Asymptotic Relative Efficiency

Intuition Beginner

Two honest procedures can both home in on the same true value as the data pile up, yet one of them gets there faster. Asymptotic relative efficiency is the number that says how much faster. It answers a practical question: if procedure A and procedure B both work in the long run, how many more observations does the worse one need to be as precise as the better one?

The key fact is that a good estimator's spread shrinks in a steady way as the sample grows — quadruple the data and the spread halves. Each estimator carries a fixed asymptotic variance, a size for that spread per unit of data. Compare two estimators by dividing their asymptotic variances. If one has half the variance of the other, it extracts the same precision from half the data, so it is twice as efficient. The ratio is the exchange rate between sample sizes.

This matters because the most precise estimator is not always the one you want. The average is unbeatable when the data are tidy and bell-shaped, but a single wild outlier can drag it anywhere. The median ignores the wild value and barely moves. So you trade a little efficiency on tidy data for safety on messy data, and the relative-efficiency number is the price tag on that trade.

The one-sentence takeaway: asymptotic relative efficiency is the ratio of two estimators' long-run spreads, read as the ratio of sample sizes they need to match each other's precision.

Visual Beginner

Picture two estimators of the same quantity, each producing a bell-shaped spread of guesses centered on the truth. Estimator A's bell is narrow; estimator B's bell is wide. Same truth, same data count, different sharpness. The relative efficiency of B to A is how much extra data B needs so its wide bell shrinks down to match A's narrow one.

comparison	what the ratio means	reading
B is half as efficient as A	B's spread is twice A's at the same data	B needs about twice the data of A
B is as efficient as A	spreads match at the same data	same sample size
B is more efficient than A	B's spread is smaller at the same data	B needs less data than A

The takeaway: a smaller asymptotic variance means a sharper bell, and the relative-efficiency number converts the sharpness gap into a sample-size gap.

Worked example Beginner

You measure the center of a bell-shaped population — heights, say — and you cannot decide whether to report the sample average or the sample median. For bell-shaped data both home in on the same center. Which is sharper, and by how much?

Step 1. The spread of the average. For bell-shaped data with spread $\sigma$ (the population standard deviation), the average from $n$ measurements has asymptotic variance ${\sigma }^2$ per unit of data. Write this as $V_{\text{mean}}={\sigma }^2$.

Step 2. The spread of the median. For the same bell-shaped data the median has a larger asymptotic variance, $V_{\text{median}}=\frac{\pi }{2}{\sigma }^2\approx 1.571{\sigma }^2$. The median is wider because it throws away the exact sizes of the measurements and keeps only their order.

Step 3. The efficiency ratio. The asymptotic relative efficiency of the median to the mean is the ratio of variances the better way up: $\frac{V_{\text{mean}}}{V_{\text{median}}}=\frac{{\sigma }^2}{(\pi /2){\sigma }^2}=\frac{2}{\pi }\approx 0.637$.

Step 4. Read it as a sample-size exchange. An efficiency of $0.637$ means the median delivers only about $64\%$ of the information the mean gets from the same data. To match the precision of the mean from $100$ heights, the median needs about $100/0.637\approx 157$ heights.

Step 5. The flip side. Now imagine the data have occasional wild values. The mean's spread balloons, while the median's barely changes. The same comparison can swing above $1$, and then the median wins.

What this tells us: on tidy bell-shaped data the median pays a roughly $36\%$ efficiency penalty — it needs about half again as much data — but that penalty buys robustness against wild values.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $X_1,\dots ,X_n$ are i.i.d. and convergence in distribution $\stackrel{d}{\to }$, in probability $\stackrel{\mathbb{P}}{\to }$, and the $o_{\mathbb{P}}/O_{\mathbb{P}}$ calculus are as in 45.04.01; asymptotic variances are computed by the delta method 45.04.02. Two estimator sequences $S_n,T_n$ of the same scalar parameter $\theta$ are asymptotically normal at rate $\sqrt{n}$ if

$$\sqrt{n}(S_n-\theta )\stackrel{d}{\to }N(0,v_S(\theta )),\sqrt{n}(T_n-\theta )\stackrel{d}{\to }N(0,v_T(\theta )),$$

with asymptotic variances $v_S(\theta ),v_T(\theta )>0$.

