Abstract
Takeuchi (2013) showed that the saddlepoint uniquely determines its distribution under the existence of the analytic characteristic functions. In this paper we shall express the saddlepoint as an envelope (sp-curve) on the statistical model manifold ℳ of the normal distributions. The length of the sp-curve on ℳ is shown to be not depending on the coordinates, and is crucial for the asymptotic normality of the standardized sample mean. The sp-transform is defined to get the sp-curves, and is shown to be bijective. With using this property, we can say that any distribution which has saddlepoint can be generated by a family of the normal distributions.
