Saddlepoint approximations

Saddlepoint approximations for densities and tail-area probabilities of certain statistics turn out to be surprisingly accurate down to very small sample sizes. Although being asymptotic in spirit (with respect to the sample size, for example) they sometimes give accurate approximations even down to a sample size of one. Moreover, the error of approximation is relative rather than absolute and hence they perform very well in the tails where other competitive methods usually fail. These are the important definite advantages of saddlepoint approximations in Statistics. I have applied saddlepoint methods to derive the joint density of slope and intercept in the classical Linear Structural Relationship model.A simulation study supports the good performance for sample sizes such as 5 or 10. Similar in spirit to the Saddlepoint method is the Wiener germ approximation which I have applied to approximating the non-central chi-square distribution and its quantiles. Here the approximation is asymptotic with respect to the degrees of freedom of the distribution.It turns out to be very accurate even down to one degree of freedom and performs better than any other approximation of the non-central chi-square that has been suggested in key reference books like the Johnson and Kotz manual on continuous univariate distributions.

When a saddlepoint approximation is "inverted", it could be used for quantile evaluation.This is an interesting alternative to the standard Cornish-Fisher method for quantile evaluation. We have suggested an explicit approximation of the inversion of the saddlepoint approximation for the purpose of quantile evaluation and have demonstrated numerically its superior performance in comparison to the Cornish-Fisher method.

(Higher order) Edgeworth expansions deliver better approximations to the limiting distribution of a statistic in comparison to the first order approximation delivered by the normal distribution. Using these higher order expansions can be beneficial especially when the sample size is small to moderate. Although these expansions are well known for some standard situations, they may be non-trivial to derive for more complicated estimators such as the kernel estimator of the p-th quantile. We have studied the Edgeworth expansion of the kernel estimator of the p-th quantile and have demonstrated analytically and numerically the size and order of the improvement achieved in comparison to the normal approximation.