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 $\Sigma$ and distribution $P$ under elliptical symmetry (α=2) is:

$$SD(\Sigma;P) = \inf_{u\in S^{d-1}} \min \left\{ P(u^T (X-\mu_\text{hs}) \leq \sqrt{u^T\Sigma u}),\; P(u^T(X-\mu_\text{hs}) \geq \sqrt{u^T\Sigma u}) \right\}$$

Maximizing this yields the scatter halfspace median. For $\alpha \neq 2$, quadratic forms do not respect the underlying geometry. The α-scatter halfspace depth is defined as:

$$SD_\alpha(\Sigma;P) = \inf_{u\in S^{d-1}} \min \left\{ P(u^T (X-\mu_\text{hs}) \leq \|\Sigma^{1/2} u\|_\alpha),\; P(u^T(X-\mu_\text{hs}) \geq \|\Sigma^{1/2} u\|_\alpha) \right\},$$

where $\Sigma^{1/2}$ is the matrix square root in $\mathcal{S}_+^d$ and $\|\cdot\|_\alpha$ 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 $\Sigma_{hs,\alpha}(P)$ is any maximizer of $SD_\alpha(\cdot;P)$:

$$\Sigma_{hs,\alpha}(P) \in \arg \max_{\Sigma \in \mathcal{S}_+^d} SD_\alpha(\Sigma;P)$$

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

$$\Sigma_{hs,\alpha}(P) = \sigma^2 I_d, \quad \sigma = F^{-1}(3/4)$$

where $F$ denotes the common marginal distribution of $X_1$. Uniqueness and sphericity derive from affine and sign-permutation invariance, restricting maximizers to scalar multiples of the identity. The value $\sigma$ is chosen to balance the probabilities in the halfspace depth definition, solving $F(\sigma d^{1/2-1/\alpha}) - 1/2 = 1 - F(\sigma)$.

4. Robustness, Equivariance, and Breakdown Properties

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

• Affine Equivariance: For nonsingular $A$ and any location $\mu$, $$\Sigma_{hs,\alpha}(P_{AX+\mu}) = A\Sigma_{hs,\alpha}(P_X) A^T$$.
• Sign-Permutation Invariance: For α-symmetric $P$, $\Sigma_{hs,\alpha}$ 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 $(\Sigma, P) \mapsto SD_\alpha(\Sigma;P)$ 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 ($P' = (1-\epsilon)P + \epsilon Q$), the sample α-scatter median $\widehat{\Sigma}_{hs,\alpha,n}$ achieves minimax-optimal error bounds. With enough samples ($n$ large), there exist constants $C_1, C_2>0$ such that for any $\delta \in (0,1/2)$:

$$P\Bigg\{ \sup_{u\in S^{d-1}} \Bigg| \frac{(\widehat{\Sigma}_{hs,\alpha,n}^{1/2}u)_\alpha}{\|u\|_\alpha} - \sigma \Bigg| \leq C\left(\epsilon + \sqrt{d/n} + \sqrt{\ln(1/\delta)/n}\right) \Bigg\} \geq 1-2\delta$$

Thus, the estimator converges at rate $O(\epsilon + \sqrt{d/n})$ (up to $\log\delta$ factors) in the pseudometric $$\sup_u(|\|\Sigma^{1/2}u\|_\alpha/\|u\|_\alpha - \|\Sigma'^{1/2}u\|_\alpha/\|u\|_\alpha|)$$ (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 $\Sigma_{hs,\alpha}(P)$. In practice:

• One discretizes the sphere $S^{d-1}$ to approximate the depth infimum across directions $u$.
• Empirical versions of the depth can be evaluated using plug-in sample statistics and quantile estimation for $F$ at $3/4$.

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.