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Keywords:

  • convex programme;
  • dual block coordinate relaxation;
  • extreme value theory;
  • 1-penalization;
  • smoothing;
  • total variation;
  • universal penalty parameter

Abstract.  We propose a non-linear density estimator, which is locally adaptive, like wavelet estimators, and positive everywhere, without a log- or root-transform. This estimator is based on maximizing a non-parametric log-likelihood function regularized by a total variation penalty. The smoothness is driven by a single penalty parameter, and to avoid cross-validation, we derive an information criterion based on the idea of universal penalty. The penalized log-likelihood maximization is reformulated as an ℓ1-penalized strictly convex programme whose unique solution is the density estimate. A Newton-type method cannot be applied to calculate the estimate because the ℓ1-penalty is non-differentiable. Instead, we use a dual block coordinate relaxation method that exploits the problem structure. By comparing with kernel, spline and taut string estimators on a Monte Carlo simulation, and by investigating the sensitivity to ties on two real data sets, we observe that the new estimator achieves good L1 and L2 risk for densities with sharp features, and behaves well with ties.