Density Estimation, Stochastic Processes and Prior Information

Summary A method is proposed for the non-parametric estimation of a probability density, based upon a finite number of observations and prior information about the smoothness of the density. A logistic density transform and a reproducing inner product from the first-order autoregressive stochastic process are employed to represent prior information that the derivative of the transform is unlikely to change radically within small intervals. The posterior estimate of the density possesses a continuous second derivative; it typically satisfies the frequentist property of asymptotic consistency. A direct analogy is demonstrated with a smoothing method for the time-dependent Poisson process; this is similar in spirit to the normal theory Kalman filter. A procedure for grouped observations in a histogram provides an alternative to the histospline method of Boneva, Kendall and Stefanov. Five practical examples are presented, including two investigations of normality, an analysis of pedestrian arrivals at a Pelican crossing and a histogram smoothing method for mine explosions data.