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What is the influence function?

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Answer

  • For a functional statistic T(F): IF(x  T,  F)  =  limε→0\mathrm{IF}(x\;T, \;F)\; = \;\mathrm{lim}_{{\varepsilon} \to 0} (T((1−ε)F  +  εδx)  −  T(F)T((1 - {\varepsilon})F\; + \;{\varepsilon}{\delta}_{x})\; - \;T(F)) / ε.
  • Measures how much T changes if a single point x is added.
  • Uses: (1) derive asymptotic variance via Var⁡(n  T)\operatorname{Var}( \sqrt n\;T) → E[IF2]E[\mathrm{IF}^{2}]; (2) build robust M-estimators (Huber) with bounded IF; (3) identify influential observations.
  • Foundation of modern robust statistics and semi-parametric efficiency theory.
Check yourself — multiple choice
  • Random
  • Directional derivative of T at F toward δx{\delta}_{x}; used for asymptotic variance, robustness (bounded IF), influential observation detection
  • Same as leverage
  • Not real

Influence function: asymptotic variance + robustness + influential-point detection.

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