  A deep generative model yields an implicit estimator for the unknown
distribution or density function of the observation. This paper investigates
some statistical properties of the implicit density estimator pursued by
VAE-type methods from a nonparametric density estimation framework. More
specifically, we obtain convergence rates of the VAE-type density estimator
under the assumption that the underlying true density function belongs to a
locally H\"{o}lder class. Remarkably, a near minimax optimal rate with respect
to the Hellinger metric can be achieved by the simplest network architecture, a
shallow generative model with a one-dimensional latent variable.
