Why Equivariance and Calibration Are Separate
The minimum covariance determinant estimator (MCD) is designed to find a location and scatter from a subset whose covariance determinant is small. Two questions must be kept distinct when interpreting its output. Affine equivariance asks whether the estimates transform correctly when the coordinates are shifted, rescaled, or mixed. Calibration asks whether the raw scatter has the scale of the target population scatter under a specified model. An estimator can satisfy the first property while its uncorrected scatter is systematically too small.
The distinction matters because the MCD deliberately uses a subset rather than all observations. Under a Gaussian model, observations retained near the center have less dispersion than the full population. A scatter correction restores the intended scale; it does not change which subset minimizes the determinant.
Finite-Sample Definition and Affine Equivariance
The lower bound \(h\geq p+1\) permits a subset covariance to have full rank, although particular data can still produce singular subsets. If the minimizing subset is not unique, the MCD may be set-valued. Statements about a single selected estimate below assume a unique minimizing subset, or a tie-breaking rule that respects the transformation in question.
Proof. For any candidate subset \(H\), linearity of the mean gives \(\bar{y}_H=A\bar{x}_H+b\). Therefore \(y_i-\bar{y}_H=A(x_i-\bar{x}_H)\), and substitution into the covariance definition gives \(S_H(y)=AS_H(x)A^{\mathsf T}\). Taking determinants, $$ \det S_H(y)=\det(A)^2\det S_H(x). $$ The factor \(\det(A)^2\) is positive and is the same for every candidate subset. Consequently a subset minimizes the transformed determinant exactly when it minimizes the original determinant. Applying the mean and scatter identities to the selected subset proves both transformation equations. \(\square\)
This argument also shows why a nonsingular coordinate change preserves the minimizing subsets. It does not guarantee that an arbitrary tie-breaking procedure will select corresponding subsets when several minimizers exist. A rule based on the original coordinate ordering, for example, might not be affine-equivariant in a tied case.
Worked Example: Transforming a Raw MCD Estimate
Take three points in \(\mathbb{R}^2\), with \(h=n=3\): \(x_1=(0,0)^{\mathsf T}\), \(x_2=(2,0)^{\mathsf T}\), and \(x_3=(0,1)^{\mathsf T}\). Since all observations must be used, the only candidate subset is the MCD subset.
Their mean is \(\bar{x}=(2/3,1/3)^{\mathsf T}\). Using denominator \(3\), the covariance is $$ S=\begin{pmatrix}8/9&-2/9\\-2/9&2/9\end{pmatrix}, \qquad \det(S)=\frac{16-4}{81}=\frac{4}{27}. $$
Now transform the data by \(y=Ax+b\), where \(A=\begin{pmatrix}2&1\\0&1\end{pmatrix}\) and \(b=(1,-1)^{\mathsf T}\). Since \(\det(A)=2\), the transformed mean is $$ A\bar{x}+b=(5/3,1/3)^{\mathsf T}+(1,-1)^{\mathsf T}=(8/3,-2/3)^{\mathsf T}. $$ The transformed covariance is \(ASA^{\mathsf T}\), and its determinant is \(\det(A)^2\det(S)=4(4/27)=16/27\). This verifies both the scatter transformation and the determinant scaling.
Gaussian Consistency Calibration
Affine equivariance reduces the Gaussian calibration calculation to a spherical model. Suppose \(X\sim N_p(\mu,\Sigma)\), with \(\Sigma\) positive definite. For a retained fraction \(\alpha\), the Gaussian population MCD retains the central ellipsoid $$ \{x:(x-\mu)^{\mathsf T}\Sigma^{-1}(x-\mu)\leq q\}, \qquad q=F_{\chi_p^2}^{-1}(\alpha), $$ where \(F_{\chi_p^2}\) is the distribution function of a chi-squared random variable with \(p\) degrees of freedom. This is the Gaussian population MCD characterization; the calculation below derives the scatter scale of that retained central region.
Write \(Z=\Sigma^{-1/2}(X-\mu)\). Then \(Z\sim N_p(0,I_p)\), and \(R^2=Z^{\mathsf T}Z\sim\chi_p^2\). Conditional on \(R^2\leq q\), symmetry gives \(E[Z\mid R^2\leq q]=0\). Rotational symmetry also makes the conditional covariance a scalar multiple of \(I_p\). Its trace is the conditional expectation of \(R^2\), so the scalar is \(E[R^2\mid R^2\leq q]/p\).
The conditional mean is \(\mu\), so the Gaussian population location needs no multiplicative scale correction. For scatter, the factor is less than one for central trimming: retaining only the central fraction reduces radial second moments. The factor can also be expressed as $$ c_{p,\alpha}=\frac{F_{\chi_{p+2}^2}(q)}{\alpha}. $$ Indeed, if \(W\sim\chi_p^2\), its density \(f_p\) satisfies \(w f_p(w)=p f_{p+2}(w)\). Integrating from \(0\) to \(q\) gives \(E[W\,\mathbf{1}\{W\leq q\}]=pF_{\chi_{p+2}^2}(q)\). Dividing by \(pP(W\leq q)=p\alpha\) yields the expression.
