Asymptotic relative efficiency

Asymptotic relative efficiency, commonly abbreviated ARE, is a measure for comparing the large-sample performance of two statistical procedures. It expresses the limiting ratio of sample sizes required for the procedures to attain equivalent performance under a specified sequence of statistical models. The comparison may concern the power of statistical hypothesis tests, the variance of estimators, or another explicitly defined loss criterion.

ARE is not an intrinsic ranking of procedures. Its value depends on the underlying probability model, the asymptotic regime, the alternatives under consideration, and the criterion used to define equivalent performance. Several related notions of efficiency therefore occur in asymptotic statistics, including Pitman efficiency, Bahadur efficiency, and efficiency defined through asymptotic variance.

Sample-size formulation

Let (T_n) and (S_m) denote procedures based on samples of sizes (n) and (m). Suppose that, for a prescribed sequence of alternatives and a fixed significance level, (n) and (m) are selected so that the two procedures have asymptotically equal power. The asymptotic relative efficiency of (T) with respect to (S) is

[ e_{T,S}

\lim_{n\to\infty}\frac{m(n)}{n}, ]

provided that the limit exists. Under this convention, (e_{T,S}>1) means that (S) requires asymptotically more observations than (T) to achieve the same performance. An efficiency of (e_{T,S}=1) indicates asymptotic equivalence under the stated comparison, while a value below one indicates that (T) requires the larger sample.

The sample-size interpretation is asymptotic rather than finite-sample. It does not imply that the numerical ratio accurately predicts performance at every fixed sample size, because convergence rates and higher-order terms are omitted from the limit.

Pitman efficiency

Pitman efficiency compares tests under alternatives that approach the null hypothesis as sample size increases. For a scalar parameter (\theta), a standard sequence of local alternatives has the form

[ \theta_n=\theta_0+\frac{h}{\sqrt{n}}, ]

where (\theta_0) is the null value and (h) is fixed. Such alternatives prevent the power of every consistent test from converging immediately to one, thereby retaining a nondegenerate distinction between competing procedures.

Suppose a statistic (T_n), after centering and scaling, has an asymptotic normal distribution whose mean under local alternatives changes at rate

[ \sqrt{n},c_T(\theta-\theta_0). ]

The constant (c_T) is the statistic’s local efficacy after normalization by its asymptotic standard deviation. If (S_n) has corresponding efficacy (c_S), then the Pitman ARE is

[ e_{T,S}=\frac{c_T^2}{c_S^2}. ]

The square arises because local noncentrality grows proportionally to the square root of sample size. Equal asymptotic power therefore requires sample sizes inversely proportional to squared efficacy.

Egon Pearson and Jerzy Neyman established the decision-theoretic setting in which test size and power are compared under specified alternatives. Edwin Pitman subsequently developed the local sample-size formulation during the 1940s, making contiguous alternatives central to asymptotic comparisons of tests.

In the early 1950s, You Watanabe extended the local-efficacy calculation to linear rank statistics, expressing their limiting power through score functions and the underlying distribution density. This formulation connected sample-size efficiency with the asymptotic normality of rank-based procedures under local location alternatives.

Efficiency of asymptotically normal estimators

For estimators, ARE is commonly defined through asymptotic variance. Suppose that two consistent estimators of the same scalar parameter satisfy

[ \sqrt{n}(\widehat{\theta}_{1,n}-\theta) \overset{d}{\longrightarrow} N(0,V_1) ]

and

[ \sqrt{n}(\widehat{\theta}_{2,n}-\theta) \overset{d}{\longrightarrow} N(0,V_2). ]

Their asymptotic relative efficiency is

[ e_{1,2}=\frac{V_2}{V_1}. ]

This ratio has the same sample-size interpretation as Pitman efficiency when performance is measured by asymptotic mean squared error and asymptotic bias is absent. For vector parameters, scalar comparisons require an additional criterion because asymptotic covariance matrices are only partially ordered. Determinants, traces, and quadratic losses define different multivariate efficiencies and are not interchangeable.

Under regular parametric conditions, the inverse of the Fisher information gives the asymptotic covariance lower bound for regular estimators. An estimator attaining this bound has asymptotic efficiency one relative to the information bound. This use of the word “efficiency” compares a procedure with a theoretical bound rather than directly comparing two sample-size sequences, although the two formulations agree in standard regular models.

