Signed-Rank Tests Under Central Symmetry

Why Combine Direction and Radial Rank?

A one-sample location test asks whether a distribution is centered at a specified point. In several dimensions, a departure from that point can occur in any direction, and observations can lie at very different distances from it. A signed-rank procedure combines these features: each observation contributes its spatial direction, weighted according to the rank of its distance from the hypothesized center.

The key calibration idea is not that the directions have one universal distribution. Rather, under central symmetry, reversing any observation while holding its unoriented direction and radius fixed is equally likely. This gives a conditional sign-flip reference distribution. It is distribution-free with respect to the radial and angular distributions in this conditional sense, but the resulting reference distribution can depend on the observed geometry.

Definitions and Test Statistic

Choose a fixed, nondecreasing score function \(a_n\) of the rank. The choice \(a_n(r)=r\) gives the usual Wilcoxon-type radial scores; multiplying every score by the same positive constant does not change the quadratic statistic below. The multivariate signed-rank vector and its sign-invariant quadratic form are

For this formula, assume \(V_n\) is positive definite. This holds, for example, when the weighted directions span \(\mathbb{R}^d\). If they do not, one can use the Moore–Penrose inverse \(V_n^+\) and interpret the statistic on the span of the observed directions. The ranks and \(V_n\) are unchanged if any \(U_i\) is replaced by \(-U_i\); the signed vector \(S_n\), and therefore \(T_n\), can change.

Why Central Symmetry Gives Fair Signs

The null hypothesis is that \(X_i-\theta_0\) is centrally symmetric about zero: for every measurable set \(A\subseteq\mathbb{R}^d\), \(P(Z_i\in A)=P(-Z_i\in A)\). For the exact conditional argument, suppose also that \(P(Z_i=0)=0\). Central symmetry does not require spherical symmetry or a density. It says that a displacement and its negative have the same probability.

For a nonzero vector \(z\), regard \(z\) and \(-z\) as one unoriented pair, denoted \([z]=\{z,-z\}\). Given this pair, the observed orientation has two possibilities. Central symmetry makes their conditional probabilities equal. Independence of the observations then makes the orientation choices independent conditional on all the pairs. In particular, conditioning on the pairs fixes every distance, rank, and unoriented direction while leaving independent fair signs.

To justify the theorem, the map \(z\mapsto[z]\) is unchanged by \(z\mapsto-z\). The conditional distribution of \(Z_i\) given \([Z_i]\) is supported on the two distinct points in that pair. Symmetry exchanges those points while leaving the conditioning event unchanged, so their conditional probabilities are equal. Independence implies that, conditional on all the pairs, the orientations remain independent: the joint law of the independent observations and their pair-valued transforms factors across \(i\). Finally, ranks and \(V_n\) depend only on the pairs, while the signed vector becomes \(\sum_i\varepsilon_i a_n(R_i)U_i\). This proves the stated conditional reference distribution. \(\square\)

Conditional Calibration

For the observed data, hold the radial ranks and unoriented directions fixed and enumerate the \(2^n\) sign vectors. For each \(\varepsilon=(\varepsilon_1,\ldots,\varepsilon_n)\in\{-1,1\}^n\), calculate

A nonrandomized upper-tail conditional \(p\)-value is the proportion of sign vectors for which \(T_n(\varepsilon)\geq T_n^{\mathrm{obs}}\). It is conditionally valid: under the null, the observed sign vector is uniform among the \(2^n\) possibilities, so the proportion of reference outcomes at least as extreme as the observed outcome is superuniform. Discreteness and ties can make the test conservative. Randomizing at the boundary can attain an exact conditional level when desired.

This is not generally a single reference distribution shared by every centrally symmetric population. The observed directions determine \(V_n\) and the values of the quadratic statistic under sign reversal. Exactness means that, conditional on the observed unoriented configuration, the null signs are fair. It does not mean that the unconditional distribution of \(T_n\) is identical for every angular distribution.

What the Test Does and Does Not Assume

The method is useful when a one-sample location question is genuinely multivariate and a distributional model such as multivariate normality is not warranted. Radial ranks limit the influence of the numerical size of a distance: for increasing rank scores, an observation’s weight is determined by its position among the distances rather than by the distance itself. Spatial directions retain the orientation of each displacement, allowing contributions in different directions to combine into a vector.

The exact calibration rests on central symmetry, not merely on a zero mean. For example, let a one-dimensional random variable equal \(1\) with probability \(3/4\) and \(-3\) with probability \(1/4\). Its mean is \((3/4)(1)+(1/4)(-3)=0\), but \(P(X=1)\ne P(X=-1)\), and the distribution is not centrally symmetric. Conditional on an absolute value of 1, the sign is not a fair coin. The sign-flip reference distribution is therefore not justified for this model.

The zero-observation condition in the theorem avoids an undefined spatial direction. If an observation equals the hypothesized center, its direction carries no sign information; a procedure may omit such observations or define an explicit convention for them, but the resulting calibration must account for that choice. In finite samples, ties in nonzero radii are different: average ranks (or another fixed tie rule depending only on the radii) preserve sign-flip validity.

Common Mistakes

• Using a mean-zero assumption in place of central symmetry. Mean zero does not make conditional signs fair. The required null is invariance of the full distribution under reflection through the hypothesized center.
• Claiming one universal null distribution for the quadratic statistic. The conditional sign distribution is exact, but its values depend on the observed directions and ranks.
• Recomputing radial ranks after a sign flip. A sign flip changes \(U_i\) to \(-U_i\), not \(D_i\). The ranks and \(V_n\) must remain fixed throughout conditional enumeration.
• Treating a zero residual as a unit direction. \(Z_i/\|Z_i\|\) is undefined when \(Z_i=0\); state how such observations are handled rather than assigning an arbitrary direction.