Influence Function Of The MCD Estimator

Why the MCD Influence Function Matters

The minimum covariance determinant (MCD) retains a fixed fraction of observations and chooses a subset whose covariance determinant is small. That trimming is central to the estimator’s resistance to extreme observations, but it also means its local response depends on whether a contaminating point falls inside or outside the selected ellipsoid. The influence function makes this response precise.

We derive the population influence functions at an elliptical distribution. The location formula shows directly why a point sufficiently far from the center has no first-order effect on location. The scatter formula has two parts: a radial effect on overall size and a trace-free effect on shape. Its shape coefficient also reveals how the geometry of the selected boundary feeds back into the scatter estimate.

Population MCD and Its Influence Function

For a functional \(G\), its influence function at \(F\) is the first-order change under point-mass contamination:

when this derivative exists; \(\Delta_z\) is a point mass at \(z\). We first work in standardized coordinates. Let \(Y\) have a spherical distribution centered at zero, let \(R=\|Y\|\) have density \(g\), and choose \(q>0\) such that \(P(R\le q)=\alpha\). Assume the distribution has a density, \(g\) is positive and continuous at \(q\), the relevant truncated second moments are finite, and the local MCD solution is differentiable. Define

where \(p\) is the dimension and \(I(\cdot)\) is the indicator function. We also require \(\alpha-qg(q)/p\ne0\) and \(1-\lambda\ne0\). At this spherical model, the raw MCD functionals are \(T_\alpha(F)=0\) and \(S_\alpha(F)=sI_p\).

The raw scatter influence function is

where

Deriving the Location Formula

At the spherical model, the selected region is the ball of radius \(q\). Write its center as \(t\), and consider the trimmed first-moment equation

The direct effect of contamination at \(y\) on this equation, at \(t=0\), is \(yI(r\le q)\). The derivative of the integral with respect to the center has two contributions. Differentiating \(x-t\) gives \(-\alpha I_p\). Translating the boundary contributes \(qg(q)I_p/p\): on the boundary \(x=qu\), and spherical symmetry gives the angular average \(E[uu']=I_p/p\). Thus the derivative with respect to \(t\) is \((-\alpha+qg(q)/p)I_p\). Setting the first-order change in the equation to zero yields the stated formula. The denominator condition ensures that this linear equation has a unique solution. \(\square\)

The indicator is decisive: if \(r>q\), the location influence is zero. A contaminating point outside the selected region does not enter the trimmed first moment to first order. The boundary still affects the denominator, however, because a small shift in the center changes which observations lie near the trimming boundary.

Deriving the Scatter Formula

Decompose a symmetric matrix into a scalar part and a trace-free part. The scalar part changes overall size; the trace-free part changes shape without changing the first-order trace. This separates two effects that would otherwise be hidden in a single matrix expression.

For the scalar part, contamination directly changes the conditional second moment by \(r^2I(r\le q)/(p\alpha)-s\). The trimming probability must remain \(\alpha\). If the boundary radius changes by \(dq\), its probability changes by \(g(q)dq\), while contamination changes the probability by \(I(r\le q)-\alpha\). Therefore

Moving the boundary changes the scalar conditional second moment by \(q^2g(q)dq/(p\alpha)\). Adding this to the direct effect gives exactly \(a(r)\).

For the trace-free part, the direct effect is \(I(r\le q)(yy'-r^2I_p/p)/\alpha\). A trace-free perturbation \(A\) of the scatter deforms the boundary in direction \(u\) by a radial amount proportional to \(q\,u'Au/2\). The spherical fourth-moment identity

shows that the boundary deformation feeds back a trace-free contribution \(\lambda A\), with \(\lambda=q^3g(q)/(s\alpha p(p+2))\). The first-order scatter equation is therefore the direct effect plus \(\lambda\) times the unknown trace-free response. Solving it divides the direct effect by \(1-\lambda\). Translation of the selected region contributes no first-order term to scatter at a spherical model: the corresponding boundary integrals are odd in the angular direction. Combining the scalar and trace-free pieces proves the formula. The condition \(1-\lambda\ne0\) makes this linear equation solvable. \(\square\)

Worked Example: Bivariate Normal MCD at Half Trimming

Elliptical Models and Affine Coordinates

Suppose \(X=\mu+\Sigma^{1/2}Y\), where \(\Sigma\) is positive definite and \(Y\) has the spherical distribution used above. The MCD functionals are affine equivariant: their location and scatter transform as \(\mu\) and \(\Sigma\) do. For \(x=\mu+\Sigma^{1/2}y\), the influence functions are therefore

The raw scatter at this elliptical model is \(s\Sigma\), not generally \(\Sigma\). If a fixed consistency multiplier \(c\) is applied to the raw scatter, its influence function is multiplied by \(c\) as well.

Boundedness, Interpretation, and a Common Pitfall

For location, the influence function is zero outside the selected ellipsoid and is bounded inside it, because \(\|y\|\le q\) there and the denominator is fixed and nonzero. For scatter, the trace-free term is also zero outside; the exterior response is the constant scalar matrix \((q^2/p-s)I_p\). Inside, both terms are bounded because the selected region is bounded. Thus, under the stated regularity conditions, both influence functions are bounded for \(r\ne q\); the theorem makes no pointwise assertion at \(r=q\).

Bounded influence is a local robustness property, not a guarantee that every finite-sample computation is stable or that all distributions have a unique MCD solution. Nor does the exterior scatter response mean an extreme point has no effect at all: it can alter the boundary needed to maintain the fixed trimming proportion.

• Pitfall: treating exterior scatter influence as zero. The direct second-moment contribution is zero outside the selected region, but the trimming boundary moves to preserve probability \(\alpha\). The scalar boundary term remains, giving \(q^2/p-s\), not zero.
• Pitfall: using the raw scale incorrectly. The quantity \(s\) is the conditional radial second moment per coordinate, \(E[R^2I(R\le q)]/(p\alpha)\). It is not the untrimmed variance or the unnormalized truncated second moment.
• Pitfall: omitting the shape feedback. The trace-free scatter response is divided by \(1-\lambda\). Keeping only the direct effect ignores the change in the selected ellipsoid as its scatter shape changes.