This study examines an adaptive log-density estimation method with an \(\ell _1\)-type penalty. The proposed estimator is guaranteed to be a valid density in the sense that it is positive and integrates to one. The smoothness of the estimator is controlled in a data-adaptive way via \(\ell _1\) penalization. The advantages of the penalized log-density estimator are discussed with an emphasis on wavelet estimators. Theoretical properties of the estimator are studied when the quality of fit is measured by the Kullback–Leibler divergence (relative entropy). A nonasymptotic oracle inequality is obtained assuming a near orthogonality condition on the given dictionary. Based on the oracle inequality, selection consistency and minimax adaptivity are proved under some regularity conditions. The proposed method is implemented with a coordinate descent algorithm. Numerical illustrations based on the periodized Meyer wavelets are performed to demonstrate the finite sample performance of the proposed estimator.

中文翻译：

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自适应对数密度估计

这项研究研究了具有\（\ ell _1 \）型罚分的自适应对数密度估计方法。就其为正并与一个整数相乘的意义而言，可以保证拟议的估计量为有效密度。估计器的平滑度通过\（\ ell _1 \）以数据自适应方式控制惩罚。讨论了惩罚对数密度估计器的优点，重点放在小波估计器上。当通过Kullback-Leibler散度（相对熵）测量拟合质量时，研究估计器的理论性质。假设给定字典的近似正交条件，则获得非渐近的oracle不等式。基于预言不等式，证明了在一定规律性条件下的选择一致性和最小极大适应性。所提出的方法是通过坐标下降算法实现的。进行了基于周期Meyer小波的数值说明，以证明所提出估计器的有限样本性能。