Density Estimation Using Multiscale Local Polynomial Transforms

The estimation of a density function with an unknown number of singularities or discontinuities is a typical example of a multiscale problem, with data observed at nonequispaced locations. The data are analyzed through a multiscale local polynomial transform (MLPT), which can be seen as a slightly overcomplete, non-dyadic alternative for a wavelet transform, equipped with the benefits from a local polynomial smoothing procedure. In particular, the multiscale transform adopts a sequence of kernel bandwidths in the local polynomial smoothing as resolution level-dependent, user-controlled scales. The MLPT analysis leads to a reformulation of the problem as a variable selection in a sparse, high-dimensional regression model with exponentially distributed responses. The variable selection is realized by the optimization of the l1-regularized maximum likelihood, where the regularization parameter acts as a threshold. Fine-tuning of the threshold requires the optimization of an information criterion such as AIC. This paper develops discussions on results in [9].