Bootstrap confidence regions based on M-estimators under nonstandard conditions

Abstract

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