The Theory And Applications Of The Saddlepoint Approximation

Saddlepoint approximations are powerful tools for obtaining accurate expressions for densities and distribution functions.The advantage of the saddlepoint approximations are: the computation speed is quick, the formula is succinct, the effect of the approximation is Accurate especially under the situations of the small samples and thetail probability, it can improve the errors of approximation to o{n^-1) and eveno(n^-2/3)in some cases.So it has the widespread application in statistics , finance ,medicine etc.In this article , the author first apply the Laplace Approximation to get the approximation of the marginal densities,then apply the saddlepoint approximation to get the density of MLE of the Pareto distribution and the approximation of the density and the distribution of Non-central chi-squared distribution.Finally the author apply the saddlepoint approximation to the Infectious diseases model to simplify the calculation.