Small-Sample Confidence Intervals

Abstract

Abstract In this article we present a technique for constructing one- or two-sided confidence intervals, which are second-order correct in terms of coverage, for either parametric or nonparametric problems. The construction is valid in the presence of nuisance parameters. The situation we consider is this: there are p parameters and we want a confidence interval for some function of them, possibly one of the parameters itself. The p parameters are estimated by M-estimates, which means they are obtained as a solution of a system of equations. Maximum likelihood estimates are included as a special case. The essential intermediate result, given in Equation (3.4), says that the estimated parameter of interest, θ, can be written as a mean, up to order O p (1/√n). The representation (3.4) is attained when θ = θ(η) is a smooth function of the parameters η and is the solution of a well-behaved system of equations. We avoid the use of pivots and strive to obtain accurate coverage. Confidence intervals are constructed from a series of tests for the natural parameter of a one-parameter exponential family. We use the Lugannani and Rice (1980) tail area approximation to calculate bootstrap P values for the test statistic.