VARIANCE REDUCTION BY SMOOTHING REVISITED
Andrzej S. Kozek
Brian Jersky
Abstract: Smoothing is a common method used in nonparametric statistics and on many
occasions it has been noted that it may result in an asymptotic variance reduction or increase
of efficiency. Another well-known effect associated with smoothing is that it introduces a
small bias. In the first part of the paper we show that if the influence function of a
Hadamard-differentiable statistical functional or its derivative have jumps, then functionals of
a kernel-smoothed cumulative distribution function may have lower asymptotic
variance than the variance of the original functional. This extends and unifies previous
results and shows detailed conditions under which the asymptotic variance reduction
by smoothing can be achieved. The smoothing however introduces a small bias
of order , where is a smoothing parameter. In the second part of the
paper we discuss the optimal balance between the bias and variance reduction.
2000 AMS Mathematics Subject Classification: Primary: 62F12; Secondary: 62E20,
62G20, 62G35.
Keywords and phrases: Smoothing; influence curve; variance reduction; robustness;
statistical functionals.