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

TL;DR

This paper introduces the tie-respecting bootstrap (TRB), a robust method for estimating distributions of eigenvalues and their functions, especially when eigenvalues are tied, improving over traditional bootstrap approaches.

Contribution

The paper proposes a new tie diagnostic and the TRB method, which is more robust and broadly applicable for eigenvalue distribution estimation with tied eigenvalues.

Findings

TRB is robust against the choice of the probability level $eta$.

TRB outperforms the $m$-out-of-$n$ bootstrap in tied eigenvalue problems.

Applicable to both finite-dimensional and functional data.

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, $\beta$, 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 $\beta$. 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.