Nonlinear kernel density estimation for binned data: convergence in entropy

Abstract

A method is proposed for creating a smooth kernel density estimate from a sample of binned data. Simulations indicate that this method produces an estimate for relatively finely binned data which is close to what one would obtain using the original unbinned data. The kernel density estimate $\hat {f}$, is the stationary distribution of a Markov process resembling the Ornstein-Uhlenbeck process. This $\hat {f}$, may be found by an iteration scheme which converges at a geometric rate in the entropy pseudo-metric, and hence in $L^1$, and transportation metrics. The proof uses a logarithmic Sobolev inequality comparing relative Shannon entropy and relative Fisher information with respect to $\hat {f}$.