Worked Example: A Two-Dimensional Gaussian at Half Retention
Let \(p=2\) and \(\alpha=1/2\). A \(\chi_2^2\) variable has density \(f(w)=\frac12 e^{-w/2}\) for \(w\geq0\), so its median \(q\) solves \(1-e^{-q/2}=1/2\). Hence \(q=2\log 2\).
The truncated first moment is $$ E[W\mathbf{1}\{W\leq q\}] =\int_0^q \frac{w}{2}e^{-w/2}\,dw =2\left[1-(1+q/2)e^{-q/2}\right] =1-\log 2. $$ Since \(P(W\leq q)=1/2\), the conditional mean is \(E[W\mid W\leq q]=2(1-\log 2)\). Dividing by \(p=2\) gives $$ c_{2,1/2}=1-\log 2\approx0.3068528,\qquad c_{2,1/2}^{-1}=\frac{1}{1-\log 2}\approx3.259. $$ Thus the raw population scatter is \((1-\log 2)\Sigma\), and multiplying it by the reciprocal restores \(\Sigma\). The displayed reciprocal is calculated from the unrounded expression; rounding the factor first can slightly change the last digits.
Finite-Sample Corrections
The population consistency factor describes the limiting target under a model and a retained fraction. It is not generally the exact finite-sample correction. For finite \(n,p,h\), the raw MCD scatter has additional sampling bias. Under a Gaussian model and an affine-equivariant construction, its expected scatter, when finite, has the form \(b_{n,p,h}\Sigma\) for a scalar \(b_{n,p,h}\). A finite-sample correction divides by this scalar. The value depends on \(n,p,h\), the precise estimator definition, and the covariance denominator.
For the usual MCD with \(h<n\), these finite-sample factors are commonly obtained from a specified calibration calculation or simulation for the chosen convention. They should not be silently replaced by the asymptotic consistency factor. The following special case gives an exact factor without simulation.
Worked Example: Exact Finite-Sample Correction When \(h=n\)
If \(h=n\), there is only one subset, so the raw MCD scatter is the covariance with denominator \(n\): $$ S=\frac{1}{n}\sum_{i=1}^n(X_i-\bar{X})(X_i-\bar{X})^{\mathsf T}. $$ For independent observations with \(E[X_i]=\mu\) and \(\operatorname{Cov}(X_i)=\Sigma\), the standard identity \(E[\sum_i(X_i-\bar{X})(X_i-\bar{X})^{\mathsf T}]=(n-1)\Sigma\) gives \(E[S]=(n-1)\Sigma/n\). Therefore the exact finite-sample correction is \(n/(n-1)\).
For example, with \(n=5\), the raw scatter has expectation \(4\Sigma/5\), and multiplying it by \(5/4\) gives an unbiased scatter matrix. This is a finite-sample correction for this particular \(h=n\) estimator; it is not the Gaussian central-trimming consistency factor for \(h<n\).
Under a centrally symmetric model, the raw MCD location is centered at the population location when its expectation exists and the estimator is defined symmetrically. Reflection of every observation about \(\mu\) maps the MCD location to \(2\mu-\hat{\mu}_{\mathrm{MCD}}\); the reflected data have the same distribution. Taking expectations gives \(E[\hat{\mu}_{\mathrm{MCD}}]=2\mu-E[\hat{\mu}_{\mathrm{MCD}}]\), hence \(E[\hat{\mu}_{\mathrm{MCD}}]=\mu\). This centering statement is not a claim that every finite-sample MCD location has small variance or that a location scale correction should be applied.
Common Mistakes and Practical Interpretation
- Assuming affine equivariance means correct scale. Equivariance specifies how an estimate changes with coordinates; Gaussian trimming can still make the raw scatter too small.
- Using the wrong correction target. A population consistency factor and a finite-sample bias factor answer different questions. State the model, \(n,p,h\), and covariance convention before applying a factor.
- Forgetting the squared determinant. Scatter transforms as \(A\hat{\Sigma}A^{\mathsf T}\), so its determinant is multiplied by \(\det(A)^2\), not by \(\det(A)\).
- Mixing denominators. The definition here uses \(h\). Replacing it by \(h-1\) changes the raw scatter and therefore changes any correction factor.
- Applying the scatter correction to location. The Gaussian retained-region location remains \(\mu\); the scalar consistency correction is for scatter.
Calibration makes robust scatter estimates interpretable on the scale of the target distribution, while affine equivariance ensures that the answer does not depend on an arbitrary nonsingular choice of coordinates. In reporting an MCD result, identify whether the scatter is raw, consistency-corrected, or finite-sample corrected. Without that information, two numerical scatter estimates may differ simply because they target different scales.
Check Your Understanding
Answer each question by identifying the transformation or correction being used.
- Under \(y_i=Ax_i+b\), what are the transformed MCD location and scatter, and how does the candidate-subset determinant change?
- Why does multiplying every candidate determinant by the same positive factor preserve the minimizing subset?
- For the two-dimensional Gaussian example with half retention, what is the exact consistency factor, and why is the correction applied to scatter rather than location?
- With denominator \(n\) and \(h=n\), derive the finite-sample correction in terms of \(n\).
- Why should a finite-sample factor for \(h<n\) not automatically be replaced by the population consistency factor?