Möbius inversion and the iterated bootstrap

Abstract

Estimating nonlinear functionals of probability distributions from samples is a fundamental statistical problem. The "plug-in" estimator obtained by applying the target functional to the empirical distribution of samples is biased. Resampling methods such as the bootstrap derive artificial datasets from the original one by resampling. Comparing the outcome of the plug-in estimator in the original and resampled datasets allows estimating and thus correcting the bias. In the asymptotic setting, iterations of this procedure attain an arbitrarily high order of bias correction, but finite sample results are scarce. This work develops a new theoretical understanding of bootstrap bias correction by viewing it as an iterative linear solver for the combinatorial operation of Möbius inversion. It sharply characterizes the regime of linear convergence of the bootstrap bias reduction for moment polynomials. It uses these results to show its superalgebraic convergence rate for band-limited functionals. Finally, it derives a modified bootstrap iteration enabling the unbiased estimation of unknown order-$m$ moment polynomials in $m$ bootstrap iterations.

Figures (6)

• Figure 1: Biased plug-in estimators. Nonlinear functions $\Phi$ of moments of empirical distributions $\mathbold{X}^1 \sim_N \mathbold{X}^0$ are biased estimators of the same function of moments of the population distribution $\mathbold{X}^0$.
• Figure 1: Bootstrap bias correction. In outcomes where the empirical distribution $\mathbold{X}^1$ represents the population $\mathbold{X}^0$ well, the bias in the plugin estimate using a resampled $\mathbold{X}^2 \sim_{N} \mathbold{X}^1$ reveals the bias due to plugin estimation. Subtracting it from the estimates allows accounting for the alternative outcomes where $\mathbold{X}^1$ does not represent $\mathbold{X}^0$ well. This is the fundamental mechanism underlying bootstrap bias correction.
• Figure 1: Möbius inversion on the partition lattice. The matrix $\overline{\mathbold{S}}$ acts on lattice functions, implementing the discrete antiderivative. Its inverse, the Möbius matrix, implements a lattice version of differentiation. In this work we draw Hasse diagrams of the partition lattice with the finest partition at the top and the coarsest partition at the bottom, which is the opposite of the usual convention. This is to match the order of rows and columns of the associated matrices, which is given by the standard partial order of the partition lattice.
• Figure 1: Multiindices and partitions. To each multiindex $\mathbold{i} \in \{1, \ldots, N\}^m$, such as the the three examples $(3, 3, 5), (1, 1, 1,), (2, 4, 4) \in \{1, \ldots, 5\}^3$, we can associate a partition $\varsigma\left(\mathbold{i}\right) \in \Pi(m)$ obtained by assigning to the same part those indices that are mapped to the same value. For $\pi \in \Pi(m)$, we write $\mathbold{i} \in \mathbb{I}^{=}_{N}\left(\pi\right)$ if the multiindex is equivalent to $\pi$ in the sense that $\varsigma(\mathbold{i}) = \pi$. We write $\mathbold{i} \in \mathbb{I}^{\geq}_{N}\left(\pi\right)$ if the multiindex is coarser than $\pi$ in the sense that $\varsigma(\mathbold{i}) \geq \pi$, meaning that $\varsigma(\mathbold{i})$ can be obtained by merging parts of $\pi$.
• Figure 1: General moment polynomials fit into our framework by considering Möbius inversion on partially ordered sets obtained as unions of partition lattices of different order, with possibly duplicated variables. As explained in the caption of \ref{['fig:moebius']}, we draw the Hasse diagram in the finest-on-top orientation.

Theorems & Definitions (31)

• Lemma 4.1
• Proof 1
• Lemma 4.2
• Proof 2
• Lemma 4.3
• Proof 3
• Theorem 4.4
• Proof 4
• Corollary 4.5
• Theorem 4.6