Bootstrap confidence regions based on M-estimators under nonstandard conditions

Suppose that a confidence region is desired for a subvector . of a multidimensional parameter .=(.,.), based on an M-estimator .^n=(.^n,.^n) calculated from a random sample of size n. Under nonstandard conditions .^n often converges at a nonregular rate rn, in which case consistent estimation of the distribution of rn(.^n..), 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) .^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.