Bartlett's bias correction to the profile score function is a saddlepoint correction

Abstract

SUMMARY Bartlett's bias correction to the profile score function is proved equal, to first order, to the leading correction term of the double saddlepoint approximation to the conditional score function in canonical exponential families. Gart (1985, 1987) has pointed out the effectiveness of Bartlett's bias correction to the profile score function in approximate conditional binomial likelihood analysis. While the leading term in Bartlett's (1953, 1955) expansion of the bias function ignores terms of order O(n-2), the first-term correction works well for conditional analysis because, as shown in this note, in canonical exponential families it agrees with the first correction term of the double saddlepoint approximation to the conditional score function, and that approximation is accurate up to terms of order O(n')