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arXiv · 0906.2128

Tie-respecting bootstrap methods for estimating distributions of sets and functions of eigenvalues

Peter Hall·Young K. Lee·Byeong U. Park·Debashis Paul
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Abstract

Bootstrap methods are widely used for distribution estimation, although in some problems they are applicable only with difficulty. A case in point is that of estimating the distributions of eigenvalue estimators, or of functions of those estimators, when one or more of the true eigenvalues are tied. The $m$-out-of-$n$ bootstrap can be used to deal with problems of this general type, but it is very sensitive to the choice of $m$. In this paper we propose a new approach, where a tie diagnostic is used to determine the locations of ties, and parameter estimates are adjusted accordingly. Our tie diagnostic is governed by a probability level, $β$, which in principle is an analogue of $m$ in the $m$-out-of-$n$ bootstrap. However, the tie-respecting bootstrap (TRB) is remarkably robust against the choice of $β$. This makes the TRB significantly more attractive than the $m$-out-of-$n$ bootstrap, where the value of $m$ has substantial influence on the final result. The TRB can be used very generally; for example, to test hypotheses about, or construct confidence regions for, the proportion of variability explained by a set of principal components. It is suitable for both finite-dimensional data and functional data.

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Peter Hall, Young K. Lee, Byeong U. Park, Debashis Paul. 2009-06-11. Tie-respecting bootstrap methods for estimating distributions of sets and functions of eigenvalues. https://doi.org/10.3150/08-bej154

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