Definition (asymptotic relative efficiency of estimators). The asymptotic relative efficiency (ARE) of $S$ relative to $T$ at $\theta$ is the ratio of asymptotic variances, smaller variance in the numerator,

$$\mathrm{ARE}(S,T)=\frac{v_T(\theta )}{v_S(\theta )}.$$

Thus $\mathrm{ARE}(S,T)>1$ when $S$ is the more precise estimator. The ARE is symmetric in the reciprocal sense $\mathrm{ARE}(S,T)=1/\mathrm{ARE}(T,S)$ and depends on $\theta$ in general.

Definition (sample-size interpretation). Let $n_S,n_T$ be the sample sizes at which $S$ and $T$ attain a common target precision (a common asymptotic variance of the estimate $S_{n_S}$, $T_{n_T}$, i.e. $v_S(\theta )/n_S=v_T(\theta )/n_T$). Then

$$\frac{n_S}{n_T}=\frac{v_S(\theta )}{v_T(\theta )}=\frac{1}{\mathrm{ARE}(S,T)},$$

so $\mathrm{ARE}(S,T)$ is the limiting fraction of $T$'s sample size that $S$ requires for the same precision. An ARE of $1/2$ for $S$ means $S$ needs twice the data.

Definition (efficacy of a test). Consider a sequence of one-sided level-$\alpha$ tests of $H_0:\theta ={\theta }_0$ based on a statistic $W_n$ with ${\mathbb{E}}_{\theta }{W}_n={\mu }_n(\theta )$ and standard deviation ${\sigma }_n(\theta )$, asymptotically normal under ${\theta }_0$ and under local alternatives. The efficacy of the test at ${\theta }_0$ is

$$c=\underset{n\to \infty }{\lim }\frac{{\mu }_n^{\prime }({\theta }_0)}{\sqrt{n}{\sigma }_n({\theta }_0)},$$

the standardised slope of the mean of $W_n$ at ${\theta }_0$ (Noether's normalisation, so that $c$ is the noncentrality acquired per unit of local shift $h$ along ${\theta }_0+h/\sqrt{n}$). Under regularity the local power along ${\theta }_0+h/\sqrt{n}$ tends to $\Phi (ch-z_{1-\alpha })$, with $\Phi$ the standard normal distribution function.

Definition (Pitman asymptotic relative efficiency of tests). For two consistent level-$\alpha$ test sequences with the same limiting size and efficacies $c_1,c_2$ at ${\theta }_0$, the Pitman ARE is the limiting ratio of the sample sizes $n_1,n_2$ giving equal asymptotic power against the same contiguous alternative,

$${\mathrm{ARE}}_{\mathrm{Pitman}}({\text{test}}_1,{\text{test}}_2)=\lim \frac{n_2}{n_1}={\left(\frac{c_1}{c_2}\right)}^2,$$

the squared ratio of efficacies. The square enters because power depends on the noncentrality $ch$ and the local shift $h$ scales as $n^{-1/2}$, so matching noncentrality matches $c^{2}n$.

Counterexamples to common slips Intermediate+

• ARE is the variance ratio, not the variance difference, and the orientation matters. $\mathrm{ARE}(S,T)=v_T/v_S$ puts the competitor in the numerator; writing $v_S/v_T$ silently reports the efficiency of $T$ relative to $S$. The two are reciprocals, not equal.

• The Pitman ARE squares the efficacy ratio; the estimation ARE does not square anything extra. For tests the efficacy $c$ is a standardised slope, and power tracks $c^2$, so the ARE is $(c_1/c_2{)}^2$. Treating the efficacy ratio itself as the ARE drops the square and gives the wrong sample-size exchange.

• An ARE below $1$ against the MLE is a theorem, not a quirk. By the asymptotic efficiency of 45.04.03 the MLE has the smallest regular asymptotic variance, so $\mathrm{ARE}(\text{any regular estimator},\text{MLE})\le 1$. An ARE reported above $1$ against the MLE signals either a non-regular (superefficient) competitor on a null set or an arithmetic slip.

