Asymptotic distribution theory of statistical functionals: The compact derivative approach for robust estimators

SummaryDerivatives of statistical functionals have been used to derive the asymptotic distributions ofL-,M- andR-estimators. This approach is often heuristic because the types of derivatives chosen have serious limitations. The Gâteaux derivative is too weak and the Fréchet derivative is too strong. In between lies the compact derivative. This paper obtains strong results in a rigorous manner using the compact derivative onC0(R). This choice of space allows results for a broader class of functionals than previous choices, and the fact that $$\left\{ {\sqrt n \left( {\tilde F_n - F} \right)} \right\}$$ is often tight provides the compact set required. A major result is the derivation of the compact derivative of the inverse c.d.f. when the range space is endowed with the uniform norm. It has applications to the asymptotic theory ofL-,M- andR-estimators. We illustrate the power of this result by applications toL-estimators in settings including the one sample problem, data grouped by quantiles, and censored survival time data.

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