Bootstrap confidence regions based on M-estimators under nonstandard conditions

Suppose that a confidence region is desired for a subvector $\theta $ of a multidimensional parameter $\xi =(\theta ,\psi )$, based on an M-estimator $\hat{\xi }_{n}=(\hat{\theta }_{n},\hat{\psi }_{n})$ calculated from a random sample of size $n$. Under nonstandard conditions $\hat{\xi }_{n}$ often converges at a nonregular rate $r_{n}$, in which case consistent estimation of the distribution of $r_{n}(\hat{\theta }_{n}-\theta )$, a pivot commonly chosen for confidence region construction, is most conveniently effected by the $m$ out of $n$ bootstrap. The above choice of pivot has three drawbacks: (i) the shape of the region is either subjectively prescribed or controlled by a computationally intensive depth function; (ii) the region is not transformation equivariant; (iii) $\hat{\xi }_{n}$ may not be uniquely defined. To resolve the above difficulties, we propose a one-dimensional pivot derived from the criterion function, and prove that its distribution can be consistently estimated by the $m$ out of $n$ bootstrap, or by a modified version of the perturbation bootstrap. This leads to a new method for constructing confidence regions which are transformation equivariant and have shapes driven solely by the criterion function. A subsampling procedure is proposed for selecting $m$ in practice. Empirical performance of the new method is illustrated with examples drawn from different nonstandard M-estimation settings. Extension of our theory to row-wise independent triangular arrays is also explored.