• ARE depends on the underlying distribution, so a single number is incomplete. The median-to-mean ARE is $2/\pi$ at the normal but exceeds $1$ for heavy-tailed errors. Quoting "the ARE of the median is $0.64$" without naming the error law states only the Gaussian instance.

Key theorem with proof Intermediate+

The signature result computes the asymptotic relative efficiency of the sample median against the sample mean as a location estimator, and exhibits the dependence on the error density that makes it the prototype for every robust-versus-efficient trade-off. The proof combines the central limit theorem on the mean with the asymptotic normality of the sample median, whose variance is read off from the density at the center.

Theorem (ARE of the median versus the mean). Let $X_1,\dots ,X_n$ be i.i.d. from a distribution symmetric about $\theta$ with finite variance ${\sigma }^2$, continuous density $f$, and $f(\theta )>0$. The sample mean ${\bar{X}}_n$ and the sample median ${\hat{m}}_n$ both estimate $\theta$, with

$$\sqrt{n}({\bar{X}}_n-\theta )\stackrel{d}{\to }N(0,{\sigma }^2),\sqrt{n}({\hat{m}}_n-\theta )\stackrel{d}{\to }N\left(0,\frac{1}{4f(\theta {)}^2}\right).$$

Consequently the asymptotic relative efficiency of the median relative to the mean is

$$\mathrm{ARE}({\hat{m}}_n,{\bar{X}}_n)=\frac{{\sigma }^2}{1/(4f(\theta {)}^2)}=4{\sigma }^{2}f(\theta {)}^2.$$

At the normal $N(\theta ,{\sigma }^2)$ this equals $2/\pi \approx 0.637$; for the Laplace (double-exponential) law it equals $2$; and for sufficiently heavy-tailed laws it exceeds $1$.

Proof. The mean limit is the central limit theorem 37.03.02 with $\theta =\mathbb{E}{X}_1$ and asymptotic variance $\mathrm{Var}(X_1)={\sigma }^2$. For the median, by symmetry $\theta$ is the population median $F^{-1}(1/2)$. The sample median is the empirical $1/2$-quantile, and the sample-quantile asymptotics give, for a continuous density positive at the quantile, $\sqrt{n}({\hat{m}}_n-\theta )\stackrel{d}{\to }N(0,p(1-p)/f(\theta {)}^2)$ at quantile level $p$; at $p=1/2$ this is $N(0,(1/2)(1/2)/f(\theta {)}^2)=N(0,1/(4f(\theta {)}^2))$. The ARE is the ratio of asymptotic variances with the median's competitor (the mean) in the numerator:

$$\mathrm{ARE}({\hat{m}}_n,{\bar{X}}_n)=\frac{v_{\text{mean}}}{v_{\text{median}}}=\frac{{\sigma }^2}{1/(4f(\theta {)}^2)}=4{\sigma }^{2}f(\theta {)}^2.$$

For the normal, ${\sigma }^2$ is the variance and $f(\theta )=1/\sqrt{2\pi {\sigma }^2}$ at the center, so $4{\sigma }^{2}f(\theta {)}^2=4{\sigma }^2\cdot \frac{1}{2\pi {\sigma }^2}=\frac{2}{\pi }$. For the Laplace with density $\frac{1}{2b}{e}^{-|x-\theta |/b}$, the variance is ${\sigma }^2=2b^2$ and $f(\theta )=1/(2b)$, so $4{\sigma }^{2}f(\theta {)}^2=4\cdot 2b^2\cdot \frac{1}{4b^2}=2$. Heavier tails inflate ${\sigma }^2$ while leaving $f(\theta )$ comparatively large, pushing $4{\sigma }^{2}f(\theta {)}^2$ above $1$. $\square$

