Some Geometry for the Maximal Invariant in Linear Regression

The maximal invariant forms the basis of a well established theory on hypothesis testing on the covariance structure in linear regression, see Lehman (1997). This paper examines the geometry of the maximal invariant. In particular it derives explicit expressions for both Fisher information and statistical curvature, see Efron (1975). The results apply for any sample size, for any sufficiently differentiable covariance structure and across a variety of sample densities. The results are illustrated for regressions involving autoregressive and moving average errors. Specifically, the effects of different specifications of the mean and of non-stationarity and non-invertibility may be quantified.