### Abstract

Common high-dimensional methods for prediction rely on having either a sparse signal model, a model in which most parameters are zero and there are a small number of nonzero parameters that are large in magnitude, or a dense signal model, a model with no large parameters and very many small nonzero parameters. We consider a generalization of these two basic models, termed here a "sparse + dense" model, in which the signal is given by the sum of a sparse signal and a dense signal. Such a structure poses problems for traditional sparse estimators, such as the lasso, and for traditional dense estimation methods, such as ridge estimation.We propose a new penalization-based method, called lava, which is computationally efficient.With suitable choices of penalty parameters, the proposed method strictly dominates both lasso and ridge. We derive analytic expressions for the finite-sample risk function of the lava estimator in the Gaussian sequence model. We also provide a deviation bound for the prediction risk in the Gaussian regression model with fixed design. In both cases, we provide Stein's unbiased estimator for lava's prediction risk. A simulation example compares the performance of lava to lasso, ridge and elastic net in a regression example using data-dependent penalty parameters and illustrates lava's improved performance relative to these benchmarks.

Original language | English (US) |
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Pages (from-to) | 39-76 |

Number of pages | 38 |

Journal | Annals of Statistics |

Volume | 45 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2017 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- High-dimensional models
- Nonsparse signal recovery
- Penalization
- Shrinkage

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## Cite this

*Annals of Statistics*,

*45*(1), 39-76. https://doi.org/10.1214/16-AOS1434