Infinitesimal Contamination and Influence
A location or scatter procedure is a rule that maps a population distribution to a parameter. To assess its local sensitivity, replace a small fraction of the population by observations fixed at a point \(y\). The first-order change in the parameter describes how strongly, and in what direction, that point affects the procedure. In multivariate problems this change is itself directional: location influence is a vector, while scatter influence is a matrix.
The distribution \(F\) is the reference model, so the influence function depends on both \(y\) and \(F\). It is a local, infinitesimal sensitivity measure—not the change caused by a large contamination fraction, and not by itself a guarantee of good performance at a particular sample size. The derivative may fail to exist at some contaminating points; when it does, its size and orientation describe the functional’s first-order response.
Deriving Influence from an Estimating Equation
A useful general derivation applies when a location functional is defined as a root of an estimating equation. Let \(\psi(x,t)\in\mathbb{R}^p\), and suppose \(T=T(F)\) satisfies \(\Psi(t,F)=E_F[\psi(X,t)]=0\). Write \(J=E_F[\partial\psi(X,t)/\partial t^\mathsf{T}]_{t=T}\), the \(p\times p\) Jacobian with respect to \(t\).
It is \((1-\varepsilon)E_F[\psi(X,T_\varepsilon)]+\varepsilon\psi(y,T_\varepsilon)=0\).
The derivative of the first term is \(J T'_0\); the derivative of its factor \(1-\varepsilon\) contributes \(-E_F[\psi(X,T)]=0\). The point-mass term contributes \(\psi(y,T)\).
Thus \(J T'_0+\psi(y,T)=0\). Since \(J\) is nonsingular, \(T'_0=-J^{-1}\psi(y,T)\).
The assumptions matter. Nonsingularity ensures that a small change in the estimating equation corresponds locally to a well-defined change in its root. The integrable bound permits differentiating the expectation; without suitable regularity, a formal derivative inside an integral need not be valid. This result is a general calculation tool, not a claim that every location functional is defined by a smooth estimating equation.
Worked Example: The Coordinatewise Median
Let \(F\) be the bivariate standard normal distribution, so each coordinate has density \(\phi\) and median zero. Define \(T(F)\) by taking the median of each marginal distribution. Contaminate at \(y=(2,-5)^\mathsf{T}\), whose coordinates are both nonzero.
For a univariate marginal with median \(m_\varepsilon\), the contaminated distribution function is \(F_\varepsilon(m)=(1-\varepsilon)F(m)+\varepsilon\mathbf{1}\{y_j\leq m\}\). Since \(y_j\neq0\), for sufficiently small \(\varepsilon\) the indicator at \(m_\varepsilon\) is locally constant, with value \(\mathbf{1}\{y_j\leq0\}\). Differentiating \(F_\varepsilon(m_\varepsilon)=1/2\) at zero gives \(\phi(0)m'_0+\mathbf{1}\{y_j\leq0\}-1/2=0\).
Because \(\phi(0)=1/\sqrt{2\pi}\), the first coordinate of the influence is \(+(1/2)/\phi(0)=\sqrt{\pi/2}\), while the second is \(-(1/2)/\phi(0)=-\sqrt{\pi/2}\). Therefore \(\operatorname{IF}(y;T,F)=(\sqrt{\pi/2},-\sqrt{\pi/2})^\mathsf{T}\).
The magnitude of each coordinate of \(y\) does not enter this answer: for this example, its sign determines the coordinatewise median’s first-order response. The calculation uses \(y_j\neq0\); at a contaminating coordinate equal to the population median, the atom is located at the quantile itself and the derivative requires separate treatment.
A Directional Example: The Spatial Median
The spatial median provides a genuinely vector-valued location functional. For a distribution with a unique spatial median, define \(T(F)\) as a solution of \(E_F[(X-T)/\|X-T\|]=0\). The vector inside the expectation is the unit direction from \(T\) to \(X\). At \(X=T\), it can be assigned any value if \(F\) has no atom there. The following calculation uses \(F=N_p(0,I_p)\), \(p>1\), and a contaminating point \(y\neq0\).
To derive the formula, put \(u=x/\|x\|\) and \(r=\|x\|\). The derivative with respect to \(t\) of \((x-t)/\|x-t\|\), evaluated at \(t=0\), is \(-(I_p-uu^\mathsf{T})/r\). Rotational symmetry gives \(E[uu^\mathsf{T}]=I_p/p\), so the estimating-equation Jacobian is \(J=-\frac{p-1}{p}E[1/R]I_p\). The point-contamination contribution is \(\psi(y,0)=y/\|y\|\). Applying the implicit-equation result yields the stated influence function.
Worked Example: Spatial-Median Influence in Two Dimensions
Take \(p=2\) and \(F=N_2(0,I_2)\). The radius \(R\) has density \(r\exp(-r^2/2)\) for \(r>0\), and therefore \(E[1/R]=\int_0^\infty \exp(-r^2/2)\,dr=\sqrt{\pi/2}\). For \(y=(3,4)^\mathsf{T}\), its radius is \(5\), so \(y/\|y\|=(3/5,4/5)^\mathsf{T}\). The influence is \(\frac{2}{\sqrt{\pi/2}}(3/5,4/5)^\mathsf{T} =2\sqrt{2/\pi}(3/5,4/5)^\mathsf{T}\), approximately \((0.958,1.277)^\mathsf{T}\).
Changing \(y=(3,4)^\mathsf{T}\) to \((30,40)^\mathsf{T}\) leaves the influence unchanged, because the direction is the same. Changing the direction changes the vector: a point on the first coordinate axis produces influence along that axis. This is bounded in magnitude as \(y\) moves farther away, but it is not direction-free.
