Local Bahadur Efficiency in Asymptotic Testing

• Local Bahadur efficiency is defined as the exponential decay rate of the tail probability for a test statistic under the alternative hypothesis relative to optimal information bounds.
• It is pivotal in evaluating goodness-of-fit tests, especially for U- and V-statistics, where integral-type procedures generally outperform supremum-based methods.
• The framework connects large deviations theory with minimax optimality, offering practical guidelines for designing tests sensitive to small, contiguous departures from the null hypothesis.

Local Bahadur efficiency is a fundamental concept in asymptotic hypothesis testing, quantifying the exponential rate at which the tail probability of a test statistic under the alternative hypothesis decays relative to the best possible rate established by the Kullack–Leibler divergence. The notion is central in the comparative evaluation of goodness-of-fit, normality, exponentiality, uniformity, and related tests, especially for small, contiguous departures from the null hypothesis. While rooted in large deviations theory, local Bahadur efficiency connects directly to minimax theory and provides a rigorous performance metric for test selection and the design of optimal procedures.

1. Formal Definition and Rate Functions

Let $\{T_n\}$ be a sequence of test statistics for testing a null hypothesis $H_0$ against a parametric alternative $H_1: G(\cdot;\theta)$, with $\theta\to0$ corresponding to $H_0$. A key ingredient is the large-deviation rate function under $H_0$: $$\lim_{n\to\infty} \frac1n \ln \Pr_{H_0}(T_n \geq t) = -f_T(t), \qquad t \in I,$$ where $f_T(\cdot)$ is continuous on interval $I$.

If under $G(\cdot;\theta)$ one has $H_0$0 as $H_0$1, then the Bahadur exact slope at parameter $H_0$2 is defined by: $H_0$3

The Kullback–Leibler “distance” between the alternative and the composite null is

$H_0$4

The local Bahadur efficiency is then

$H_0$5

ensuring that the rate of exponential decay for any test does not exceed that of the information bound.

2. Local Efficiency for U- and V-Statistic-Based Goodness-of-Fit Tests

For statistics expressible as U- or V-statistics (including many new characterization-based goodness-of-fit tests), the following canonical structure holds:

• The statistic $H_0$6 can be represented as a V- or U-statistic of degree $H_0$7, with a symmetric kernel $H_0$8.
• Its projection under $H_0$9 is $H_1: G(\cdot;\theta)$0, and the variance $H_1: G(\cdot;\theta)$1.
• Under standard regularity, the limiting distribution under $H_1: G(\cdot;\theta)$2 is normal, and the large deviations rate function near $H_1: G(\cdot;\theta)$3 is $H_1: G(\cdot;\theta)$4 for $H_1: G(\cdot;\theta)$5, $H_1: G(\cdot;\theta)$6 determined by $H_1: G(\cdot;\theta)$7 and the kernel degree.

For smooth or locally contiguous alternatives $H_1: G(\cdot;\theta)$8, $H_1: G(\cdot;\theta)$9 converges in probability to $\theta\to0$0 for an explicit $\theta\to0$1.

In such cases, local Bahadur efficiency simplifies to

$\theta\to0$2

This structure underlies nearly all modern efficiency computations across goodness-of-fit settings (Milošević, 2015, Nikitin et al., 2016, Volkova, 2014, Volkova, 2014, Nikitin et al., 2012, Milošević, 2015).

3. Empirical Results for Major Test Classes

The local Bahadur efficiency, $\theta\to0$3, has been systematically computed for a wide range of test statistics and alternatives. The following table presents representative values for selected test families and alternatives, synthesized from computational tables and explicit formulae in the literature:

Test family & Statistic	Alternatives & Local Bahadur Efficiency $\theta\to0$4
Exponentiality (Integral, (Nikitin et al., 2016, Milošević, 2015, Volkova, 2014))	Weibull: $\theta\to0$5 (details vary)
Exponentiality (Kolmogorov, (Nikitin et al., 2016, Milošević, 2015))	Weibull, Makeham, Gamma
Pareto (Integral, (Volkova, 2014))	Ley–Paindaveine 1, 2 & log-Weibull
Normality (Cramér–von Mises (Milošević et al., 2021))	Lehmann, Ley–Paindaveine, contamination
Normality (Epps–Pulley (Ebner et al., 2021))	Lehmann, Ley–Paindaveine, contamination
Weighted ECF (BHEP, GE (Meintanis et al., 2022))	Normal, logistic, contamination
Integrated GOF under skew (Durio et al., 2016)	Skewness, various $H_0$9

Integral-type (CvM, “smooth”) statistics consistently dominate Kolmogorov-type (supremum) statistics in local Bahadur efficiency. For exponentiality, integral tests based on special characterizations reach $H_0$2 in the $H_0$3–$H_0$4 range for classical alternatives, exceeding older procedures (e.g. KS, memory-loss, Gini). In normality, Anderson–Darling and BHEP tests attain $H_0$5 near or above $H_0$6 in favorable cases, with tuning parameter optimization further enhancing efficiency (Ebner et al., 2021, Meintanis et al., 2022).