Bridge. This theorem builds toward the whole comparative theory of robust and nonparametric procedures, and the efficacy machinery it foreshadows appears again in the Pitman efficiency of the sign and Wilcoxon tests below, where the same density-at-the-center quantity $f(\theta )$ controls the noncentrality. The foundational reason ARE depends on the error law is that the mean weights every observation by its value while the median weights only ordering, so a single comparison cannot be law-free: this is exactly the trade-off that motivates robust estimation, where one accepts an ARE below $1$ at the Gaussian model to buy an ARE above $1$ under contamination. The construction generalises the efficiency comparison of 45.04.03 from "estimator versus the optimal MLE" to "estimator versus estimator," and it is dual to the Cramér-Rao computation in that the MLE supplies the variance floor $I(\theta {)}^{-1}$ against which every ARE in the chapter is benchmarked. Putting these together, the median-versus-mean ratio $4{\sigma }^{2}f(\theta {)}^2$ is the prototype, and the bridge is the passage from a variance ratio for estimators to the efficacy-squared ratio for tests, which the delta-method linearisation of 45.04.02 makes a single principle applied to estimates and to test statistics alike.

Exercises Intermediate+

Advanced results Master

The scalar ARE extends to the multivariate Loewner-order comparison and to the testing side through the Pitman-Noether efficacy theorem; the efficiency bound against the MLE is the convolution theorem in disguise, and the Hodges-Lehmann lower bound delimits the cost of distribution-freeness.

Theorem 1 (the efficacy theorem; Pitman-Noether). Under Noether's regularity — asymptotic normality of $W_n$ under ${\theta }_0$ and under contiguous alternatives ${\theta }_0+h/\sqrt{n}$, with ${\mu }_n(\theta )$ differentiable at ${\theta }_0$ and ${\sigma }_n({\theta }_0)$ continuous — the local power of the level-$\alpha$ test based on $W_n$ tends to $\Phi (ch-z_{1-\alpha })$ with efficacy $c=\lim {\mu }_n^{\prime }({\theta }_0)/(\sqrt{n}{\sigma }_n({\theta }_0))$, and the Pitman ARE of two such test sequences equals $(c_1/c_2{)}^2$. The squared efficacy plays for tests the role the inverse asymptotic variance plays for estimators: the efficacy of the score (likelihood-ratio) test equals the square root of the Fisher information, so ${\mathrm{ARE}}_{\mathrm{Pitman}}(\text{any regular test},\text{score test})\le 1$, the testing analogue of the estimation bound ^{[Noether — On a theorem of Pitman]}.

Theorem 2 (ARE against the MLE never exceeds one; the Loewner-order form). For any regular asymptotically normal estimator $S$ of a scalar $\theta$, $\mathrm{ARE}(S,{\hat{\theta }}_n^{\mathrm{MLE}})=I(\theta {)}^{-1}/v_S(\theta )\le 1$, with equality iff $S$ is asymptotically efficient. In ${\mathbb{R}}^k$ the scalar variance ratio is replaced by the Loewner order on asymptotic covariance matrices: $V_S⪰I(\theta {)}^{-1}$, and the ARE for a smooth scalar functional $g(\theta )$ is the ratio of delta-method variances $\nabla g{I}^{-1}\nabla g^{\mathsf{T}}/(\nabla g{V}_S\nabla g^{\mathsf{T}})$, which lies in $(0,1]$ direction by direction. This is the convolution theorem of 45.04.03 read as a statement about ratios rather than differences ^{[van der Vaart — Asymptotic Statistics]}.

Theorem 3 (the Wilcoxon-to-$t$ efficiency and its lower bound; Hodges-Lehmann). For testing a location shift with error density $f$ of finite variance ${\sigma }^2$, the Pitman ARE of the Wilcoxon rank test to the $t$-test is $12{\sigma }^2(\int f^2{)}^2$, equal to $3/\pi \approx 0.955$ at the normal, $1.5$ at the Laplace, and unbounded above for sufficiently heavy tails. Over all finite-variance $f$ it satisfies $12{\sigma }^2(\int f^2{)}^2\ge 108/125=0.864$, the minimum attained at the truncated-parabola (Epanechnikov) density. The corresponding Hodges-Lehmann estimator — the median of pairwise Walsh averages — inherits the same ARE relative to the mean, so the bound transfers from the test to the point estimator ^{[Hodges, J. L. & Lehmann, E. L. — The efficiency of some nonparametric competitors of the t-test]}.

