Uniform mean estimation for monotonic processes

TL;DR

This paper introduces a method for constructing uniform confidence bands for the mean of monotonic stochastic processes, such as CDFs, using coin-betting techniques that adapt to variance and leverage monotonicity for tight, anytime-valid intervals.

Contribution

It presents a novel approach combining coin-betting and monotonicity to derive tight, adaptive, uniform confidence bands for monotonic processes like CDFs.

Findings

Achieves state-of-the-art performance in simulations.

Provides anytime-valid confidence intervals that adapt to variance.

Simplifies confidence band computation for empirical CDFs.

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.