Abstract We study a penalized logspline density estimation method using a total variation penalty. The B-spline basis is adopted to approximate the logarithm of density functions. Total variation of derivatives of splines is penalized to impart a data-driven knot selection. The proposed estimator is a bona fide density function in the sense that it is positive and integrates to one. We devise an efficient coordinate descent algorithm for implementation and study its convergence property. An oracle inequality of the proposed estimator is established when the quality of fit is measured by the Kullback–Leibler divergence. Based on the oracle inequality, it is proved that the estimator achieves an optimal rate of convergence in the minimax sense. We also propose a logspline method for the bivariate case by adopting the tensor-product B-spline basis and a two-dimensional total variation type penalty. Numerical studies show that the proposed method captures local features without compromising the global smoothness.

中文翻译：

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使用总变异惩罚的惩罚对数样条密度估计

摘要 我们研究了一种使用总变异惩罚的惩罚对数样条密度估计方法。采用B样条基逼近密度函数的对数。样条导数的总变化受到惩罚，以提供数据驱动的结选择。建议的估计量是一个真正的密度函数，因为它是正的并且集成为 1。我们设计了一种有效的坐标下降算法来实现并研究其收敛性。当拟合质量由 Kullback-Leibler 散度测量时，建议的估计量的预言不等式成立。基于oracle不等式，证明了估计器在极大极小意义下达到了最优收敛速度。我们还通过采用张量积 B 样条基和二维全变分型惩罚，为双变量情况提出了对数样条方法。数值研究表明，所提出的方法可以在不影响全局平滑度的情况下捕获局部特征。