General saddlepoint approximations in the bootstrap

In Easton and Ronchetti (1986), a method of general saddlepoint approximations is proposed and shown useful, especially in the case of small sample sizes. A possible improvement of the method is suggested to prevent its potential deficiencies and increase its applicability. Easton and Ronchetti's method and its modified version are extended to bootstrap applications. These results provide a satisfactory answer to Davison and Hinkley's (1988) open question on the bootstrap distribution in the AR(1) model.