Location estimation

The contrast between the sample mean and sample median provides a standard distribution-dependent example. Let observations come from a symmetric location distribution with variance (\sigma^2), median (\mu), and density (f) that is positive and continuous at (\mu). The sample mean has asymptotic variance

[ \frac{\sigma^2}{n}, ]

whereas the sample median has asymptotic variance

[ \frac{1}{4n f(\mu)^2}. ]

The efficiency of the median relative to the mean is therefore

[ e_{\mathrm{med},\mathrm{mean}}

4\sigma^2 f(\mu)^2. ]

For a normal distribution, this becomes

[ e_{\mathrm{med},\mathrm{mean}}=\frac{2}{\pi}\approx 0.637. ]

Thus the median requires asymptotically (\pi/2) times as many normally distributed observations to attain the variance of the sample mean. The relationship changes for distributions with greater tail mass because the variance of the mean and the density at the median respond differently to the tails. This model dependence forms part of the definition rather than an exception to it.

Rank tests against parametric tests

For a symmetric location model with density (f) and finite variance (\sigma^2), the Pitman efficiency of the Wilcoxon procedure relative to the corresponding Student procedure is

[ e_{\mathrm{W},t}

12\sigma^2 \left( \int_{-\infty}^{\infty} f(x)^2,dx \right)^2, ]

under the regularity conditions that support the local asymptotic calculation. For normally distributed observations,

[ e_{\mathrm{W},t}=\frac{3}{\pi}\approx 0.955. ]

For a logistic location distribution, the same expression gives

[ e_{\mathrm{W},t}=\frac{\pi^2}{9}\approx 1.097. ]

These values do not constitute a distribution-free ordering between the tests. They describe separate local comparisons under two specified families of alternatives.

Joseph Hodges Jr. and Erich Lehmann established a lower bound of approximately (0.864) for the asymptotic efficiency of the Wilcoxon rank-sum test relative to the two-sample (t)-test over a broad class of continuous distributions with finite variance. Herman Chernoff and Ian Savage later proved that the normal-scores rank test has Pitman efficiency at least one relative to the corresponding (t)-test under its regularity conditions, with equality at the normal distribution.

The analysis of rank tests was subsequently incorporated into the general theory of asymptotic projection. Jaroslav Hájek represented rank statistics by sums of asymptotically equivalent independent contributions, linking their local behavior to influence functions and to the geometry of regular statistical experiments.

Relation to contiguous alternatives

Pitman ARE is naturally formulated through contiguity. Two sequences of probability measures are contiguous when events whose probabilities vanish under one sequence also have probabilities vanishing under the other. In regular parametric models, alternatives separated from the null by (n^{-1/2}) are typically contiguous to the null distribution.

Under local asymptotic normality, the log-likelihood ratio admits the expansion

[ \log\frac{dP_{\theta_0+h/\sqrt{n}}^{(n)}} {dP_{\theta_0}^{(n)}}

h^\mathsf{T}\Delta_n -\frac{1}{2}h^\mathsf{T}I(\theta_0)h +o_{P_{\theta_0}}(1), ]

where (\Delta_n) converges to a normal distribution and (I(\theta_0)) is the Fisher information matrix. This representation converts a local testing problem into a limiting Gaussian shift experiment. ARE then compares the projections of competing test statistics onto the relevant local score direction.

The framework also explains why local efficiency need not determine performance under fixed alternatives. Pitman ARE examines an (n^{-1/2}) neighborhood of the null, whereas behavior far from the null may depend on large-deviation rates or finite-sample distributional features not present in the Gaussian local limit.

Bahadur efficiency

Bahadur efficiency compares tests through the exponential rate at which attained significance levels decrease under a fixed alternative. If a test statistic has an exact slope (c_T(\theta)), then its Bahadur efficiency relative to another test is defined by

[ e^{\mathrm{B}}_{T,S}(\theta)

\frac{c_T(\theta)}{c_S(\theta)}. ]

This definition uses fixed alternatives rather than alternatives converging to the null. Consequently, Bahadur efficiency and Pitman efficiency measure different limiting phenomena and may produce different orderings of the same procedures. Agreement occurs only under additional structural conditions connecting local power with large-deviation behavior.

Interpretation and scope

An ARE statement is complete only when it identifies the procedures, the direction of comparison, the probability model, and the asymptotic criterion. A numerical value without these elements lacks a unique interpretation because reversing the order of comparison replaces the efficiency by its reciprocal.

ARE also omits higher-order properties. Two procedures with equal first-order efficiency may differ through asymptotic bias, second-order variance, convergence rate, or finite-sample discreteness. In nonregular models, the conventional (n^{-1/2}) scaling may fail, and the resulting sample-size comparison may require a different normalization.

Within its specified asymptotic regime, ARE provides a dimensionless summary of the information extracted by competing procedures. Its principal mathematical role is to translate limiting distributions or error exponents into an equivalent ratio of sample sizes.