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.
References
Alemayehu, D. (1988). Bootstrapping the latent roots of certain random matrices. Comm. Statist. Simulation Comput. 17 857–869.
Andrews, D.W.K. (2000). Inconsistency of the bootstrap when a parameter is on the boundary of the parameter space. Econometrica 68 399–405.
Beran, R. (1988). Balanced simultaneous confidence sets. J. Amer. Statist. Assoc. 83 679–686.
Beran, R. (1997). Diagnosing bootstrap success. Ann. Inst. Statist. Math. 49 1–24.
Beran, R. and Srivastava, M.S. (1985). Bootstrap tests and confidence regions for functions of a covariance matrix. Ann. Statist. 13 95–115.
Besse, P. (1992). PCA stability and choice of dimensionality. Statist. Probab. Lett. 13 405–410.
Besse, P. and Ramsay, J.O. (1986). Principal components analysis of sampled functions. Psychometrika 51 285–311.
Bickel, P.J. (2003). Unorthodox bootstraps. J. Korean Statist. Soc. 32 213–224.
Bickel, P.J., Götze, F. and Van Zwet, W. (1997). Resampling fewer than n observations: gains, losses, and remedies for losses. Statist. Sinica 7 1–31.
Bosq, D. (1989). Propriétés des opérateurs de covariance empiriques d’un processus stationnaire hilbertien. C.R. Acad. Sci. Paris Sér. I 309 873–875.
Bosq, D. (2000). Linear Processes in Function Spaces. Theory and Applications. Springer Lecture Notes in Statistics 149. New York: Springer.
Breiman, L. (1996). Bagging predictors. Machine Learning 24 123–140.
Dauxois, J., Pousse, A. and Romain, Y. (1982). Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference. J. Multivariate Anal. 12 136–154.
Dümbgen, L. (1993). On nondifferentiable functions and the bootstrap. Probab. Theory Related Fields 95 125–140.
Dümbgen, L. (1995). Minimax tests for convex cones. Ann. Inst. Statist. Math. 47 155–165.
Eaton, M.L. and Tyler, D.E. (1991). On Wielandt’s inequality and its application to the asymptotic distribution of the eigenvalues of a random symmetric matrix. Ann. Statist. 19 260–271.
Hall, P., Härdle, W. and Simar, L. (1993). On the inconsistency of bootstrap distribution estimators. Comput. Statist. Data Anal. 16 11–18.
Hall, P. and Hosseini-Nasab, M. (2006). On properties of functional principal components analysis. J. Roy. Statist. Soc. Ser. B 68 109–126.
Hall, P. and Jing, B.-Y. (1995). Uniform coverage bounds for confidence intervals and Berry–Esseen theorems for Edgeworth expansion. Ann. Statist. 23 363–375.
Hall, P., Lee, Y.K., Park, B.U. and Paul, D. (2007). Tie-respecting bootstrap methods for estimating distributions of sets and functions of eigenvalues. Available at http://stat.snu.ac.kr/theostat/papers/bernoulli-trb.pdf.
Nagao, H. (1988). On the jackknife statistics for eigenvalues and eigenvectors of a correlation matrix. Ann. Inst. Statist. Math. 40 477–489.
Ramsay, J.O. and Dalzell, C.J. (1991). Some tools for functional data analysis (with discussion). J. Roy. Statist. Soc. Ser. B 53 539–572.
Ramsay, J.O. and Silverman, B.W. (2002). Applied Functional Data Analysis: Methods and Case Studies. New York: Springer.
Ramsay, J.O. and Silverman, B.W. (2005). Functional Data Analysis, 2nd ed. New York: Springer.
Rice, J.A. and Silverman, B.W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. J. Roy. Statist. Soc. Ser. B 53 233–243.
Romano, J. (2004). On non-parametric testing, the uniform behaviour of the t-test, and related problems. Scand. J. Statist. 31 567–584.
Schott, J.R. (2006). A high-dimensional test for the equality of the smallest eigenvalues of a covariance matrix. J. Multivariate. Anal. 97 827–843.
Zhang, J. and Boos, D.D. (1993). Testing hypotheses about covariance matrices using bootstrap methods. Comm. Statist. Theory Methods 22 723–739.
