Saddlepoint approximations for estimating equations

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