BOOTSTRAP CONFIDENCE REGIONS BASED ON M-ESTIMATORS UNDER NONSTANDARD CONDITIONS

• 成果类型: Article
• 署名作者: Lee, Stephen M. S.; Yang, Puyudi
• 署名单位: University of Hong Kong; University of California System; University of California Davis
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1803
• 发表日期: 2020
• 页码: 274-299
• 关键词: simultaneous inference; generalized bootstrap; asymptotic properties; parameter; CHOICE; depth; sets

摘要

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