Density estimation via exponential model selection

We address the problem of estimating some unknown density on a bounded interval using some exponential models of piecewise polynomials. We consider a finite collection of such models based on a family of partitions. And we study the maximum-likelihood estimator built on a data-driven selected model among this collection. In doing so, we validate Akaike's criterion if the partitions that we consider are regular and we modify it if the partitions are irregular. We deduce the rate of convergence of the squared Hellinger risk of our estimator in the regular case when the logarithm of the density belongs to some Besov space.