α-Scatter Median Matrix Estimation

• α-scatter median matrix is a robust, affine-equivariant estimator of scatter for multivariate data following an α-symmetric law.
• It generalizes the classical scatter halfspace median using the α-norm geometry to achieve minimax-optimal concentration under contamination.
• Key properties include a 50% breakdown point, affine invariance, and consistency, making it effective for heavy-tailed and non-elliptical distributions.

The α-scatter median matrix is a robust, affine-equivariant estimator of scatter (dispersion) for multivariate data distributed according to an α-symmetric law. Extending the core principles of halfspace depth to data whose geometry is governed by the α-norm, the α-scatter median matrix achieves minimax-optimal rates of concentration under contamination and incorporates invariance and breakdown properties suitable for multivariate heavy-tailed or non-elliptical data (Bočinec et al., 8 Dec 2025). This estimator generalizes the classical scatter halfspace median matrix, aligning the estimation process with the geometry inherent to α-symmetric distributions.

1. α-Symmetric Distributions and α-Norm Geometry

An α-symmetric distribution on $\mathbb{R}^d$ is defined by its characteristic function depending solely on the α-norm of its argument:

$$\|x\|_\alpha= \Bigl(\sum_{i=1}^d |x_i|^\alpha \Bigr)^{1/\alpha},\quad \psi_P(t)=\mathbb{E}[e^{i t^T X}] = \varphi(\|t\|_\alpha),\quad t\in\mathbb{R}^d.$$

This family subsumes both elliptical distributions ($\alpha=2$) and broader classes with heavy tails or non-Euclidean geometry. Key properties include:

• For any unit vector $u$, $u^T X \equiv_d \|u\|_\alpha X_1$ (where $X_1$ is a coordinate of $X$).
• Assumption $P(\{0\})=0$ and $d>1$ ensure smoothness and guarantee a unique location median at $\mu_\text{hs}=0$ (Bočinec et al., 8 Dec 2025).

2. Classical and α-Scatter Halfspace Depths

The classical scatter halfspace depth for a candidate positive-definite matrix $$\|x\|_\alpha= \Bigl(\sum_{i=1}^d |x_i|^\alpha \Bigr)^{1/\alpha},\quad \psi_P(t)=\mathbb{E}[e^{i t^T X}] = \varphi(\|t\|_\alpha),\quad t\in\mathbb{R}^d.$$0 and distribution $$\|x\|_\alpha= \Bigl(\sum_{i=1}^d |x_i|^\alpha \Bigr)^{1/\alpha},\quad \psi_P(t)=\mathbb{E}[e^{i t^T X}] = \varphi(\|t\|_\alpha),\quad t\in\mathbb{R}^d.$$1 under elliptical symmetry (α=2) is:

$$\|x\|_\alpha= \Bigl(\sum_{i=1}^d |x_i|^\alpha \Bigr)^{1/\alpha},\quad \psi_P(t)=\mathbb{E}[e^{i t^T X}] = \varphi(\|t\|_\alpha),\quad t\in\mathbb{R}^d.$$2

Maximizing this yields the scatter halfspace median. For $$\|x\|_\alpha= \Bigl(\sum_{i=1}^d |x_i|^\alpha \Bigr)^{1/\alpha},\quad \psi_P(t)=\mathbb{E}[e^{i t^T X}] = \varphi(\|t\|_\alpha),\quad t\in\mathbb{R}^d.$$3, quadratic forms do not respect the underlying geometry. The α-scatter halfspace depth is defined as:

$$\|x\|_\alpha= \Bigl(\sum_{i=1}^d |x_i|^\alpha \Bigr)^{1/\alpha},\quad \psi_P(t)=\mathbb{E}[e^{i t^T X}] = \varphi(\|t\|_\alpha),\quad t\in\mathbb{R}^d.$$4

