Exact saddlepoint approximations

The renormalized saddlepoint approximation to the probability density of an estimator often has a surprisingly low relative error over the whole admissible range of the parameter. In particular it is known to be exact for certain densities. This raises the question of how to characterize the class of such exact cases. The density of the mean of a univariate random sample is discussed. It is shown that the normal, gamma and inverse normal are the only possible densities for which the renormalized saddlepoint approximation reproduces exactly the density of the mean.