A Note on Differentials and the CLT and LIL for Statistical Functions, with Application to M-Estimates

A parameter expressed as a functional $T(F)$ of a distribution function (df) F may be estimated by the "statistical function" T(Fₙ) based on the sample df Fₙ. For analysis of the estimation error T(Fₙ) - T(F), we adapt the differential approach of von Mises (1947) to exploit stochastic properties of the Kolmogorov-Smirnov distance ₓ|Fₙ(x) - F(x)|. This leads directly to the central limit theorem (CLT) and law of the iterated logarithm (LIL) for T(Fₙ) - T(F). The adaptation also incorporates innovations designed to broaden the scope of statistical application of the concept of differential. Application to a wide class of robust-type M-estimates is carried out.