Sparse Density Estimation with Measurement Errors
This work addresses density estimation with measurement errors, which is a domain-specific problem in statistics and meteorology, offering incremental improvements over existing methods.
The paper tackles the problem of estimating an unknown density from data with measurement errors by proposing a weighted Elastic-net penalized method for sparse coefficients estimation, achieving significant improvement in numerical experiments and superior shape detection in a meteorology dataset compared to conventional approaches.
This paper aims to build an estimate of an unknown density of the data with measurement error as a linear combination of functions from a dictionary. Inspired by the penalization approach, we propose the weighted Elastic-net penalized minimal $\ell_2$-distance method for sparse coefficients estimation, where the adaptive weights come from sharp concentration inequalities. The optimal weighted tuning parameters are obtained by the first-order conditions holding with a high probability. Under local coherence or minimal eigenvalue assumptions, non-asymptotical oracle inequalities are derived. These theoretical results are transposed to obtain the support recovery with a high probability. Then, some numerical experiments for discrete and continuous distributions confirm the significant improvement obtained by our procedure when compared with other conventional approaches. Finally, the application is performed in a meteorology data set. It shows that our method has potency and superiority of detecting the shape of multi-mode density compared with other conventional approaches.