Recent inference theory and enigmatic examples

1.1. Exponential and general models. For an exponential model, suppose φ and s are the canonical parameter and canonical variable, with {1, φ1(θ), . . . , φp(θ)} linearly independent. . If not an exponential model, a third order approximation is available using φ′(θ) = (∂/∂V )`(θ; y)|y0 where V = (v1, . . . , yp) are inference directions at y. If these directions V are not obvious, they can be extracted by differentiating a full quantile function at the observed data: V = (∂/∂θ)q(θ; y)|θ̂0,y0 ; see Fraser and Reid [3] 1.2. Density for canonical variable. The density for the canonical variable s can be presented in saddlepoint format to third order as