Canonical Parameterizations and Zero Parameter-Effects Curvature

Abstract

SUMMARY An equation derived by Hougaard (1982) characterizes certain canonical parameterizations of curved exponential families. The geometrical interpretation of Hougaard's equation is presented and, with this background, the relationship of work on Gaussian non-linear regression, by Bates and Watts (1980) and others, to that on general asymptotic inference, by Efron (1975) and Amari (1982a, b), is discussed. Hougaard's δ-parameterization is that in which there is zero “parameter-effects curvature” with respect to Amari's (1982b) α-connection geometry, when α= 1–2δ. For the generalized linear models of Nelder and Wedderburn (1972) there is a separate “parameter-effects” array for each α, and each is a convex combination of the exponential and mixture “parameter-effects” arrays. On the other hand, in non-linear regression, for any family of symmetric errors there is only one array and it is a constant multiple of that for the Gaussian family.