Theorem 4 (the influence-function representation of estimator ARE). For an asymptotically linear estimator $S_n$ with influence function ${\psi }_S$, so that $\sqrt{n}(S_n-\theta )=n^{-1/2}{\sum }_i{\psi }_S(X_i)+o_{\mathbb{P}}(1)$ with $\mathbb{E}{\psi }_S=0$, the asymptotic variance is $v_S=\mathbb{E}[{\psi }_S^2]$, and the ARE of two such estimators is the ratio of squared-influence expectations $\mathbb{E}[{\psi }_T^2]/\mathbb{E}[{\psi }_S^2]$. The mean has $\psi (x)=x-\theta$; the median has $\psi (x)=\mathrm{sign}(x-\theta )/(2f(\theta ))$; the trimmed mean has a clipped linear influence function. The ARE is thereby a contrast of how each estimator weights observations, and the bounded influence function of the median or trimmed mean is precisely what caps the damage of an outlier and trades Gaussian efficiency for robustness ^{[van der Vaart — Asymptotic Statistics]}.

Theorem 5 (Bahadur and Hodges-Lehmann efficiencies; the criterion is not unique). Pitman efficiency fixes the alternative locally and compares sample sizes for equal power; Bahadur efficiency fixes the power and alternative and compares the exponential rates at which the $p$-values vanish under the alternative, and Hodges-Lehmann efficiency fixes the size and compares the rates of the type-II error. The three need not agree numerically, and the Pitman ARE is the one governed by the efficacy because it is the local-alternative limit where every regular statistic is asymptotically normal; the Bahadur ARE depends on large-deviation rates and can rank tests differently for fixed (non-local) alternatives. The choice of efficiency criterion is a modelling decision about which regime of alternatives matters ^{[Lehmann — Elements of Large-Sample Theory]}.

Synthesis. The foundational reason a single number summarises the long-run comparison of two procedures is that, at rate $\sqrt{n}$, every regular estimator is asymptotically normal and every consistent test acquires a normal-shift power curve, so all that survives in the limit is one variance per estimator and one efficacy per test — and the ratio of those scalars is the asymptotic relative efficiency, read as a sample-size exchange. The central insight is that estimation and testing carry the same content twice: the estimator ARE $v_T/v_S$ and the test ARE $(c_1/c_2{)}^2$ are the same comparison seen through the delta-method linearisation of 45.04.02, and the median's $4{\sigma }^{2}f(\theta {)}^2$ against the mean is literally the sign test's efficacy-squared against the $t$, the density-at-the-center $f(\theta )$ being the shared invariant. This is dual to the Cramér-Rao and convolution theory of 45.04.03: where that unit proves $I(\theta {)}^{-1}$ is the unbeatable floor, ARE measures how far above the floor a competitor sits, so $\mathrm{ARE}(\cdot ,\mathrm{MLE})\le 1$ is the convolution theorem re-expressed as a ratio.

Putting these together, ARE is the precise reason robust and nonparametric methods are not merely defensive: the Wilcoxon test gives up at most $13.6\%$ to the $t$-test at the parabolic worst case yet can be unboundedly more efficient under heavy tails, an asymmetry that converts the efficiency calculus into an argument for distribution-free inference. The bridge from this chapter to the rest of statistics is exactly that asymmetry — the efficiency lost at the idealised Gaussian model is small and bounded, while the efficiency gained against the contamination that real data carry is large and unbounded, which is why efficiency comparisons, not point estimates alone, drive the choice of method.

Full proof set Master

The median-versus-mean ARE is proved in the Key theorem; the sign, Wilcoxon, and Hodges-Lehmann-bound computations are Exercises 5-8. The structural propositions are recorded here.

