## Second‐order inference for generalized least squares

### Abstract

Confidence regions for generalized least squares are commonly derived from a measure of departure calculated on the tangent plane at the MLE or on the tangent plane at the true value; the first gives approximate confidence regions, the second exact. For surfaces with curvature, indeed with varying curvature, the exact regions typically are not likelihood regions and can include parameter values of highest and of lowest likelihood. This paper develops an alternative approach to deriving exact confidence regions and uses both surface curvature and distance from the surface as supporting ingredients. For this, conditionality is invoked in two ways beyond that supported by the usual conditionality principle. For the case of normal error the ordinary chi‐squared departure is replaced by a Von Mises‐type angular (or cosine) departure which is assessed using curvature properties in the data direction and radial distance of the data from the regression surface. For the usual linear model (constant curvature equal to zero) the method coincides with the ordinary tests and confidence regions; for the case of constant nonzero curvature, the method generalizes to spheres and sphere‐cylinders the Fisher (Statistical Methods and Scientific Inference, 1956) analysis of a rotationally symmetric normal on ℝ2 with mean constrained to a circle. The effects of conditioning are indicated by a computer plot for obtaining 95% confidence.

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