Differentiability of Statistical Functionals and Consistency of the Jackknife

In statistical applications the unknown parameter of interest can frequently be defined as a functional θ=T(F), where F is an unknown population. Statistical inferences about θ are usually made based on the statistic T(Fₙ), where Fₙ is the empirical distribution. Assessing T(Fₙ) (as an estimator of θ) or making large sample inferences usually requires a consistent estimator of the asymptotic variance of T(Fₙ). 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(Fₙ) are L-, M-estimators and some test statistics.