Proposition 1 (the sample-size-ratio characterisation of estimator ARE). Let $\sqrt{n}(S_n-\theta )\stackrel{d}{\to }N(0,v_S)$ and $\sqrt{n}(T_n-\theta )\stackrel{d}{\to }N(0,v_T)$ with $v_S,v_T>0$. If $n_S,n_T\to \infty$ are chosen so that the estimator variances match, $v_S/n_S\to \kappa$ and $v_T/n_T\to \kappa$ for a common target $\kappa >0$, then $n_S/n_T\to v_S/v_T=1/\mathrm{ARE}(S,T)$.

Proof. From $v_S/n_S\to \kappa$ and $v_T/n_T\to \kappa$ with $\kappa >0$, divide: $\frac{v_S/n_S}{v_T/n_T}\to 1$, hence $\frac{n_S}{n_T}=\frac{v_S}{v_T}\cdot \frac{v_T/n_T}{v_S/n_S}\to \frac{v_S}{v_T}\cdot 1=\frac{v_S}{v_T}$. By definition $\mathrm{ARE}(S,T)=v_T/v_S$, so $n_S/n_T\to 1/\mathrm{ARE}(S,T)$. The limit is the asymptotic fraction of $T$'s sample size that $S$ needs for the same precision. $\square$

Proposition 2 (the efficacy formula gives the local power, and the Pitman ARE squares the efficacy ratio). Under Noether's conditions, the level-$\alpha$ one-sided test rejecting for large $W_n$ has, along ${\theta }_0+h/\sqrt{n}$, asymptotic power $\Phi (ch-z_{1-\alpha })$, and two such tests with efficacies $c_1,c_2$ have ${\mathrm{ARE}}_{\mathrm{Pitman}}=(c_1/c_2{)}^2$.

Proof. Standardise $Z_n=(W_n-{\mu }_n({\theta }_0))/{\sigma }_n({\theta }_0)$, asymptotically $N(0,1)$ under ${\theta }_0$, so the test rejects when $Z_n>z_{1-\alpha }$. Under ${\theta }_n={\theta }_0+h/\sqrt{n}$ the mean shifts to ${\mu }_n({\theta }_n)\approx {\mu }_n({\theta }_0)+{\mu }_n^{\prime }({\theta }_0)h/\sqrt{n}$, so $Z_n$ has asymptotic mean ${\mu }_n^{\prime }({\theta }_0)h/(\sqrt{n}{\sigma }_n({\theta }_0))\to ch$ and unit variance (contiguity preserves the variance to first order). Hence the power tends to $\mathbb{P}(N(ch,1)>z_{1-\alpha })=\Phi (ch-z_{1-\alpha })$. For two tests to have equal asymptotic power against the same fixed alternative $\delta$, write the alternative for test $j$ at sample size $n_j$ as $h_j=\delta \sqrt{n_j}$; equal power requires equal noncentrality $c_1{h}_1=c_2{h}_2$, i.e. $c_1\delta \sqrt{n_1}=c_2\delta \sqrt{n_2}$, giving $n_2/n_1=(c_1/c_2{)}^2$. $\square$

Proposition 3 (ARE against the MLE is at most one). For any regular asymptotically normal estimator $S_n$ of a scalar $\theta$ in a family with finite positive Fisher information $I(\theta )$, $\mathrm{ARE}(S_n,{\hat{\theta }}_n^{\mathrm{MLE}})\le 1$, with equality iff $v_S=I(\theta {)}^{-1}$.

Proof. The MLE has asymptotic variance $I(\theta {)}^{-1}$ 45.04.03. By the convolution theorem of 45.04.03, any regular limit law of $\sqrt{n}(S_n-\theta )$ factors as $N(0,I(\theta {)}^{-1})\ast M$, so its variance is $v_S=I(\theta {)}^{-1}+\mathrm{Var}(M)\ge I(\theta {)}^{-1}$. Therefore $\mathrm{ARE}(S_n,{\hat{\theta }}_n^{\mathrm{MLE}})=I(\theta {)}^{-1}/v_S\le 1$, with equality iff $\mathrm{Var}(M)=0$, i.e. $M={\delta }_0$ and $S_n$ is asymptotically efficient. $\square$

Proposition 4 (the median's asymptotic variance from the influence function). For i.i.d. data with continuous density $f$ positive at the median $\theta =F^{-1}(1/2)$, the sample median is asymptotically linear with influence function $\psi (x)=\mathrm{sign}(x-\theta )/(2f(\theta ))$, so $v_{\text{median}}=\mathbb{E}[{\psi }^2]=1/(4f(\theta {)}^2)$.

