Density deconvolution under a k-monotonicity constraint

Maximum likelihood estimation of a k-monotone density which can be represented as a scale mixture of beta densities with parameters 1 and k is well-studied when no measurement error is present. In this paper, we further study the maximum likelihood method of density deconvolution under a k-monotonicity constraint. Using the mixture representation of a k-monotone density straightforwardly is prevented due to the likelihood unboundedness problem. To counter the problem, a reparameterization trick is applied by creating a scalar tuning parameter. When the tuning parameter is known, we establish the identifiability of the reparameterized mixture model for the observed data and the consistency of the maximum likelihood estimator of a k-monotone density in the presence of measurement error. We also provide a method to choose the tuning parameter. Numerical studies are used to illustrate the proposed estimation and selection procedure.