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.
"Tie-respecting bootstrap methods for estimating distributions of sets and functions of eigenvalues." Bernoulli 15 (2) 380 - 401, May 2009. https://doi.org/10.3150/08-BEJ154