© 1993 by Biometrika Trust
Analytical approximations to conditional distribution functions
1Department of Statistics, Sequoia Hall, Stanford University Stanford, California 94305-4065, U.S.A.
2Statistical Laboratory, University of Cambridge 16 Mill Lane, Cambridge CB2 1SB, England
Conditional inference plays a central role in statistics, but determination of relevant conditional distributions is often difficult. We develop analytical procedures that are accurate and easy to apply for approximating conditional distribution functions. For a continuous random vector X = (X1,..., Xp), we estimate the conditional distribution function of Y1 given Y2,..., Yk(k 
p), where each Yi is a smooth function of X. Previous approaches have dealt with the cases where the variable whose conditional distribution is sought is a linear function of means, and where there are p – 1 conditioning variables. However, sometimes the statistic of interest is a nonlinear function of means and it is advantageous to condition on a lower-dimensional ancillary statistic. Our procedure first involves approximating the marginal density function for Y1,..., Yk, by an approach of Phillips (1983) and Tierney, Kass & Kadane (1989). An accurate approximation to the required conditional probability is then obtained by applying a marginal tail probability approximation of DiCiccio & Martin (1991) to the conditional density of Y1 given Y2,..., Yk. Our method is illustrated in several examples, including one which uses a saddlepoint approximation for the density of X, and the method is applied for conditional bootstrap inference.
Key Words: Ancillary statistic • Conditional bootstrap • Laplace's method • Marginal density • Saddlepoint approximation • Tail probability approximation