Uniform Confidence Band for Optimal Transport Map on One-Dimensional Data

TL;DR

This study derives a limit distribution of a uniform norm of an estimation error for the OT map, and then develops a uniform confidence band based on it, and develops a smoothed bootstrap method with its validation and guarantee on an asymptotic coverage probability of the confidence band.

Abstract

We develop a statistical inference method for an optimal transport map between distributions on real numbers with uniform confidence bands. The concept of optimal transport (OT) is used to measure distances between distributions, and OT maps are used to construct the distance. OT has been applied in many fields in recent years, and its statistical properties have attracted much interest. In particular, since the OT map is a function, a uniform norm-based statistical inference is significant for visualization and interpretation. In this study, we derive a limit distribution of a uniform norm of an estimation error for the OT map, and then develop a uniform confidence band based on it. In addition to our limit theorem, we develop a bootstrap method with kernel smoothing, then also derive its validation and guarantee on an asymptotic coverage probability of the confidence band. Our proof is based on the functional delta method and the representation of OT maps on the reals.

Figures (5)

• Figure 1: Plots of the coverage probabilities (left) and the median of average widths (right) of the simulated uniform confidence bands on $[-2.5,2.5]$ as functions of sample size $n$.
• Figure 2: Examples of $(1-\alpha)$-level uniform confidence bands on $[-2.5,2.5]$, for three different values of $\alpha$ and three different values of the bandwidth parameter $\beta$, based on specific samples of size $200, 700$ and $2000$.
• Figure 3: Analysis of the distribution shifts in ages of death from the year 2001 to 2021 using our uniform confidence bands.
• Figure 4: Plots of the average of the coverage probabilities (left) and the median of average widths (right) of the simulated pointwise confidence intervals over $[-2.5,2.5]$ as functions of sample size $n$.
• Figure 5: Examples of $(1-\alpha)$-level pointwise confidence intervals over $[-2.5,2.5]$, for three different values of $\alpha$, based on specific samples of size $200, 700$ and $2000$.

Theorems & Definitions (24)

• Remark 1: Choice of kernels and bandwidth
• Proposition 1: Bahadur Representation
• Lemma 2
• Remark 2: Comparison with a pointwise confidence interval
• Remark 3: Relation to ROC curves
• Theorem 3: Bootstrap consistency
• Corollary 4: Asymptotic Validity of Bootstrap Confidence Band
• Lemma 5
• Proposition 6: Bahadur Representation of 1D Transport Map
• proof : Proof of Lemma \ref{['lemma:hadamarddiff']}