[Paper Review] Test Martingales for bounded random variables

Harrie Hendriks|arXiv (Cornell University)|Jan 29, 2018

Probability and Risk Models11 references3 citations

TL;DR

This paper introduces test martingales for bounded random variables to enable sequential hypothesis testing and confidence interval construction for the mean under minimal distributional assumptions. By leveraging supermartingale properties and a convex combination of exponential tilting, the method achieves power-1 tests and valid confidence intervals, with applications in financial auditing and statistical inference.

ABSTRACT

Given a random sample from a random variable $T$ which is bounded from above, $T\leτ$ a.s., we define processes that are positive supermartingales if $E(T)\geμ$. Such processes are called test martingales. Tests of the supermartingale hypothesis implicitly test the hypothesis $H_0:E(T)\geμ$. We construct test martingales that lead to tests with power 1. We also construct confidence upper bounds. We extend the techniques to testing $H_0:E(T)=μ$ and constructing confidence intervals. In financial auditing random sampling is proposed as one of the possible techniques to gather enough assurance to be able to state that there is no 'material' misstatement in a financial report. The goal of our work is to provide a mathematical context that could represent such process of gathering assurance by means of repeated random sampling.

Motivation & Objective

• To develop a statistical framework for sequential testing of the mean of a bounded random variable using test martingales.
• To construct confidence upper bounds and two-sided confidence intervals for the expectation under minimal assumptions.
• To ensure the method achieves power 1 for testing hypotheses about the mean, even with nonparametric distributions.
• To provide a mathematically rigorous yet accessible tool for applied statisticians and auditors, particularly in financial sampling contexts.
• To extend existing martingale-based inference to nonparametric settings where only bounded support is assumed.

Proposed method

• Define test martingales as positive supermartingales under the null hypothesis $\mathbf{E}(T) \geq \mu$, constructed via exponential tilting of bounded random variables.
• Use a convex combination of test martingales parameterized by $c \in [-1,1]$ to form a robust family $M_n^\mu(\pi) = \int_{-1}^1 M_n^\mu(c) \pi(c)\,dc$.
• Leverage the convexity of $M_n^\mu(\pi)$ in $\mu$ to ensure that the set $\{\mu \mid \forall n: M_n^\mu(\pi) < 1/\alpha\}$ is an interval, forming a valid confidence region.
• Apply the maximal inequality for supermartingales to guarantee that the true mean lies in the confidence interval with probability at least $1 - \alpha$.
• Construct test martingales using the form $M_n^\mu(c) = \prod_{i=1}^n (1 - c(T_i - \mu)/(\tau_1 - \mu))$ for $c \geq 0$ and $M_n^\mu(c) = \prod_{i=1}^n (1 - c(T_i - \mu)/(\mu - \tau_0))$ for $c \leq 0$, ensuring supermartingale behavior under $\mathbf{E}(T) \geq \mu$.
• Allow adaptive design by reweighting the test martingale family at intermediate times using observed data, while preserving measurability and supermartingale properties.

Experimental results

Research questions

• RQ1 Can test martingales be constructed for bounded random variables that yield tests with power 1 under the null hypothesis $\mathbf{E}(T) \geq \mu$?
• RQ2 How can confidence upper bounds and two-sided confidence intervals be constructed using test martingales without parametric assumptions?
• RQ3 What is the role of convexity in the parameter $\mu$ of the test martingale to ensure interval confidence regions?
• RQ4 How can the method be adapted sequentially during sampling without violating martingale or measurability constraints?
• RQ5 Can the framework be applied to real-world problems such as financial audit sampling with minimal distributional assumptions?

Key findings

• The constructed test martingales achieve power 1 for testing $H_0: \mathbf{E}(T) \geq \mu$, ensuring no Type II error under the null.
• Confidence intervals $\{\mu \mid \forall n: M_n^\mu(\pi) < 1/\alpha\}$ are valid with coverage probability at least $1 - \alpha$, and are convex sets due to the convexity of $M_n^\mu(\pi)$ in $\mu$.
• The sample mean $\overline{t}_n$ always lies within the confidence region $\{\mu \mid M_n^\mu(\pi) < 1/\alpha\}$, ensuring practical relevance.
• For $T \sim \mathrm{Beta}(2,98)$, the average sample number was 117.79 with a mean of 69.99 and standard deviation 5.07, showing efficient stopping behavior.
• The method supports adaptive design: at time $n$, the test martingale can be reweighted via $M_{n+\ell}^\mu = M_n^\mu N_\ell^\mu$ using new test families based on observed data, preserving supermartingale structure.
• The framework is robust to model misspecification and applies to nonparametric settings, such as financial misstatement sampling where $T(\omega) \in [0,1]$.