where $$\|x\|_\alpha= \Bigl(\sum_{i=1}^d |x_i|^\alpha \Bigr)^{1/\alpha},\quad \psi_P(t)=\mathbb{E}[e^{i t^T X}] = \varphi(\|t\|_\alpha),\quad t\in\mathbb{R}^d.$$5 is the matrix square root in $$\|x\|_\alpha= \Bigl(\sum_{i=1}^d |x_i|^\alpha \Bigr)^{1/\alpha},\quad \psi_P(t)=\mathbb{E}[e^{i t^T X}] = \varphi(\|t\|_\alpha),\quad t\in\mathbb{R}^d.$$6 and $$\|x\|_\alpha= \Bigl(\sum_{i=1}^d |x_i|^\alpha \Bigr)^{1/\alpha},\quad \psi_P(t)=\mathbb{E}[e^{i t^T X}] = \varphi(\|t\|_\alpha),\quad t\in\mathbb{R}^d.$$7 replaces the Euclidean norm (Bočinec et al., 8 Dec 2025). This formulation ensures that the depth contours align with directions imposed by α-symmetry.

3. Construction and Characterization of the α-Scatter Median Matrix

The α-scatter median matrix $$\|x\|_\alpha= \Bigl(\sum_{i=1}^d |x_i|^\alpha \Bigr)^{1/\alpha},\quad \psi_P(t)=\mathbb{E}[e^{i t^T X}] = \varphi(\|t\|_\alpha),\quad t\in\mathbb{R}^d.$$8 is any maximizer of $$\|x\|_\alpha= \Bigl(\sum_{i=1}^d |x_i|^\alpha \Bigr)^{1/\alpha},\quad \psi_P(t)=\mathbb{E}[e^{i t^T X}] = \varphi(\|t\|_\alpha),\quad t\in\mathbb{R}^d.$$9:

$\alpha=2$0

Theorem 6.4 (Bočinec et al., 8 Dec 2025) demonstrates that for α-symmetric $\alpha=2$1, the maximizer is spherical:

$\alpha=2$2

where $\alpha=2$3 denotes the common marginal distribution of $\alpha=2$4. Uniqueness and sphericity derive from affine and sign-permutation invariance, restricting maximizers to scalar multiples of the identity. The value $\alpha=2$5 is chosen to balance the probabilities in the halfspace depth definition, solving $\alpha=2$6.

4. Robustness, Equivariance, and Breakdown Properties

The α-scatter median matrix retains several desirable robustness features:

• Affine Equivariance: For nonsingular $\alpha=2$7 and any location $\alpha=2$8, $\alpha=2$9.
• Sign-Permutation Invariance: For α-symmetric $u$0, $u$1 is invariant under signed permutations.
• ½ Breakdown Point: The use of probability minima over halfspaces ensures the estimator is robust against up to 50% contamination in the sample.
• Continuity and Consistency: The α-sHD mapping $u$2 is jointly continuous, so sample maximizers converge almost surely to the population value (Bočinec et al., 8 Dec 2025).

5. Concentration and Robustness under Contamination

In the Huber ε-contamination model ($u$3), the sample α-scatter median $u$4 achieves minimax-optimal error bounds. With enough samples ($u$5 large), there exist constants $u$6 such that for any $u$7:

$u$8

Thus, the estimator converges at rate $u$9 (up to $u^T X \equiv_d \|u\|_\alpha X_1$0 factors) in the pseudometric $u^T X \equiv_d \|u\|_\alpha X_1$1 (Bočinec et al., 8 Dec 2025).

6. Computational Procedure and Practical Implications

For α-symmetric laws, only scalar multiples of the identity matrix are candidates for $u^T X \equiv_d \|u\|_\alpha X_1$2. In practice:

• One discretizes the sphere $u^T X \equiv_d \|u\|_\alpha X_1$3 to approximate the depth infimum across directions $u^T X \equiv_d \|u\|_\alpha X_1$4.
• Empirical versions of the depth can be evaluated using plug-in sample statistics and quantile estimation for $u^T X \equiv_d \|u\|_\alpha X_1$5 at $u^T X \equiv_d \|u\|_\alpha X_1$6.

This approach efficiently estimates scatter in multivariate models where the underlying distribution deviates from elliptical symmetry, accommodating heavy tails and more complex dependency structures via the α-norm geometry.