Variance Estimation in a Model With Gaussian Submodels

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.