ANALYTICAL APPROXIMATIONS TO CONDITIONAL DISTRIBUTION-FUNCTIONS

Conditional inference plays a central role in statistics, but determin ation of relevant conditional distributions is often difficult. We dev elop analytical procedures that are accurate and easy to apply for app roximating conditional distribution functions. For a continuous random vector X = (X(1),..., X(p)), we estimate the conditional distribution function of Y-1 given Y-2,..., Y-k (k less than or equal to p), where each Y-i is a smooth function of X. Previous approaches have dealt wi th the cases where the variable whose conditional distribution is soug ht is a linear function of means, and where there are p-1 conditioning variables. However, sometimes the statistic of interest is a nonlinea r function of means and it is advantageous to condition on a lower-dim ensional ancillary statistic. Our procedure first involves approximati ng the marginal density function for y(1),...,Y-k, by an approach of P hillips (1983) and Tierney, Kass & Kadane (1989). An accurate approxim ation to the required conditional probability is then obtained by appl ying a marginal tail probability approximation of DiCiccio and Martin (1991) to the conditional density of Y-1 given Y-2,...,Y-k. Our method is illustrated in several examples, including one which uses a saddle point approximation for the density of X, and the method is applied fo r conditional bootstrap inference.