One-sample location tests based on center-outward signs and ranks

TL;DR

The paper develops multivariate one-sample location tests based on center-outward ranks and signs for centrally symmetric distributions, proposing two procedures (random signs and symmetrized-sample) and deriving their asymptotic distributions. It establishes connections to the univariate sign and Wilcoxon tests in the one-dimensional case and demonstrates, via comprehensive simulations, that the center-outward approaches offer competitive power and robustness to non-Gaussian and heavy-tailed distributions, with grid design significantly influencing performance. The work provides distribution-free, affine-equivariant testing tools grounded in optimal transport concepts, and discusses practical trade-offs between computational cost and statistical power, including sensitivity analyses to symmetry violations. Overall, the center-outward framework delivers flexible, high-signal tests for multivariate location that outperform classical Hotelling-style tests in non-elliptical settings and offer a principled alternative when symmetry is plausible but Gaussian assumptions fail.

Abstract

A multivariate one-sample location test based on the center-outward ranks and signs is considered, and two different testing procedures are proposed for centrally symmetric distributions. The first test is based on a random division of the data into two samples, while the second one uses a symmetrized sample. The asymptotic distributions of the proposed tests are provided. For univariate data, two variants of the symmetrized test statistic are shown to be equivalent to the standard sign and Wilcoxon test respectively. The small sample behavior of the proposed techniques is illustrated by a simulation study that also provides a power comparison for various transportation grids.

Figures (5)

• Figure 1: Power of the considered tests for the standard normal distribution $\mathsf{N}_d(\boldsymbol{0},\boldsymbol{I})$ in $\mathbb{R}^d$ for $d\in\{2,4,6\}$ (in columns) shifted by $\delta \cdot \boldsymbol{s}$ for $\boldsymbol{s}=(1,\dots,1)^\top$ (dir$=1$, first row) or $\boldsymbol{s}=(1,0,\dots,0)^\top$ (dir$=2$, second row) for sample sizes $n\in\{150,300\}$ and various $\delta\geq 0$.
• Figure 2: Power of the considered tests for $t_1$ distribution in $\mathbb{R}^d$ for $d\in\{2,4,6\}$ (in columns) shifted by $\delta \cdot \boldsymbol{s}$ for $\boldsymbol{s}=(1,\dots,1)^\top$ (dir$=1$, first row) or $\boldsymbol{s}=(1,0,\dots,0)^\top$ (dir$=2$, second row) for sample sizes $n\in\{150,300\}$ and various $\delta\geq 0$.
• Figure 3: Power for the "double exponential" distribution in $\mathbb{R}^d$ for $d\in\{2,4,6\}$ (in columns) shifted by $\delta \cdot \boldsymbol{s}$ for $\boldsymbol{s}=(1,\dots,1)^\top$ (dir$=1$, first row) or $\boldsymbol{s}=(1,0,\dots,0)^\top$ (dir$=2$, second row) for sample sizes $n\in\{150,300\}$ and various $\delta\geq 0$.
• Figure 4: Proportion of rejections of the null hypothesis of zero location for the c-o test with random signs (RAN), symmetrized sample (SYM), Hotelling's $T^2$ test (HOT), and spatial rank test (SPAT) for skew data in dimension $d=2$. The columns correspond to distributional parameter $\alpha$: the larger $\alpha$, the more skewed distribution, see details in the main text.
• Figure 5: Power curves of two multivariate one-sample tests (symmetrized c-o test and spatial rank test) compared to the power of $d$ univariate marginal Wilcoxon tests (MARG) combined via the Bonferroni correction for "double exponential" distribution in $\mathbb{R}^d$ for $d\in\{2,4,6\}$ (in columns). A centered sample is shifted by $\delta \cdot \boldsymbol{s}$ for $\boldsymbol{s}=(1,\dots,1)^\top$ (dir$=1$, first row) or $\boldsymbol{s}=(1,0,\dots,0)^\top$ (dir$=2$, second row) for sample sizes $n\in\{150,300\}$.

Theorems & Definitions (16)

• remark 1
• proposition 1
• proof
• proposition 2
• remark 2
• remark 3
• proposition 3
• proof
• remark 4
• proposition 4