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Uniform mean estimation for monotonic processes

Eugenio Clerico, Hamish E Flynn, Patrick Rebeschini

TL;DR

This work tackles the problem of uniformly estimating the mean of monotonic stochastic processes, such as CDFs, from i.i.d. data by constructing anytime-valid, variance-adaptive confidence bands that hold over the entire domain. It blends the coin-betting concentration machinery with a continuous PAC-Bayes-style union bound that leverages monotonicity to achieve uniform validity across $y$. For CDF estimation, the authors exploit the piecewise-constant nature of the empirical CDF to obtain simple, computable bounds that demonstrate state-of-the-art performance in simulations. They present relaxations based on binary relative entropy and variance-adaptive refinements, discuss asymptotic rates (notably $O\left(\sqrt{\frac{\log T}{T}}\right)$), and note potential improvements via advanced betting schemes. Overall, the approach yields interpretable, tight confidence bands with practical implications for uniform inference on monotone mean functions and related risk measures.

Abstract

We consider the problem of deriving uniform confidence bands for the mean of a monotonic stochastic process, such as the cumulative distribution function (CDF) of a random variable, based on a sequence of i.i.d.~observations. Our approach leverages the coin-betting framework, and inherits several favourable characteristics of coin-betting methods. In particular, for each point in the domain of the mean function, we obtain anytime-valid confidence intervals that are numerically tight and adapt to the variance of the observations. To derive uniform confidence bands, we employ a continuous union bound that crucially leverages monotonicity. In the case of CDF estimation, we also exploit the fact that the empirical CDF is piece-wise constant to obtain simple confidence bands that can be easily computed. In simulations, we find that our confidence bands for the CDF achieve state-of-the-art performance.

Uniform mean estimation for monotonic processes

TL;DR

This work tackles the problem of uniformly estimating the mean of monotonic stochastic processes, such as CDFs, from i.i.d. data by constructing anytime-valid, variance-adaptive confidence bands that hold over the entire domain. It blends the coin-betting concentration machinery with a continuous PAC-Bayes-style union bound that leverages monotonicity to achieve uniform validity across . For CDF estimation, the authors exploit the piecewise-constant nature of the empirical CDF to obtain simple, computable bounds that demonstrate state-of-the-art performance in simulations. They present relaxations based on binary relative entropy and variance-adaptive refinements, discuss asymptotic rates (notably ), and note potential improvements via advanced betting schemes. Overall, the approach yields interpretable, tight confidence bands with practical implications for uniform inference on monotone mean functions and related risk measures.

Abstract

We consider the problem of deriving uniform confidence bands for the mean of a monotonic stochastic process, such as the cumulative distribution function (CDF) of a random variable, based on a sequence of i.i.d.~observations. Our approach leverages the coin-betting framework, and inherits several favourable characteristics of coin-betting methods. In particular, for each point in the domain of the mean function, we obtain anytime-valid confidence intervals that are numerically tight and adapt to the variance of the observations. To derive uniform confidence bands, we employ a continuous union bound that crucially leverages monotonicity. In the case of CDF estimation, we also exploit the fact that the empirical CDF is piece-wise constant to obtain simple confidence bands that can be easily computed. In simulations, we find that our confidence bands for the CDF achieve state-of-the-art performance.

Paper Structure

This paper contains 16 sections, 4 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Notation
  4. Setting
  5. Method
  6. Coin-betting
  7. Concentration via coin-betting
  8. Leveraging monotonicity
  9. Relaxations
  10. Binary relative entropy
  11. Variance-dependence
  12. Asymptotic rates
  13. Application to CDF estimation
  14. Technicalities
  15. Conclusion
  16. ...and 1 more sections

Key Result

Lemma 1

For any fixed $C\geq 0$ and any $T$, both $y\mapsto\psi_{T,+}^{-1}(y,C)$ and $y\mapsto\psi_{T,-}^{-1}(y,C)$ are non-decreasing. For any fixed $y\in\mathcal{Y}$ and any $T$, $C\mapsto\psi_{T,+}^{-1}(y,C)$ is non-decreasing and $C\mapsto\psi_{T,-}^{-1}(y,C)$ is non-increasing.

Figures (2)

  • Figure 1: CDF confidence bands from $T=1000$ i.i.d. observations sampled from three different distributions supported in $[0,1]$. Our method to produce the highlighted bands ensures that they contain the whole black line (the true CDF $F$) with probability at least $0.95$ on the random datasets used to generate them. From left to right (in black): $F(y)=y$, $F(y)=\sin(\pi\sqrt{y}/2)^6$, and $F(y)=(3 + \mathbb{I}\{y\geq0.6\})/4 + \sin(2\pi y^{0.9})^3/10$. The confidence bands reported are those discussed in \ref{['sec:cdf']}, and the code used for the numerical evaluation of the binary relative entropy is from clerico2022conditionally.
  • Figure 2: Confidence bands ($\delta=0.05$) for $F(y) = \sin(\frac{\pi}{2}\sqrt y)^6$. The difference between the confidence bands and the true CDF is reported on the vertical axis. The confidence bands via our approach are plotted with a solid line. The dashed lines denote the confidence bands from Theorem 2 in howard2022sequential, $\|\hat{F}_T-F\|_\infty \leq 0.85\sqrt{\tfrac{1}{T}(\log\log(eT) + 0.8\log(1612/\delta))}$. The dotted lines depict the variance-adaptive implicit confidence bands from Theorem 5 in howard2022sequential.

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof