 Solution

Several finite-sample counterparts of the influence function should be distinguished. The empirical influence function substitutes the empirical distribution: ${\mathrm{EIF}}_n(x)=\mathrm{IF}(x;T,F_n)$. For a smooth M-estimator it isIt requires a well-defined derivative and nonsingular denominator. An unsmoothed empirical quantile functional, for example, need not admit the population-style density derivative.

The actual one-observation sensitivity curve is a finite difference:It records contamination of size $1/(n+1)$ rather than an infinitesimal perturbation. For a smooth functional it approaches the population influence curve as the sample grows. A deletion counterpart uses $F_{n,-i}=(n{F}_n-{\delta }_{X_i})/(n-1)$, givingThe jackknife resampling pseudovalue $J_i=nT(F_n)-(n-1)T(F_{n,-i})$ is therefore approximately $T(F_n)+{\mathrm{EIF}}_n(X_i)$. The jackknife variance estimate $(n-1)n^{-1}{\sum }_i[T(F_{n,-i})-{\overline{T}}_{(-)}{]}^2$ corresponds asymptotically to $n^{-2}{\sum }_i{\mathrm{EIF}}_n(X_i{)}^2$ for centered empirical influences.

In linear regression, Cook's distance measures the fitted-value effect of deleting a case. For $p$ coefficients, residual $e_i$, leverage $h_{ii}$ and variance estimate ${\hat{\sigma }}^2$, its usual expression is $D_i=e_i^2{h}_{ii}/[p{\hat{\sigma }}^2(1-h_{ii}{)}^2]$. It illustrates that a case can be influential through both a large residual and high leverage. Finite perturbations and nonlinear refitting can differ materially from a first-order influence approximation; these diagnostics should not be conflated.