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 .