Iterated modified histograms as dynamical systems

Abstract

A modified histogram estimate depending on a reference density has recently been proved to have good consistency properties according to several information theoretic criteria. In the present paper, this modified histogram is studied as a dynamical system in a functional space with the reference density as initial state. Properties of stationary densities of the system are investigated. It is proved, under appropriate assumptions, that the trajectories are almost surely stationary. Indeed, this result opens up the way for optimization of the estimate with respect to the reference density. Results of simulations are presented.

Keywords:

• Modified histogram estimate
• Dynamical system
• Stationary density
• Quantile