Finite-sample nonparametric mean tests: Leave-one-out duality and asymptotic optimality

Technical Summary: Finite-sample nonparametric mean tests: Leave-one-out duality and asymptotic optimality

Problem Statement
The paper addresses the problem of finite-sample valid hypothesis testing for the mean of nonnegative random variables. Specifically, given i.i.d. observations $X_1, \dots, X_n$ from an unknown distribution on $\mathbb{R}_+$ with mean $\mu$, the goal is to test the one-sided hypothesis:
$$H_0: \mu \le 1 \quad \text{versus} \quad H_1: \mu > 1.$$
Standard nonparametric inference for the mean is generally impossible without distributional restrictions, as established by Bahadur and Savage [1956], who showed that for the class of all distributions with a finite mean, any uniformly level-$\alpha$ test has power at most $\alpha$ at every alternative. This impossibility arises because the mean is non-robust; a small probability mass far in the unfavorable direction can drastically alter the mean while having negligible effect on finite-sample statistics.

The authors restrict the problem to nonnegative random variables ($X_i \ge 0$). While this allows for the construction of valid tests, existing tools are limited. Standard permutation and rank tests do not apply directly. Normal approximations and bootstrap methods offer only asymptotic validity. Concentration-based methods (e.g., Hoeffding, Bernstein) typically require known bounded support, variance control, or stronger moment assumptions, which are often unavailable in settings with heavy-tailed or highly skewed nonnegative data (e.g., insurance losses, revenue per customer).

Methodology: Leave-One-Out Dual Certificate Framework
The core methodological contribution is a "leave-one-out dual certificate" framework. The authors define a leave-one-out null class $\mathcal{P}^n_{loo}$, which includes distributions where the conditional mean satisfies $E_P[X_i | X_{-i}] \le 1$ almost surely. This class is broader than the standard i.i.d. null but allows for a rigorous duality argument.

To prove that a statistic $T: \mathbb{R}^n_+ \to [0, 1]$ is a valid p-value (i.e., $P(T(X) \le \alpha) \le \alpha$ for all $P \in \mathcal{P}^n_{loo}$), the authors formulate an optimization problem over the class of feasible distributions. They utilize the Lagrangian dual of this problem. By introducing nonnegative Lagrange multipliers $\lambda^{(i)}_\alpha$, they derive a pointwise inequality condition (denoted as cert):
$$\alpha + \sum_{i=1}^n \lambda^{(i)}_\alpha(x_{-i})(x_i - 1) \ge \mathbb{1}\{T(x) \le \alpha\} \quad \forall x \in \mathbb{R}^n_+.$$
If such nonnegative functions $\lambda^{(i)}_\alpha$ exist, weak duality guarantees that $T$ is a valid p-value. This approach reduces the infinite-dimensional optimization over distributions to finding functions satisfying pointwise inequalities.

A key extension of this framework is a "dual switching" principle (Theorem 5.1). If two p-values $S$ and $T$ admit dual certificates with specific monotonicity and ordering properties, their pointwise minimum $\min\{S, T\}$ is also a valid p-value. This allows for the combination of different testing strategies to improve power without sacrificing validity.

Key Contributions and Results

1. Validation of the Nonparametric Likelihood Ratio ($T_{nplr}$):
   The paper provides a rigorous proof of the finite-sample validity of the statistic $T_{nplr}$ proposed by Wang and Zhao [2003]. This statistic is defined via a nonparametric likelihood ratio:
   $$T_{nplr}(x) = \min_{t \in [0,1]} \prod_{i=1}^n (1 - t + t x_i)^{-1}.$$
   While Wang and Zhao conjectured its validity, and concurrent work by Ming et al. [2026] verified it via a different representation, this paper establishes validity under the leave-one-out null using the dual certificate framework.

2. Generalized Binomial P-value ($T_{bin+}$):
   The authors introduce a new statistic, $T_{bin+}$, which extends the Clopper-Pearson exact binomial test to general nonnegative random variables. It is constructed by considering a "worst-case" scenario where observations take values in $\{0, x_{max}\}$. The statistic is defined as:
   $$T_{bin+}(x) = \frac{\int_{-1/(x_{max}-1)}^0 \prod_{i \ne j} (1 + s(x_i - 1)) ds}{\int_{-1/(x_{max}-1)}^1 \prod_{i \ne j} (1 + s(x_i - 1)) ds},$$
   where $j$ is an index such that $x_j = x_{max}$. The paper proves $T_{bin+}$ is a valid p-value for $\mathcal{P}^n_{loo}$ and provides an efficient representation involving integrals of product functions.

3. Combined Statistic:
   Using the dual switching principle, the authors show that the pointwise minimum $\min\{T_{nplr}, T_{bin+}\}$ is a valid p-value. This combined statistic is more powerful than either component alone, as $T_{nplr}$ is more sensitive to evidence spread across multiple observations, while $T_{bin+}$ is more sensitive to evidence concentrated in a few large outliers.

4. Asymptotic Optimality:
   The paper establishes sharp optimality results in two regimes under the i.i.d. null $\mathcal{P}^n_{iid}$:

   • Zero-One Detectability: In the fully nonparametric regime without moment assumptions, the squared Hellinger distance $H^2(Q_n, \mathcal{P}_0)$ determines detectability. If $n H^2 \to \infty$, the tests $T_{nplr}$, $T_{bin+}$, and their minimum achieve asymptotic power 1. If $n H^2 \to 0$, no test achieves power greater than $\alpha$.
   • Local Asymptotic Power: Under regular local alternatives approaching the null at rate $n^{-1/2}$, the optimal nonparametric power coincides with that of a Gaussian mean-shift experiment. The paper proves that $T_{bin+}$ and $\min\{T_{nplr}, T_{bin+}\}$ attain this optimal power. In contrast, $T_{nplr}$ is shown to be strictly conservative and fails to attain the optimal local asymptotic power.

Significance and Claims
The paper claims to resolve the open question of the finite-sample validity of the nonparametric likelihood ratio statistic $T_{nplr}$ under a broad leave-one-out null. It introduces a new statistic, $T_{bin+}$, which fills a gap in nonparametric testing for nonnegative data by extending exact binomial methods to unbounded supports.

The significance of the work lies in:

• Theoretical Rigor: Providing a unified dual framework that proves validity without requiring bounded support or variance assumptions, which are standard in concentration inequalities but often unrealistic in practice.
• Optimality: Demonstrating that while $T_{nplr}$ is valid, it is suboptimal in local asymptotic regimes, whereas $T_{bin+}$ achieves the nonparametric power lower bound.
• Practical Utility: The combination of these statistics yields a method that is both finite-sample valid and asymptotically optimal, offering substantial power gains over existing methods (such as CDF-band methods, betting-based methods, and empirical Bernstein bounds) in numerical experiments.

The authors emphasize that their results are specific to nonnegative random variables and the leave-one-out null structure, distinguishing their approach from sequential betting methods (e.g., Waudby-Smith and Ramdas [2024]) which apply to a larger sequential null but may not offer the same finite-sample power properties under the i.i.d. assumption.