A model-based approach to density estimation in sup-norm
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911021498630144 |
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| author | Maillard, Guillaume |
| author_facet | Maillard, Guillaume |
| contents | We define a general method for finding a quasi-best approximant in sup-norm to a target density belonging to a given model, based on independent samples drawn from distributions which average to the target (which does not necessarily belong to the model). We also provide a general method for selecting among a countable family of such models. These estimators satisfy oracle inequalities in the general setting. The quality of the bounds depends on the volume of sets on which $|p-q|$ is close to its maximum, where $p,q$ belong to the model (or possibly to two different models, in the case of model selection). This leads to optimal results in a number of settings, including piecewise polynomials on a given partition and anisotropic smoothness classes. Particularly interesting is the case of the single index model with fixed smoothness $β$, where we recover the one-dimensional rate: this was an open problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20239 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A model-based approach to density estimation in sup-norm Maillard, Guillaume Statistics Theory We define a general method for finding a quasi-best approximant in sup-norm to a target density belonging to a given model, based on independent samples drawn from distributions which average to the target (which does not necessarily belong to the model). We also provide a general method for selecting among a countable family of such models. These estimators satisfy oracle inequalities in the general setting. The quality of the bounds depends on the volume of sets on which $|p-q|$ is close to its maximum, where $p,q$ belong to the model (or possibly to two different models, in the case of model selection). This leads to optimal results in a number of settings, including piecewise polynomials on a given partition and anisotropic smoothness classes. Particularly interesting is the case of the single index model with fixed smoothness $β$, where we recover the one-dimensional rate: this was an open problem. |
| title | A model-based approach to density estimation in sup-norm |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2506.20239 |