Explaining the saddlepoint approximation

Saddlepoint approximations are powerful tools for obtaining accurate expressions for densities and distribution functions. \Ve give an elementary motivation and explanation of saddlepoint approximation techniques, stressing the connection with the familiar Taylor series expansions and the Laplace approximation of integrals. Saddlepoint methods are applied to the convolution of simple densities and, using the Fourier inversion formula, the saddlepoint approximation to the density of a random variable is derived. \Ve then apply the method to densities of sample means of iid random variables, and also demonstrate the technique for approximating the density of a maximum likelihood estimator in exponential families.