Bahadur Efficiency in Statistical Testing

• Bahadur efficiency is a framework for quantifying the asymptotic performance of tests and estimators by capturing the exponential decay rate of tail probabilities as sample sizes grow.
• It employs the concept of the Bahadur slope to compare tests under local alternatives, using large deviation theory to assess statistical sensitivity.
• Applications include goodness-of-fit and normality tests, providing a basis for test tuning and benchmarking against optimal procedures like the likelihood ratio test.

Bahadur efficiency is a framework for quantifying the asymptotic performance of statistical tests and estimators, capturing the exponential decay rate of error probabilities or tail probabilities under large samples. In hypothesis testing, Bahadur efficiency expresses how rapidly the achieved significance level (type I error under alternatives) or attained power approaches its limiting value as sample size increases. It provides a unifying, parametric-free scale for comparing the asymptotic sensitivity of various test statistics and estimators, particularly in the regime of local or contiguous alternatives.

1. Definition of Bahadur Slope and Efficiency

Let $\{T_n\}$ be a sequence of statistics or estimators for testing a null hypothesis $H_0$ against an alternative $H_1$, or for estimating a parameter $\theta$. Suppose the critical values $t_{\alpha,n}$ are set such that under $H_0$, $P_0(T_n \ge t_{\alpha, n}) = \alpha + o(1)$. Under alternative $P_\theta$, define the Bahadur exact slope as: $$c(\theta) = \lim_{n \to \infty} -\frac{1}{n} \ln P_\theta(T_n \ge t_{\alpha, n}).$$ This exponent $c(\theta)$ measures the fastest possible rate at which the tail probability under $H_1$ decays to zero as $n \to \infty$ at a fixed significance threshold $t_{\alpha,n}$, or, equivalently, it quantifies the "speed" with which the test "penetrates" into the tail under the alternative as sample size grows (Berrahou et al., 2012, Ebner et al., 2021, Meintanis et al., 2022).

For two tests with exact slopes $c_1(\theta)$ and $c_2(\theta)$, their (relative) Bahadur efficiency is given by: $e(\theta) = \frac{c_1(\theta)}{c_2(\theta)},$ interpreted as the limiting ratio of sample sizes required for the two tests to attain the same level of sensitivity under $P_\theta$.

In the context of estimation, let $\{T_n\}$ be a sequence estimating $\theta$, and define the (right) tail probability $$\alpha_n(\epsilon, \theta) = P_\theta(|T_n - \theta| > \epsilon)$$. The Bahadur slope is (Akaoka et al., 2021): $$c_T(\theta) = \sup\left\{c \ge 0 : \limsup_{n \to \infty} -\frac{1}{n} \ln \alpha_n(\epsilon, \theta) \ge c \text{ for all small } \epsilon > 0 \right\}.$$

2. Large Deviation Theory and Rate Functions

The theory is built on large deviations asymptotics. If under $H_0$, the test statistic $T_n$ satisfies

$$P_0(T_n > t) \approx \exp\{- n I(t)\}, \quad n \to \infty$$

for some rate function $I(t)$ (typically continuous and quadratic near zero), then under a local or contiguous $P_\theta$, limits of $T_n$ are computed, and the Bahadur slope can be written as

$c_T(\theta) = 2 I(b_T(\theta)),$

where $b_T(\theta)$ is the limit in probability of $T_n$ under $P_\theta$ (Milošević, 2015, Berrahou et al., 2012, Durio et al., 2016, Volkova, 2014, Nikitin et al., 2012).

For local alternatives $P_\theta$ close to $P_0$ (e.g., $g(x; \theta) = f_0(x) + \theta h(x) + o(\theta)$), the expansion $c_T(\theta) = l_T \theta^2 + o(\theta^2)$ is typical, and the Kullback–Leibler divergence $K(\theta)$ between $P_\theta$ and $P_0$ similarly expands as $K(\theta) = \frac{1}{2} I \theta^2 + o(\theta^2)$. The local Bahadur efficiency is

$$e^B(T) = \frac{l_T}{I} = \lim_{\theta \to 0} \frac{c_T(\theta)}{2 K(\theta)} \le 1.$$

Attainment of $e^B(T)=1$ identifies locally asymptotically optimal tests against the considered alternatives (Milošević, 2015, Durio et al., 2016, Volkova, 2014, Berrahou et al., 2012).

3. Explicit Calculations for Common Classes

Goodness-of-fit Testing:

For a U-statistic or V-statistic-based GoF test (integral or supremum type), Bahadur efficiency is calculated by:

• Identifying the large-deviation rate function $f(a)$: $P_{H_0}(T_n > a) \sim \exp\{- n f(a)\}$.
• Determining the limit in probability $b_T(\theta)$ under the alternative.
• Setting $c_T(\theta) = 2 f(b_T(\theta))$.

Examples:

• For the $L_1$-distance test for independence, $$c_{L_1}(\theta) = \frac{\Delta^2}{2} \theta^2 + o(\theta^2)$$, with explicit $\Delta$ depending on the local alternative (Berrahou et al., 2012).
• For the integral-type tests for Pareto and power function distributions, explicit kernel projections yield numeric efficiency values against smooth perturbations (Volkova, 2014, Nikitin et al., 2012).