Proof. The empirical $1/2$-quantile satisfies the Bahadur representation ${\hat{m}}_n-\theta =-({\hat{F}}_n(\theta )-1/2)/f(\theta )+o_{\mathbb{P}}(n^{-1/2})$, where ${\hat{F}}_n$ is the empirical distribution function. Writing ${\hat{F}}_n(\theta )-1/2=n^{-1}{\sum }_i(1\{X_i\le \theta \}-1/2)$, the influence function is $\psi (x)=-(1\{x\le \theta \}-1/2)/f(\theta )=\mathrm{sign}(x-\theta )/(2f(\theta ))$ (using $1\{x\le \theta \}-1/2=-\frac{1}{2}\mathrm{sign}(x-\theta )$ for $x\ne \theta$). Since $\mathrm{sign}(X-\theta )=\pm 1$ each with probability $1/2$, $\mathbb{E}[{\psi }^2]=\mathbb{E}[\mathrm{sign}(X-\theta {)}^2]/(4f(\theta {)}^2)=1/(4f(\theta {)}^2)$, the stated asymptotic variance. $\square$

Proposition 5 (the Hodges-Lehmann lower bound $0.864$). Over all densities $f$ with finite variance, ${\inf }_{f}12{\sigma }^2(f)(\int f^2{)}^2=108/125$, attained at the Epanechnikov density.

Proof. The functional $J(f)=12{\sigma }^2(f)(\int f^2{)}^2$ is invariant under location and scale, so minimise over densities with mean $0$ and a fixed scale, varying the shape. A Lagrange calculation for the constrained minimisation of $\int f^2$ at fixed variance and fixed total mass produces an extremal $f$ that is a downward parabola where positive and zero elsewhere: $f(x)=\frac{3}{4a^3}(a^2-x^2)1\{|x|\le a\}$. For this density $\int fdx=1$, ${\sigma }^2=\int x^{2}fdx=a^2/5$, and $\int f^2=\frac{9}{16a^6}{\int }_{-a}^a(a^2-x^2{)}^{2}dx=\frac{9}{16a^6}\cdot \frac{16a^5}{15}=\frac{3}{5a}$. Hence $J(f)=12\cdot \frac{a^2}{5}\cdot \frac{9}{25a^2}=\frac{108}{125}=0.864$. A second-variation check confirms this stationary point is the minimum, and $J$ is unbounded above (heavy tails inflate ${\sigma }^2$ without shrinking $\int f^2$ proportionally), so the range of the Wilcoxon-to-$t$ ARE is $[0.864,\infty )$. $\square$

Connections Master

The asymptotic normality and efficiency of the MLE 45.04.03 supplies the benchmark against which every ARE in this unit is read: the MLE's asymptotic variance $I(\theta {)}^{-1}$ is the floor, so $\mathrm{ARE}(\cdot ,\mathrm{MLE})\le 1$ is the convolution theorem of that unit re-expressed as a ratio rather than a Loewner inequality, and the efficacy of the score test inherits the same optimality on the testing side. The median-versus-mean comparison here is the concrete instance of "efficient versus inefficient" that unit treats abstractly.

The delta method 45.04.02 is the engine that produces the asymptotic variances the ARE compares: every plug-in or transformed estimator's variance is a sandwich $\nabla g\Sigma \nabla g^{\mathsf{T}}$, and the efficacy of a test is the delta-method slope of the mean of its statistic divided by the null standard deviation. The ARE of two smooth functionals of the same estimator reduces, through the delta method, to the ratio of their sandwich variances along the relevant gradient direction.

The convolution theorem and local-asymptotic-minimax bounds 45.04.08 sharpen the estimation ARE into a precise optimality statement: the ARE against the MLE is at most one because every regular limit law is the efficient Gaussian convolved with extra noise, so the "extra noise" is exactly the efficiency deficit the ARE quantifies, and the local-asymptotic-minimax criterion is what licenses comparing test sequences along contiguous alternatives in the first place.

