On the empirical saddlepoint approximation

Abstract

The properties of the saddlepoint approximation are investigated when the required cumulant generating function is obtained empirically. Properties of the empirical moment generating function and empirical cumulant generating function and derivatives of these processes which are needed for this study are derived first, in particular their uniform consistency, moment structure, and weak convergence to normality are established. A numerical investigation exploring use of the empirical saddlepoint approximation as a tool in density estimation is discussed briefly.