Pitman Efficiency of Multivariate Rank Tests

Why Compare Rank Tests Locally?

Two multivariate rank tests can both control their null rejection probability while responding differently to small departures from the null. Pitman efficiency makes that comparison precise: it compares the tests’ power against alternatives that approach the null at rate \(n^{-1/2}\). For location tests, the comparison reveals how a test’s radial score aligns with the location information carried by the assumed radial distribution.

We develop the comparison for a spherical location model in \(p\) dimensions. This setting isolates the role of radial scores. The conclusions apply when the center and scale conventions of the test are fixed; they do not by themselves establish efficiency under every scatter structure or every alternative.

Model, Scores, and Local Alternatives

Assume throughout that the null distribution is absolutely continuous, \(P(R=0)=0\), and the location family is differentiable in quadratic mean at zero. Assume also that \(0<E[\psi(R)^2]<\infty\), so the location Fisher information is finite and nonzero. A local alternative has the form \(\theta_n=h/\sqrt n\), where \(h\in\mathbb{R}^p\) is fixed.

For a score function \(\phi:[0,1]\to\mathbb{R}\), define the radial rank-score statistic \(S_n=n^{-1/2}\sum_{i=1}^n\phi(R_{i:n}/(n+1))U_i\), where \(R_{i:n}\) is the rank of \(R_i=\|X_i\|\) among the observed radii. We use bounded, sufficiently regular scores for which replacing \(R_{i:n}/(n+1)\) by \(F_R(R_i)\) changes \(S_n\) by \(o_p(1)\) under the null and under the stated local alternatives. This rank-substitution condition holds for standard smooth score choices under the usual continuity and moment conditions. The rank version is distribution-free in its radial ordering under the spherical null; its local power still depends on the radial model.

The probability integral transform \(V=F_R(R)\) is uniform on \((0,1)\) under the null. Define \(c_\phi=E[\phi(V)^2]=\int_0^1\phi(v)^2\,dv\) and \(a_\phi=E[\phi(F_R(R))\psi(R)]\). These two quantities will determine the test’s null scale and its local response.

Local Limit and Power

Here \(I_p\) is the \(p\)-dimensional identity matrix. The joint-limit condition is standard for smooth rank scores with finite score variance; it lets us use the local likelihood expansion to obtain the alternative limit rather than trying to calculate the distribution of the statistic directly under each alternative.

Derivation. Under spherical symmetry, \(U\) is uniform on the unit sphere and independent of \(R\). Rotational symmetry gives \(E[UU^\mathsf{T}]=I_p/p\). With \(V=F_R(R)\), the limiting covariance of a single score vector is therefore \(E[\phi(V)^2UU^\mathsf{T}]=(c_\phi/p)I_p\). The rank-substitution condition transfers this covariance to the rank statistic. The central sequence for location is \(n^{-1/2}\sum_i\psi(R_i)U_i\). Its cross-covariance with the score statistic is \(E[\phi(F_R(R))\psi(R)UU^\mathsf{T}]=(a_\phi/p)I_p\). By differentiability in quadratic mean, the location model is locally asymptotically normal with this central sequence. The local-alternative change-of-measure result for jointly asymptotically normal statistics then shifts the limiting mean by the cross-covariance matrix times \(h\), giving \((a_\phi/p)h\), while leaving the limiting covariance unchanged. Finally, standardizing this normal vector gives the stated central and noncentral chi-square limits. \(\square\)

The theorem makes the efficiency comparison explicit. The radial score expressed on the uniform rank scale is \(q_f(v)=\psi(F_R^{-1}(v))\), with \(F_R^{-1}\) the radial quantile function. Since \(a_\phi=\int_0^1\phi(v)q_f(v)\,dv\), the local response is an inner product of score functions. For the efficient location score, \(a_{q_f}=c_{q_f}=E[\psi(R)^2]\).

This is a squared cosine in the space of square-integrable functions on \((0,1)\). Cauchy–Schwarz implies that the efficiency is at most one, with equality when \(\phi\) is proportional to \(q_f\) almost everywhere. If the inner product is zero, the test has zero first-order sensitivity to these local location shifts. For two non-efficient scores \(\phi_1\) and \(\phi_2\), compare \(a_{\phi_1}^2/c_{\phi_1}\) with \(a_{\phi_2}^2/c_{\phi_2}\); the common factor \(\|h\|^2/p\) cancels.

Worked Comparisons

Interpreting the Comparison

The formula separates two ingredients. The denominator \(c_\phi\) measures the null variability contributed by the chosen score. The numerator \(a_\phi^2\) measures how strongly that score aligns with the radial location score of the model. A score can have a large norm without having a large local response: components orthogonal to \(q_f\) add variability but no first-order location signal.

This is why there is no universally most efficient rank score across all radial distributions. A score tuned to one radial location score may align less well with another. The analysis here compares local location power under a specified spherical model; it does not settle robustness to contamination, performance against large shifts, or power for non-location alternatives.

Common Pitfall

• Using the wrong measure for a score norm. A rank score norm is \(c_\phi=\int_0^1\phi(v)^2\,dv\), because the null rank percentile is uniform. A moment such as \(E[q(Y)^2]\) for a nonuniform radial variable \(Y\) must instead be integrated against the density of \(Y\). Replacing one measure by the other changes the efficiency calculation.
• Comparing only score slopes. A larger cross-moment alone does not imply greater efficiency. The score’s null variance \(c_\phi\) must also be included.
• Reading Pitman efficiency as a universal ranking. The ratio is tied to local \(n^{-1/2}\) location alternatives and the radial model used to define \(q_f\). Different models or alternatives can reverse the comparison.