Analytical approximations to conditional distribution functions

Abstract

Abstract : 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 we estimate conditional tail probabilities are smooth functions 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, in many practical circumstances 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 by an approach of Phillips (1983) and Tierney, Kass and Kadane (1989). An accurate approximation to the required conditional probability is then obtained by applying a marginal tail probability approximation of DiCiccio and Martin (1991) to the conditional density. 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.