Differentiability of Statistical Functionals and Consistency of the Jackknife

Abstract

In statistical applications the unknown parameter of interest can frequently be defined as a functional $\theta=T(F)$, where F is an unknown population. Statistical inferences about $\theta$ are usually made based on the statistic $T(F_n)$, where $F_n$ is the empirical distribution. Assessing $T(F_n)$ (as an estimator of $\theta$) or making large sample inferences usually requires a consistent estimator of the asymptotic variance of $T(F_n)$. Asymptotic behavior of the jackknife variance estimator is closely related to the smoothness of the functional T. This paper studies the smoothness of T through the differentiability of T and establishes some general results for the consistency of the jackknife variance estimators. The results are applied to some examples in which the statistics $T(F_n)$ are L-, M-estimators and some test statistics.