Optimal e-values for testing the mean of a bounded random variable against a composite alternative

TL;DR

This work derives explicit (RE)GROW e-variables for testing the mean of a bounded random variable against composite alternatives, extending beyond the absolute continuity framework. It shows that the coin-betting e-class suffices to obtain optimal e-variables under mean constraints, and provides closed-form (or easily computable) parameters for GRO, GROW, and REGROW across two-sided, one-sided, and agnostic alternatives. REGROW, in particular, yields nontrivial evidence accumulation where GROW can be powerless, with worst-case alternatives interpreted as two-point distributions at the boundaries. The results illuminate how optimal e-values can be constructed in non-dominated settings and offer guidance for robust, alternative-driven evidence aggregation in sequential/aggregated testing contexts.

Abstract

We derive the unique e-values with optimal (relative) growth rate in the worst case for testing the mean of a bounded random variable, hereby contributing with the first application beyond the assumption of mutually absolutely continuous hypotheses of the (RE)GROW quality criteria for e-values originally proposed by Grünwald et al. (2024). For both criteria, we characterise explicitly the alternatives for which it is most difficult to test against, which also admit a meaningful interpretation. We give two important examples of interest where REGROW provides a powerful quality criterion to choose optimal e-variables whereas GROW leads to trivial solutions.

Figures (3)

• Figure 1: Visual intuition behind the Bernoulli example. The solid blue curve shows the optimal growth rate $\mathrm{GRO}(Q_\mu)=\mathbb E_{Q_\mu}[\log Z_\mu]=\mathrm{kl}(\mu,\mu_0)$ as a function of $\mu$. For any fixed $\varepsilon\in[0,1]$, the e-power $\mathbb E_{Q_\mu}[\log Z_\varepsilon]$ is affine in $\mu$ and coincides with the tangent to the blue curve at $\mu=\varepsilon$. The green dotted line corresponds to the trivial e-variable $Z_{\mu_0}=(1,1)^\top$, which has zero e-power for all $\mu$. It is the only tangent that is everywhere nonnegative, hence $Z_{\mu_0}$ is the GROW solution for both $\mathcal{Q}_1$ and $\mathcal{Q}_2$. For REGROW, the relevant quantity is the gap between $\mu\mapsto\mathrm{GRO}(Q_\mu)$ and its tangents. For $\mathcal{Q}_1$, only $\mu\in(\mu_0,1]$ matter, and the optimal e-variable $Z_{\varepsilon_1^\star}$ (brown dash-dotted line) equalises this gap at $\mu=\mu_0$ and $\mu=1$ ($\Delta_1$). For $\mathcal{Q}_2$, the REGROW solution is $Z_{1/2}$ (red dashed line), for which the worst cases are $\mu=0$ and $\mu=1$, where the gap to $\mathrm{GRO}(Q_\mu)$ equals $\Delta_2$.
• Figure 2: The parameter space $I_{\mu_0}^2$ with the regions where $F_{\alpha,\beta}$ is convex (red) or concave (yellow), for $\mu_1>\mu_0$. By \ref{['lemma:fc']}, $F^c_{\alpha,\beta}\equiv F_{\alpha,\beta}$ in the red area and $F^c_{\alpha,\beta}\equiv G_{\alpha,\beta}$ in the yellow area. For $\alpha>0$, the directional derivatives of the function $\beta\mapsto F^c_{\alpha,\beta}(\mu_1)$ are represented as fine arrows. Note that $\beta\mapsto F_{\alpha^\star_{\mathrm{RGW}},\beta}(\mu_1)$ has two local minima, achieved at the two black dots in the figure. Formally proving this fact is the main step to show \ref{['eq:proof_REGROW_eq3']}. In particular, the two black dots are the two unique points simultaneously achieving the sup-inf in \ref{['eq:proof_REGROW_1']}.
• Figure 3: The optimal $\alpha^\star_{\mathrm{GW}}$ from \ref{['eq:GROW_betting_parameter']} and \ref{['eq:GROW_betting_parameter_one_sided']}, and the optimal $\alpha^\star_{\mathrm{RGW}}$ and $\tilde{\alpha}^{\star}_{\mathrm{RGW}}$ given from \ref{['eq:def_characterization_alpha_star_REGROW']} and \ref{['eq:def_characterization_alpha_star_REGROW-one-sided']} as functions of $\mu_1$ (for the alternatives $\mathcal{Q} = \{\mathbb E_Q[X]=\mu_1\}$ and $\tilde{\mathcal{Q}} = \{\mathbb E_Q[X]>\mu_1\}$). Notably, $\alpha^\star_{\mathrm{GW}} \downarrow 0$ whereas $\tilde{\alpha}^{\star}_{\mathrm{RGW}}>0$ as $\mu_1\downarrow\mu_0$, showing that GROW yields a trivial solution, while REGROW is meaningful for testing $\tilde{\mathcal{P}} = \{\mathbb E_Q[X]\leq\mu_0\}$ vs. $\tilde{\mathcal{Q}} = \{\mathbb E_Q[X]>\mu_0\}$. For $\mu_1<\mu_0$ and the symmetric testing problem $\tilde{\mathcal{P}}' = \{\mathbb E_P[X] \geq \mu_0\}$ vs. $\tilde{\mathcal{Q}}' = \{\mathbb E_P[X]< \mu_1\}$, the graphs are point-symmetric with respect to $(\mu_0,0)$, up to scaling.

Theorems & Definitions (13)

• Lemma 1
• Proposition 1: Corollary of Theorem 1 in Eugenio_JAR
• Corollary 1
• Example 1: Testing a Bernoulli
• Theorem 1
• proof
• Theorem 2
• Theorem 3
• proof
• proof