Asymptotics with Increasing Dimension for Robust Regression with Applications to the Bootstrap

A stochastic expansion for $M$-estimates in linear models with many parameters is derived under the weak condition $\kappa n^{1/3}(\log n)^{2/3} \rightarrow 0$, where $n$ is the sample size and $\kappa$ the maximal diagonal element of the hat matrix. The expansion is used to study the asymptotic distribution of linear contrasts and the consistency of the bootstrap. In particular, it turns out that bootstrap works in cases where the usual asymptotic approach fails.