Explaining the saddlepoint approximation

Saddlepoint approximations are powerful tools for obtaining accurate expressions for densities and distribution functions. We give an elementary motivation and explanation of approximation techniques, starting with Taylor series expansions and progressing to the Laplace approximation of integrals. These approximations are illustrated with examples of the convolution of simple densities. We then turn to the saddlepoint approximation and, using both the Fourier inversion formula and Edgeworth expansions, we derive the saddlepoint approximation to the density of a single random variable. We next approximate the density of the sample mean of lid random variables, and also demonstrate the technique for approximating the density of a maximum likelihood estimator in exponential families.