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February 2020 Bootstrap confidence regions based on M-estimators under nonstandard conditions
Stephen M. S. Lee, Puyudi Yang
Ann. Statist. 48(1): 274-299 (February 2020). DOI: 10.1214/18-AOS1803


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.


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Stephen M. S. Lee. Puyudi Yang. "Bootstrap confidence regions based on M-estimators under nonstandard conditions." Ann. Statist. 48 (1) 274 - 299, February 2020.


Received: 1 August 2017; Revised: 1 December 2018; Published: February 2020
First available in Project Euclid: 17 February 2020

zbMATH: 07196539
MathSciNet: MR4065162
Digital Object Identifier: 10.1214/18-AOS1803

Primary: 62G09, 62G15

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.48 • No. 1 • February 2020
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