Normality Testing:

For EDF-based normality tests (Kolmogorov–Smirnov, Cramér–von Mises, Anderson-Darling), Bahadur local efficiency is given by explicit formulae involving kernel eigenvalues and projections on local alternative scores. The exact slope for the LRT is given by $2 K(\theta)$, providing a universal benchmark (Milošević et al., 2021, Ebner et al., 2021).

Characteristic Function–Based Tests:

Weighted $L^2$-type empirical characteristic function (ECF) tests have Bahadur slopes determined by the key kernel eigenvalue and alternative-specific functionals. Optimal parameter tuning (e.g., for the Epps–Pulley or energy test) yields efficiencies that may dominate or rival EDF-based tests over common alternatives (Meintanis et al., 2022, Ebner et al., 2021).

4. Extensions and Contemporary Frameworks

Moderate Deviations and Multi-parameter Settings:

Bahadur efficiency is generalized to moderate deviations. For an estimator $\hat\theta_n$ and a sequence $u_n \to 0$ with $n u_n^2 \to \infty$, the moderate-deviation Bahadur slope at $\theta_0$ is

$$B(\hat\theta_n; \theta_0) = \liminf_{n \to \infty} \frac{1}{n u_n^2 I(\theta_0) / 2} \ln P_{\theta_0}(|\hat \theta_n - \theta_0| > U_n).$$

The minimax lower bound corresponds to $-1$, mirroring the classical Hajek–Le Cam lower bound in the moderate deviation regime (Ermakov, 27 Apr 2025).

Semi-tail Units and Efficiency Differences:

A recent approach proposes expressing Bahadur slopes in “semi-tail units” ($s$-values): $s(t) = - \log_2 P_0(T_n \ge t)$. The per-sample Bahadur slope in this scale is $\sigma(\theta) = c(\theta)/(2 \ln 2)$ (Vos, 28 Jun 2025). The efficiency of test 1 vs test 2 can then be interpreted both as a ratio $c_1/c_2$ and as a difference $\sigma_1 - \sigma_2$, where each increment in semi-tail units corresponds to a halving of the probability tail, providing additivity and interpretability.

Efficiency Type	Formula	Interpretation
Ratio (classical)	$e = c_1(\theta)/c_2(\theta)$	Relative sample size rate
Difference (semi-tail)	$E_{\mathrm{diff}} = (\sigma_1 - \sigma_2)$	Added tail-halving per sample

This additive scale makes differences in efficiency directly interpretable as deeper tail penetration per observation.

5. Practical Implementation and Benchmarks

Key principles for applied Bahadur efficiency analysis:

• Local asymptotics: Focus is typically on local (contiguous) alternatives; expansions for both the test statistic’s drift and the information divergence yield quadratic approximations for $c_T(\theta)$ and $2K(\theta)$.
• Test tuning: Many test families (ECF-based, weighted, etc.) can have their efficiency optimized across alternatives via parameter selection.
• Benchmarking: The likelihood ratio test (LRT), when defined, provides the theoretical maximum $c_{LRT}(\theta) = 2 K(\theta)$ and thus local efficiency $e^B=1$; ratios to attainable test slopes quantify sub-optimality.
• Locally Asymptotically Optimal (LAO) alternatives: For a fixed test, the class of alternatives achieving $e^B=1$ is explicitly characterized via the test statistic’s kernel (U- or V-statistic projections) (Milošević, 2015, Nikitin et al., 2016, Volkova, 2014, Nikitin et al., 2012, Berrahou et al., 2012).

Practically, for traditional GoF tests, the integral-type statistics (e.g., Cramér–von Mises, Anderson–Darling, integral U-statistics) consistently demonstrate higher Bahadur efficiency than supremum-type (Kolmogorov–Smirnov, D-type) statistics across classical models (Milošević et al., 2021, Milošević, 2015).

6. Impact and Contemporary Directions

Bahadur efficiency remains a universal, pivotal concept for test comparison in asymptotic regimes, especially for:

• Composite and nonparametric hypotheses: Efficiency theory extends to composite, nonparametric, and multivariate tests using characteristic function–based, power divergence, or integrated process statistics (Harremoës et al., 2010, Meintanis et al., 2022, Vos, 28 Jun 2025).
• Multivariate and dependent settings: The framework is extended to tests for independence, multi-dimensional GoF, and models with nuisance parameter estimation (Berrahou et al., 2012, Akaoka et al., 2021, Ermakov, 27 Apr 2025).
• Fine-tuning for practical use: Numerical tabulations of Bahadur efficiencies for a wide range of tests and alternatives assist researchers in selecting optimal procedures for specific “close-to-null” regimes (Ebner et al., 2021, Meintanis et al., 2022).
• Interpretability and scale: The semi-tail (s-value) approach standardizes interpretation and enables intuitive additive comparisons across disparate statistics and scales (Vos, 28 Jun 2025).

Bahadur efficiency theory thus provides a precise, model-agnostic, and interpretable toolkit for the asymptotic evaluation and selection of hypothesis tests and estimators, and continues to be refined in contemporary theoretical and applied work.