Canonical regression models for exponential families

Abstract

Every canonical exponential family generates a regression model that is also a canonical exponential family. For example, for the normal it is the set of say n normal observations, with means and variances of the form µ N = y N w/z N w and v N = 1/z N w for 1 ≤ N ≤ n, where w is the canonical regression parameter and {y N , z N } are known vectors of the same dimension. This is a much richer model than the usual linear regression model with means and variances µ N = x N β and v N = v for 1 ≤ N ≤ n. We give the first few terms of the Edgeworth-Cornish-Fisher expansions for the distribution, density and quantiles of any smooth function of the maximum likelihood estimate of w, and the associated expansion for the confidence limit of any smooth function of w.