A stochastic algorithm for parametric and non-parametric estimation in the case of incomplete data

Our purpose is to estimate the joint distribution of probability of a pair of random variables when only one of these variables is observed. In other words, there are observed data and missing data. Our estimation method is an iterative procedure which can be seen as a stochastic version of the EM algorithm. At each iteration, we simulate the non-observed variable with the posterior distribution and estimate the joint distribution. We deal with the case of Markov random fields indexed by Z^p and study a convolution model. With this example, we show that the method can address a wide class of models, widely used in signal or image processing. In fact, we estimate a convolution filter and a noise variance as well as the parameters of a mixture of populations and a Gibbs distribution. Finally, we show that a non-parametric estimation of the probability density of the non-observed variables can be performed. Simulations and applications to real data give very satisfactory results.