A note on bootstrapping \(M\)-estimators in ARMA models (Q2703240)

Bootstraping a linear approximation to the \(M\)-estimator in autoregressive moving-average (ARMA) models, proposed by \textit{J.-P. Kreiss} and \textit{J. Franke} [J. Time Ser. Anal. 13, No. 4, 297-317 (1992; Zbl 0787.62092)], is targeted to approximate the sampling distribution of a linear approximation to the \(M\)-estimator rather than the \(M\)-estimator itself. From the first-order asymptotic consideration it is equally justified since the \(M\)-estimator and the linear approximation both have the same limiting normal distribution. However, if one considers their higher order asymptotic behaviour (say, in terms of Edgeworth expansions) such an equivalence may not exist, since the quadratic term of the Taylor series approximation of the estimator typically has a contribution even in a two-term Edgeworth expansion.NEWLINENEWLINENEWLINEThus, the authors proposal is to apply the bootstrap principle directly to the \(M\)-estimator which is what the usual (or naive) approach would dictate. Although this may impose little more burden on the computation, the results are generally more accurate than those obtained by the Kreiss and Franke bootstrap. A number of simulation results is presented to compare the two procedures for estimating the sampling distribution of an \(M\)-estimator. The theoretical asymptotic validity of the standard bootstrap applied to the \(M\)-estimator is established.