 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 submodels or when the submodels are close to each other, whereas two-step estimators are preferred when the submodels are highly distinguishable. The results have implications for model averaging and model selection issues.