= Solution
Several finite-sample counterparts of the <influence function> should be distinguished. The <empirical influence function> substitutes the empirical distribution: $\operatorname{EIF}_n(x)=\operatorname{IF}(x;T,F_n)$. For a smooth <M-estimator> it is
$$
-\left[n^{-1}\sum_i\partial_\theta\psi(X_i,T(F_n))\right]^{-1}\psi(x,T(F_n)).
$$
It 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:
$$
\boxed{\operatorname{SC}_n(x)=(n+1)\left[T\!\left(\frac{nF_n+\delta_x}{n+1}\right)-T(F_n)\right].}
$$
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 $F_{n,-i}=(nF_n-\delta_{X_i})/(n-1)$, giving
$$
T(F_{n,-i})-T(F_n)\approx-\frac{\operatorname{EIF}_n(X_i)}{n-1}.
$$
The <Jackknife resampling> pseudovalue $J_i=nT(F_n)-(n-1)T(F_{n,-i})$ is therefore approximately $T(F_n)+\operatorname{EIF}_n(X_i)$. The jackknife variance estimate $(n-1)n^{-1}\sum_i[T(F_{n,-i})-\overline T_{(-)}]^2$ corresponds asymptotically to $n^{-2}\sum_i\operatorname{EIF}_n(X_i)^2$ 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 $\widehat\sigma^2$, its usual expression is $D_i=e_i^2h_{ii}/[p\widehat\sigma^2(1-h_{ii})^2]$. 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.
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