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For the standard kernel density estimate, it is known that one can tune the bandwidth such that the expected L1 error is within a constant factor of the optimal L1 error (obtained when one is allowed to choose the bandwidth with knowledge ofthe density). In this paper, we pose the same problem f or variable bandwidth kernel estimates where the bandwidths are allowed to depend upon the location. We show in particular that for positive kernels on the real line, for any data-based bandwidth, there exists a density for which the ratio of expected L1 error over optimal L1 error tends to infinity. Thus, the problem oftuning the variable bandwidth in an optimal manner is "too hard". Moreover, from the class ofcounterexamples exhibited in the paper, it appears that placing conditions on the densities (monotonicity, convexity, smoothness) does not help.

Variable Kernel Estimates: On the Impossibility of Tuning the Parameters