4. Conditions for Attaining Efficiency Bound and Local Asymptotic Optimality

A test is locally Bahadur optimal (full efficiency, $H_0$7) if and only if the “score function” $H_0$8 for the alternative is proportional to the projecting function (kernel) of the test, up to the null-model tangent (e.g., $H_0$9 direction for exponentiality, $$\lim_{n\to\infty} \frac1n \ln \Pr_{H_0}(T_n \geq t) = -f_T(t), \qquad t \in I,$$0 for Pareto). Explicitly:

• For exponentiality integral tests, $$\lim_{n\to\infty} \frac1n \ln \Pr_{H_0}(T_n \geq t) = -f_T(t), \qquad t \in I,$$1 yields $$\lim_{n\to\infty} \frac1n \ln \Pr_{H_0}(T_n \geq t) = -f_T(t), \qquad t \in I,$$2 (Milošević, 2015, Nikitin et al., 2016, Volkova, 2014).
• For Kolmogorov-type tests, local optimality requires score alignment with the supremum kernel’s projection at the variance-maximizing $$\lim_{n\to\infty} \frac1n \ln \Pr_{H_0}(T_n \geq t) = -f_T(t), \qquad t \in I,$$3.

Such “most favorable” alternative families are constructed in every cited work and realize the minimax bound for local error exponents.

5. Extensions to Other Distributional Problems

The local Bahadur efficiency framework extends seamlessly to uniformity (Too–Lin (Milošević, 2015)), power distributions (Puri–Rubin (Nikitin et al., 2012)), integrated goodness-of-fit tests (Durio et al., 2016), and composite/empirical characteristic function-based tests (Meintanis et al., 2022). In each context, the methodology—rate function expansion, projection computation, and Kullback–Leibler evaluation—remains invariant, though technical details (e.g., kernel symmetries, eigenvalue computations) differ.

For power-divergence statistics (Harremoës et al., 2010), the Bahadur functions admit infinite relative efficiency when the likelihood-ratio test (Kullback–Leibler divergence, $$\lim_{n\to\infty} \frac1n \ln \Pr_{H_0}(T_n \geq t) = -f_T(t), \qquad t \in I,$$4) is compared with higher power divergences ($$\lim_{n\to\infty} \frac1n \ln \Pr_{H_0}(T_n \geq t) = -f_T(t), \qquad t \in I,$$5).

In estimation problems, local Bahadur efficiency is formulated for moderate deviations in parametric estimation, with the best possible exponent $$\lim_{n\to\infty} \frac1n \ln \Pr_{H_0}(T_n \geq t) = -f_T(t), \qquad t \in I,$$6 (Fisher information) achieved only by “Bahadur-optimal” estimators under regularity assumptions (Ermakov, 27 Apr 2025).

6. Practical and Theoretical Significance

Local Bahadur efficiency provides a rigorous, interpretable metric for comparing the sensitivity of statistical tests to small, contiguous alternatives, directly connecting to exponential tail probabilities and finite-sample power rankings. Tests with higher local Bahadur efficiency exhibit a steeper exponential decay of type II error, making them more desirable for detecting subtle distributional changes. The consistent superiority of smooth, integral-type procedures is a robust empirical finding, and full efficiency is attainable via careful kernel-alternative alignment. Recent developments report unexpectedly high Bahadur efficiencies for modern integrated or empirical characteristic function-based tests, especially when appropriately tuned (Durio et al., 2016, Ebner et al., 2021, Meintanis et al., 2022).

Monte Carlo studies systematically confirm that these local efficiency rankings reflect actual power ordering for moderate sample sizes, reinforcing the practical utility of the local Bahadur efficiency criterion in test selection and design.

7. Regularity Conditions and Limitations

All derivations and bounds for local Bahadur efficiency require:

• Regularity of the null and alternative families (differentiability, finiteness of Fisher information or analogous quantities).
• Nondegeneracy of kernel projections and large-deviation rate functions.
• Sufficient smoothness to expand the likelihood ratio or relevant functional under the alternative.

Violations of these conditions may lead to nonstandard rates or invalidate the efficiency comparisons. For degenerate U-statistics (e.g., some uniformity or correlation tests), second-order projections and associated boundary problems for eigenvalues may become critical (Milošević, 2015).

---

Key references:

• (Nikitin et al., 2016, Milošević, 2015, Volkova, 2014, Volkova, 2014, Nikitin et al., 2012, Milošević, 2015, Durio et al., 2016, Ebner et al., 2021, Meintanis et al., 2022, Milošević et al., 2021, Ermakov, 27 Apr 2025, Harremoës et al., 2010).