Scatter-Shape Influence Can Depend on Orientation
For a transparent scatter calculation, consider a normalized spatial-sign shape functional. Let \(T(F)\) be the spatial median and define \(C(F)=E_F[q_T(X)q_T(X)^\mathsf{T}]\), where \(q_t(x)=(x-t)/\|x-t\|\) for \(x\neq t\). This matrix has trace one, so it records shape rather than overall scale. At a spherical distribution with center zero, rotational symmetry gives \(C(F)=I_p/p\).
For \(F=N_p(0,I_p)\), \(p>1\), and \(y\neq0\), the influence is \(\operatorname{IF}(y;C,F)=\frac{yy^\mathsf{T}}{\|y\|^2}-\frac{I_p}{p}\). To see why the movement of the center does not add a first-order term, differentiate \(q_t(x)q_t(x)^\mathsf{T}\) with respect to \(t\) at zero in a direction \(h\): \(-\{(I_p-uu^\mathsf{T})hu^\mathsf{T}+uh^\mathsf{T}(I_p-uu^\mathsf{T})\}/r\). Its expectation is zero: under \(x\mapsto-x\), \(u\) changes sign, \(I_p-uu^\mathsf{T}\) and \(r\) do not, so each integrand changes sign. The derivative is integrable because its entries are bounded by a constant times \(1/R\), whose expectation is finite for \(p>1\). Thus the center’s first-order movement contributes zero. The direct point-mass contribution is \(q_0(y)q_0(y)^\mathsf{T}\), while contamination removes an \(\varepsilon\) fraction of the baseline \(I_p/p\), giving the formula.
Worked Example: A Bivariate Shape Direction
For the standard bivariate normal and \(y=(3,4)^\mathsf{T}\), the unit direction is \(q=(3/5,4/5)^\mathsf{T}\). Hence \(qq^\mathsf{T}=\begin{pmatrix}0.36&0.48\\0.48&0.64\end{pmatrix}\), and the shape influence is \(\begin{pmatrix}0.36&0.48\\0.48&0.64\end{pmatrix}-\begin{pmatrix}0.5&0\\0&0.5\end{pmatrix} =\begin{pmatrix}-0.14&0.48\\0.48&0.14\end{pmatrix}\).
This influence has trace zero, as it must for a functional constrained to have trace one. Along the contaminant direction \(q\), its eigenvalue is \(1/2\); along the perpendicular direction, its eigenvalue is \(-1/2\). Thus the point’s orientation determines which direction is relatively emphasized and which is relatively reduced. Its distance from the center does not enter this particular functional’s influence.
Affine Changes of Coordinates
Influence also respects the coordinate transformation of an affine-equivariant functional. Let \(A\) be an invertible \(p\times p\) matrix and \(b\in\mathbb{R}^p\), and let \(F^{A,b}\) be the distribution of \(AX+b\) when \(X\sim F\). If a location functional satisfies \(T(F^{A,b})=AT(F)+b\), then its influence transforms as \(\operatorname{IF}(Ay+b;T,F^{A,b})=A\operatorname{IF}(y;T,F)\). If a scatter functional satisfies \(S(F^{A,b})=AS(F)A^\mathsf{T}\), then \(\operatorname{IF}(Ay+b;S,F^{A,b})=A\operatorname{IF}(y;S,F)A^\mathsf{T}\).
For the location identity, the contaminated distribution after transformation is exactly \((1-\varepsilon)F^{A,b}+\varepsilon\Delta_{Ay+b}\). Apply affine equivariance to this distribution, subtract the uncontaminated identity \(T(F^{A,b})=AT(F)+b\), divide by \(\varepsilon\), and take the right limit as \(\varepsilon\) decreases to zero. This gives the location formula. For scatter, the same steps use scatter equivariance and yield the matrix formula. These transformations describe how influence coordinates change; they do not imply that every functional, including the normalized shape example above, is affine equivariant. \(\square\)
Interpretation and a Common Pitfall
An influence function should be read as a map from possible contaminating points to first-order parameter changes. For location, its vector direction and norm both matter. For scatter, the matrix entries describe changes to variances and cross-products; eigenvectors can reveal the orientations of increase and decrease. A radial pattern is only one possibility: the coordinatewise median responds by coordinate signs, while the spatial median responds to the full unit direction.
- Common mistake: treating influence as a function only of distance. For the spatial median at a spherical model, the formula is a constant times \(y/\|y\|\). Two contaminants at equal distances but different directions generally produce different location changes.
- Correction: Evaluate the full vector or matrix at the contaminant point. Separate radial distance from angular direction only when the derived formula justifies doing so.
- Common mistake: confusing a bounded influence function with a complete robustness guarantee. Influence is a local derivative at a specified \(F\); it does not alone describe finite contamination or breakdown.
- Correction: Interpret it as a first-order sensitivity diagnostic, and keep the reference distribution and the functional’s definition explicit.
Check Your Understanding
Use the definitions and derivations above to answer these questions.
- Write the point-contamination path \(F_{\varepsilon,y}\) and explain what the influence function differentiates.
- For an estimating-equation location functional, what nonsingularity condition is needed to obtain the derivative formula?
- Why does the spatial median’s influence at a spherical normal model depend on the direction of \(y\), but not its distance?
- For the normalized spatial-sign shape functional, why must the influence matrix have trace zero?
- State the affine transformation rule for a location influence function and identify the transformed contaminating point.