Saddlepoint approximations to Studentized bootstrap distributions based on M-estimators (Q2488395)

Let the data be i.i.d. observations \(X_1,\dots,X_n\). Assume that the statistic of interest \(T\) and some nuisance statistic \(S=(S_1,\dots,S_{d-1})\) satisfy \(d\) estimating equations \(\sum_{i=1}^n\psi(X_i,t,s)=0\). The author describes a saddlepoint approximation to the distribution of the Studentized version of \(T\) with application of the Laplace approximation to the integral of the obtained joint density \(f_{t,s}\) by \(s\). The technique is applied to derive approximations for distributions of Studentized Huber's M-estimates of location parameters. Results of approximations are assessed on some small sets of real data. The Laplace approximation is compared with the results of numerical integration.