Consistency and the modes of stochastic convergence 45.04.01 underwrite the entire apparatus: the asymptotic-linearity representations that give the influence-function form of the ARE, the contiguity arguments that keep the variance of a test statistic stable under local alternatives, and the Slutsky steps that convert the efficacy into a limiting power curve all run on the $o_{\mathbb{P}}/O_{\mathbb{P}}$ calculus and weak-convergence tools developed there.

The chi-squared limits of Wald, score, and likelihood-ratio tests 45.06.01 connect to the testing ARE through the efficacy: the score test's efficacy is $\sqrt{I(\theta )}$, so it is the Pitman-optimal local test, and the ARE of the Wald or rank tests to it measures the local power they forfeit, exactly the noncentrality-parameter comparison that governs the power of those chi-squared tests against contiguous alternatives.

Historical & philosophical context Master

The idea of comparing estimators by the ratio of the sample sizes needed for equal precision is due to Ronald A. Fisher, who in his 1920s efficiency programme defined the efficiency of an estimator as the fraction of the available information it uses and observed that the sample median is less efficient than the mean for normal data. The asymptotic-variance ratio as the operational measure of estimator efficiency was systematised in the large-sample literature that followed.

For tests, the decisive contribution is E. J. G. Pitman's 1949 unpublished Columbia lecture notes ^{[Pitman — Lecture Notes on Nonparametric Statistical Inference]}, which defined the asymptotic relative efficiency of two tests as the limiting ratio of sample sizes for equal power against a sequence of local alternatives shrinking to the null at rate $n^{-1/2}$, and identified the efficacy — the standardised slope of the test statistic's mean — as the quantity controlling it. Gottfried Noether supplied in 1955 the regularity conditions making the heuristic a theorem, proving the Pitman ARE equals the squared ratio of efficacies ^{[Noether — On a theorem of Pitman]}. The most consequential computation was Joseph Hodges and Erich Lehmann's 1956 result ^{[Hodges, J. L. & Lehmann, E. L. — The efficiency of some nonparametric competitors of the t-test]} that the Wilcoxon test's Pitman efficiency relative to the $t$-test is $3/\pi$ at the normal and never falls below $108/125=0.864$ over all finite-variance error laws, while being unbounded above under heavy tails — the quantitative case for rank-based inference that reshaped applied statistics.

Bibliography Master

@book{vanderVaart1998,
  author    = {van der Vaart, Aad W.},
  title     = {Asymptotic Statistics},
  publisher = {Cambridge University Press},
  series    = {Cambridge Series in Statistical and Probabilistic Mathematics},
  year      = {1998}
}

@book{Lehmann1999,
  author    = {Lehmann, Erich L.},
  title     = {Elements of Large-Sample Theory},
  publisher = {Springer-Verlag},
  address   = {New York},
  year      = {1999}
}

@article{HodgesLehmann1956,
  author  = {Hodges, J. L. and Lehmann, E. L.},
  title   = {The efficiency of some nonparametric competitors of the $t$-test},
  journal = {Annals of Mathematical Statistics},
  volume  = {27},
  number  = {2},
  year    = {1956},
  pages   = {324--335}
}

@article{Noether1955,
  author  = {Noether, Gottfried E.},
  title   = {On a theorem of Pitman},
  journal = {Annals of Mathematical Statistics},
  volume  = {26},
  number  = {1},
  year    = {1955},
  pages   = {64--68}
}

@unpublished{Pitman1949,
  author = {Pitman, E. J. G.},
  title  = {Lecture Notes on Nonparametric Statistical Inference},
  note   = {Lectures given at Columbia University; unpublished mimeographed notes},
  year   = {1949}
}

@article{HodgesLehmann1963,
  author  = {Hodges, J. L. and Lehmann, E. L.},
  title   = {Estimates of location based on rank tests},
  journal = {Annals of Mathematical Statistics},
  volume  = {34},
  number  = {2},
  year    = {1963},
  pages   = {598--611}
}