University of Washington - Department of Statistics
A convex-transformed density is a quasi-concave (or a quasi-convex) density which is a composition of monotone and convex functions. We consider a scale of such families of multivariate densities indexed by a parameter which is a monotone function. The exponential function corresponds to log-concave densities, while power functions correspond to heavier tailed densities or densities concentrated on the positive orthant. Many parametric and non parametric families of densities can be included in a suitable family of convex-transformed densities: normal, gamma, beta, Gumbel and other log-concave densities, multivariate Pareto, Burr, Student t, Snedecor etc. We study the properties of nonparametric estimation in the families of convex-transformed densities, including existence and consistency of the maximum likelihood estimator, and asymptotic minimax lower bounds for estimation.