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

Abstract

A parameter expressed as a functional $T(F)$ of a distribution function (df) $F$ may be estimated by the "statistical function" $T(F_n)$ based on the sample df $F_n$. For analysis of the estimation error $T(F_n) - T(F)$, we adapt the differential approach of von Mises (1947) to exploit stochastic properties of the Kolmogorov-Smirnov distance $\sup_x|F_n(x) - F(x)|$. This leads directly to the central limit theorem (CLT) and law of the iterated logarithm (LIL) for $T(F_n) - T(F)$. 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.