Minimum covariance determinant estimator: Definition & Examples

The minimum covariance determinant (MCD) estimator estimates multivariate location and scatter by finding a subset of a specified size whose covariance matrix has the smallest determinant. It gives a quant an outlier-resistant reference for comparing vectors of returns or market features: a few unusual rows need not dictate the centre and shape of the whole cloud. This is a method of robust statistics. The defining subset objective appears in Rousseeuw and Van Driessen’s FastMCD paper.

The subset is the optimisation variable

For observations $\mathbf{x}_1,\ldots,\mathbf{x}_n\in\mathbb{R}^p$ , choose a subset size $h$ . For each index set $H$ with $|H|=h$ , calculate

\bar{\mathbf{x}}_H=\frac{1}{h}\sum_{i\in H}\mathbf{x}_i, \qquad S_H=\frac{1}{h}\sum_{i\in H}(\mathbf{x}_i-\bar{\mathbf{x}}_H)(\mathbf{x}_i-\bar{\mathbf{x}}_H)^\top.

The uncorrected subset problem is

H_*\in\operatorname*{arg\,min}_{H:\,|H|=h}\det(S_H).

The determinant measures the product of directional variances. For positive definite $S_H$ , a fixed-distance ellipsoid has volume proportional to $\sqrt{\det(S_H)}$ : the objective seeks a compact joint cloud. It does not simply keep the $h$ observations nearest to the ordinary sample mean.

Here the divisor is $h$ , rather than the unbiased-sample convention

h-1

. For fixed

h

, switching conventions multiplies every determinant by the same factor

(h/(h-1))^p

and leaves the minimising subset unchanged.

Five points, all five candidate subsets

Consider these fictional, unitless two-feature observations:

A=(-1,0),\quad B=(0,-1),\quad C=(0,1),\quad D=(1,0),\quad E=(10,10).

Set $h=4$ . There are only $\binom{5}{4}=5$ candidates, so exhaustive enumeration settles the optimisation exactly.

Included observations	$\det(S_H)$ , divisor $4$
$A,B,C,D$	$0.25$
$A,B,C,E$	$13.21875$
$A,B,D,E$	$13.21875$
$A,C,D,E$	$11.96875$
$B,C,D,E$	$11.96875$

The unique minimiser excludes $E$ . Its centre and empirical scatter are

\bar{\mathbf{x}}_{H_*}=\begin{pmatrix}0\\0\end{pmatrix}, \qquad S_{H_*}=\begin{pmatrix}0.5&0\\0&0.5\end{pmatrix}, \qquad \det(S_{H_*})=0.25.

By contrast, using all five points with divisor $5$ gives centre $(2,2)$ and

S_{\mathrm{all}}=\begin{pmatrix}16.4&16\\16&16.4\end{pmatrix}.

Its implied correlation is

16/16.4=40/41\approx0.9756

, compared with zero for the selected four-point scatter. One jointly extreme row creates nearly perfect positive sample correlation in this example. Neither number establishes a causal relationship or independence.

Fictional five-point example. The all-point centre is (2,2), with scatter diagonal 16.4 and off-diagonal 16. The selected four points have centre (0,0), scatter diagonal 0.5 and determinant 0.25. Equal squared-distance contours show the stretched all-point ellipse and circular selected-subset shape; the right panel magnifies the four selected observations.

Original fictional example. Both panels use equal horizontal and vertical scales internally, with different ranges. Contours have squared distance 2; they are geometric comparisons, not probability regions or calibrated outlier thresholds.

Raw support is only one stage

The matrix

S_{H_*}

above is an uncorrected optimisation fixture. A consistency factor rescales the subset scatter for a specified reference model; a finite-sample correction may also apply. Reweighting uses distances from an initial robust fit to recompute location and scatter. These distinctions are explained in Hubert, Debruyne and Rousseeuw, §2.2.

Software terminology deserves attention. The checked scikit-learn MinCovDet documentation distinguishes raw_covariance_ and raw_support_ from the final covariance_ and support_ after correction and reweighting. Thus

0.5I

is not a promised default library result. Report the subset size, divisor, correction and reweighting choices. Exhaustive enumeration is feasible here; larger problems commonly use FastMCD’s search algorithm.

A fitted robust scatter can replace the ordinary covariance in a Mahalanobis distance. The probability calibration of those fitted distances is a separate question.

What the robustness claim requires

For nonsingular subset scatter,

h>p

is necessary; collinearity can still make it singular. Under general position—no more than

p

observations on any affine hyperplane—the raw MCD’s finite-sample replacement breakdown point is

\varepsilon_n^*=\frac{\min(n-h+1,h-p)}{n}.

Breakdown allows scatter to explode or collapse to singularity. Here no three points are collinear, and the formula gives

2/5

: two adversarial replacements can break the estimator. It does not promise tolerance of 40% contamination. Near-50% breakdown is an asymptotic statement with suitable

h

, not a universal finite-sample guarantee. Increasing

h

typically trades robustness for efficiency. See §3.2 of the MCD review.

The scikit-learn guidance targets Gaussian or unimodal symmetric data and cautions against multimodal inputs. In market data, a compact central regime and a genuine stress regime may represent different economic questions. Excluding a stress observation from a robust reference does not establish that it was a recording error or should disappear from risk analysis.

For the foundations, Expectation, Variance and Covariance explains scale, co-movement and portfolio variance; its full lesson is available with a free signed-in account. Eigenvalues and Covariance Matrices develops the directional-variance interpretation; the full lesson requires membership, with a public preview.

Sources

Peter J. Rousseeuw and Katrien Van Driessen, “A Fast Algorithm for the Minimum Covariance Determinant Estimator” (1999), abstract and opening discussion: subset objective and FastMCD motivation.
Mia Hubert, Michiel Debruyne and Peter J. Rousseeuw, “Minimum Covariance Determinant and Extensions” (2018), §§2.2 and 3.2: corrected and reweighted estimates, general position and replacement breakdown.
scikit-learn, MinCovDet API reference, description, parameters and attributes, checked 30 October 2026: model scope and raw versus final fitted outputs.

The subset is the optimisation variable

Five points, all five candidate subsets

Raw support is only one stage

What the robustness claim requires

Sources

Related terms

Correlation

Mahalanobis distance