论文标题
重新访问LE CAM方程：凸密度类别的精确最小值
Revisiting Le Cam's Equation: Exact Minimax Rates over Convex Density Classes
论文作者
论文摘要
我们研究了在凸密度类别的密度估计中得出最小值速率的经典问题。在Le Cam（1973），Birge（1983，1986），Wong and Shen（1995），Yang and Barron（1999）的基础上，我们确定了任何凸密度类别的确切（达到常数）最小值。因此，这项工作通过证明密度类的局部度量熵始终捕获这种设置下的最小值最佳速率，从而扩展了这些已知结果。我们的边界在对密度类别的丰富性较弱的假设下，在参数和非参数凸密度类别中提供了一个统一的观点。我们提出的“多阶段筛” MLE适用于任何此类凸密度类别。我们进一步证明，该估计值也适应了真正的潜在感兴趣密度。我们将风险界限应用于重新验证的已知最小值率，包括界限总变化和持有人密度类别。我们进一步说明了结果较少的类别（例如，密度的凸混合物），通过得出上限来说明结果的实用性。
We study the classical problem of deriving minimax rates for density estimation over convex density classes. Building on the pioneering work of Le Cam (1973), Birge (1983, 1986), Wong and Shen (1995), Yang and Barron (1999), we determine the exact (up to constants) minimax rate over any convex density class. This work thus extends these known results by demonstrating that the local metric entropy of the density class always captures the minimax optimal rates under such settings. Our bounds provide a unifying perspective across both parametric and nonparametric convex density classes, under weaker assumptions on the richness of the density class than previously considered. Our proposed `multistage sieve' MLE applies to any such convex density class. We further demonstrate that this estimator is also adaptive to the true underlying density of interest. We apply our risk bounds to rederive known minimax rates including bounded total variation, and Holder density classes. We further illustrate the utility of the result by deriving upper bounds for less studied classes, e.g., convex mixture of densities.
