The properties of non-parametric kernel estimators for probability density function from two special classes are investigated. Each class is parametrized with distribution smoothness parameter. One the was introduced by Rosenblatt, another one in this paper. For case known parameter, rates mean square convergence optimal (on bandwidth) found. unknown estimation procedure parameter developed and...

This paper studies the minimax rate of nonparametric conditional density estimation under a weighted absolute value loss function in multivariate setting. We first demonstrate that is impossible if one only requires pX|Z smooth x for all values z. motivates us to consider sub-class absolutely continuous distributions, restricting pX|Z(x|z) not be Hölder x, but also total variation propose corre...

In this paper we present a generalization of kernel density estimation called Convex Adaptive Kernel Density Estimation (CAKE) that replaces single bandwidth selection by a convex aggregation of kernels at all scales, where the convex aggregation is allowed to vary from one training point to another, treating the fundamental problem of heterogeneous smoothness in a novel way. Learning the CAKE ...

This paper proves that the welfare of the first price auction in Bayes-Nash equilibrium is at least a .743-fraction of the welfare of the optimal mechanism assuming agents’ values are independently distributed. The previous best bound was 1 − 1/e ≈ .63, derived in Syrgkanis and Tardos (2013) using smoothness, the standard technique for reasoning about welfare of games in equilibrium. In the wor...

We prove the existence of a Gevrey family of invariant curves for analytic reversible mappings under weaker nondegeneracy condition. The index of the Gevrey smoothness of the family could be any number μ > τ 2, where τ > m − 1 is the exponent in the small divisors condition and m is the order of degeneracy of the reversible mappings. Moreover, we obtain a Gevrey normal form of the reversible ma...

The infinite mixture of normals model has become a popular method for density estimation problems. This paper proposes an alternative hierarchical model that leads to hyperparameters that can be interpreted as the location, scale and smoothness of the density. The priors on other parts of the model have little effect on the density estimates and can be given default choices. Automatic Bayesian ...