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.