Saddlepoint approximations for estimating equations

SUMMARY The saddlepoint method is used to approximate to the distribution of an estimator defined by an estimating equation. Two different approaches are available, one of which is shown to be equivalent to the technique of Field & Hampel (1982). Two recent formulae for the tail probability, due to Lugannani & Rice (1980) and to Robinson (1982) respectively, which are uniformly accurate over the whole range of the estimator, are compared numerically with the exact results and those computed by Field & Hampel. They are found to be of comparable accuracy while avoiding the use of numerical integration. The most accurate is that of Lugannani & Rice. Hampel (1974) introduced a new technique for approximating to the probability density of an estimator defined by an estimating equation. It is an example of what he called 'small sample asymptotics' where high accuracy is achieved for quite small sample sizes n, even down to single figures. In the original version it gave an approximation to the logarithmic derivative of the density function which was integrated numerically to get the density. The distribution function was obtained by a second numerical integration, which could then be used to renormalize both the density and the distribution function. Field & Hampel (1982) develop the technique in detail and compare its performance with that of other approximation methods. As Hampel had pointed out, his approach is closely related to the saddlepoint method of Daniels (1954) which was applied to sample means and ratios of means. Following the private communication referred to by Field & Hampel (1982, p. 31) it was realized that the first numerical integration was unnecessary and that Hampel's approach could be shortened to give a direct approximation to the density which is in fact a saddlepoint approximation. The purpose of the present paper is to extend the use of the saddlepoint method to estimating equations. There are two distinct ways of doing this which lead to different approximations of similar accuracy. One appears to be more convenient for approxi- mating to tail probabilities by numerical integration, and was used to compute the saddlepoint approximations quoted in Field & Hampel's Tables 1 and 2; The other gives the density directly in a form equivalent to their equation (4 3), which is then integrated