Abstract: A saddlepoint approximation to a nonlinear function of a sum of independent and identically distributed random variables is provided, using Watson’s lemma applied to a double integral. The accuracy of the expansion depends on the smoothness of the nonlinear function. The result is applied to variance approximation, inverse moments, convergence rate, likelihood ratio statistics, and harmonic mean, and compared with results in the literature.

Key words and phrases: Asymptotic expansion, nonlinear moment, saddlepoint approximation, Watson’s lemma.