The present contribution investigates multivariate bootstrap procedures for
general stabilizing statistics, with specific application to topological data
analysis. Existing limit theorems for topological statistics prove difficult to
use in practice for the construction of confidence intervals, motivating the
use of the bootstrap in this capacity. However, the standard nonparametric
bootstrap does not directly provide for asymptotically valid confidence
intervals in some situations. A smoothed bootstrap procedure, instead, is shown
to give consistent estimation in these settings. The present work relates to
other general results in the area of stabilizing statistics, including central
limit theorems for functionals of Poisson and Binomial processes in the
critical regime. Specific statistics considered include the persistent Betti
numbers of \v{C}ech and Vietoris-Rips complexes over point sets in Rd, along with Euler characteristics, and the total edge length of the
k-nearest neighbor graph. Special emphasis is made throughout to weakening
the necessary conditions needed to establish bootstrap consistency. In
particular, the assumption of a continuous underlying density is not required.
A simulation study is provided to assess the performance of the smoothed
bootstrap for finite sample sizes, and the method is further applied to the
cosmic web dataset from the Sloan Digital Sky Survey (SDSS). Source code is
available at github.com/btroycraft/stabilizing_statistics_bootstrap.Comment: 59 pages, 3 figures. Restructured paper with alternate problem
settings moved to appendix. Rewrote data analysis and simulations study
sections to be more comprehensive, moved each to the end of the pape
