On the variance estimator and its bounds in general linear models under linear restrictions

Abstract

In the article, the error variance estimator σ ^ 2 and its finite sample size bounds are investigated in the framework of a general linear model under linear restrictions (gLMLR) where no distributions are assumed except for some moment conditions and the covariance matrix of the response may be singular, which extends the results in Lindholm and Wahl (2020). In particular, we overcome the difficulty that the matrix of quadratic form in σ ^ 2 is not idempotent. The estimator is obtained by the proposed restricted and proxy-type generalized least square procedure. Meanwhile, the estimable function of the regression coefficient β is estimated, too. Besides, the finite sample properties of these estimators and their consistency are also established. In the end, a simulation study and a real data analysis are done to examine the finite performance of these estimators.