Rank transformations in Kernel density estimation

Abstract

We introduce and study a kernel density estimator which takes into account not only the local information contained in the data, but also their global structure given through the ranks. The approach allows for an adaptive choice of the smoothing parameters which avoids estimation of higher order derivatives. Our methodology also leads to a new isoperimetric problem which seems to be of independent interest. While in traditional kernel smoothing efficiency is obtained by choosing proper kernels, this role is now played by appropriate rank transformations.

Keywords:

• kernel density estimator
• adaptive choice
• isoperimetric problem
• ranks
• empirical distribution

AMS Subject Classifications :

• 62G07
• 62G20
• 62G30

Acknowledgements

We are grateful to two anonymous referees for their helpful comments on an earlier version of the paper.