Abstract
This paper aims to estimate an unknown density of the data with measurement errors as a linear combination of functions from a dictionary. The main novelty is the proposal and investigation of the corrected sparse density estimator (CSDE). Inspired by the penalization approach, we propose the weighted Elastic-net penalized minimal -distance method for sparse coefficients estimation, where the adaptive weights come from sharp concentration inequalities. The first-order conditions holding a high probability obtain the optimal weighted tuning parameters. Under local coherence or minimal eigenvalue assumptions, non-asymptotic oracle inequalities are derived. These theoretical results are transposed to obtain the support recovery with a high probability. 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 dataset. It shows that our method has potency and superiority in detecting multi-mode density shapes compared with other conventional approaches.
Highlights
Academic Editors: Andrea Prati, Over the years, the mixture models have been extensively applied to model unknown distributional shapes in astronomy, biology, economics, and genomics
We propose an improvement of the sparse estimation strategy proposed in [9], in which Bunea et al propose a1 -type penalty [11] to obtain a sparse density estimate (SPADES)
This paper considers the estimation of density functions in the presence of a classical measurement error
Summary
Academic Editors: Andrea Prati, Over the years, the mixture models have been extensively applied to model unknown distributional shapes in astronomy, biology, economics, and genomics (see [1] and references therein). The high-dimensional inference method has been applied to the infinite mixture models with a sparse mixture of p → ∞ components, which is an interesting and challenging problem (see [9,10]). Note that the multivariate kernel density estimator can only deal with a continuous distribution, and it requires a multivariate bandwidths section, while our method is dimensional-free (the number of the required tuning parameters is only two). Considering the multi-modal density aspect of the meteorology dataset, our proposed estimator has a stronger ability to detect multiple modes for the underlying distribution compared with other methods, such as SPADES or un-weighted Elastic-net estimator.
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