Accurate confidence limits for scalar functions of vector M-estimands

This paper concerns high-order inference for scalar parameters that are estimated by functions of multivariate M-estimators. Asymptotic formulae for the bias and skewness of the studentised statistic are derived. Although these formulae appear complicated, they can be evaluated easily by using matrix operations and numerical differentiation. Various methods for constructing second-order accurate confidence limits are discussed, including a method based on skewness-reducing transformations and a generalisation of the ABC method. The use of the skewness-reducing transformations is closely related to empirical likelihood; expressing the studentised statistic in terms of a skewness-reducing reparameterisation brings the standard asymptotic intervals closer in shape to empirical likelihood intervals. The improvement in one- and two-sided coverage accuracy achieved by taking the bias and skewness into account is illustrated in numerical examples. It is found in the examples that taking skewness into account by reparameterisation or parameterisation invariance yields better coverage accuracy than correcting for skewness by polynomial expansions.