On a Convexity of Saddlepoint and its Curvature

Abstract

If a non-degenerated probability distribution F has an analytic chracteristic function, there exists a strictly increasing function, so called a saddlepoint. The saddlepoint uniguely determines the distribution by its behavior only in a neighborhood of the expectation, and is being a line if and only if F is the normal (Takeuchi (2013)). In this paper we shall show that a local convexity of the saddlepoint includes much information of the corresponding distribution, with using sp-curvature which exists even for non-absolutely continuous distributions, and also for non-parametric case. It should be noted that the sp-curvature can determine the corresponding probability distribution uniquely, with moments up to second order, and it naturally explains the performance of the asymptotic normality of a statistics.