A necessary and sufficient condition for convergence in distribution of the P-P process in $L^1[0,1]$

TL;DR

The paper characterizes the convergence in distribution of the percentile-percentile (P-P) process in $L^1[0,1]$ by a necessary and sufficient condition: $\sqrt{n}(R_n-R)$ converges iff the P-P curve $R$ is absolutely continuous, with $R(u)=F(Q(u))$ and $R_n(u)=F_n(Q_n(u))$. In the AC case the limit is a Gaussian process $\mathcal{R}(u)=\mathcal{B}_1(R(u))-r(u)\mathcal{B}_2(u)$, and the authors establish bootstrap validity via a delta-method framework and Hadamard differentiability of a composition map. The necessity part shows non-convergence whenever $R$ is not AC, using a Brownian-bridge-based argument and Lebesgue-Stieltjes measures. The paper also extends the results to independent samples with differing sizes, adjusting the limit process and bootstrap construction accordingly. Overall, it provides a rigorous, actionable criterion and a complete inferential framework for P-P plots in $L^1[0,1]$.

Abstract

We establish that the procentile-procentile (P-P) process constructed from a random sample of pairs converges in distribution in $L^1[0,1]$ if and only if the P-P curve is absolutely continuous. If the P-P process converges in distribution then it may be approximated using the bootstrap.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• proof : Proof of Theorem \ref{['thm:sufficiency']}
• Theorem 3
• Lemma 2
• proof
• Lemma 3
• proof