We consider estimation of a functional of the data distribution based on i.i.d. observations. We assume the target function can be defined as the minimizer of the expectation of a loss function over a class of $d$-variate real valued cadlag functions that have finite sectional variation norm. For all $k=0,1,\ldots$, we define a $k$-th order smoothness class of functions as $d$-variate functions on the unit cube for which each of a sequentially defined $k$-th order Radon-Nikodym derivative w.r.t. Lebesgue measure is cadlag and of bounded variation. For a target function in this $k$-th order smoothness class we provide a representation of the target function as an infinite linear combination of tensor products of $\leq k$-th order spline basis functions indexed by a knot-point, where the lower (than $k$) order spline basis functions are used to represent the function at the $0$-edges. The $L_1$-norm of the coefficients represents the sum of the variation norms across all the $k$-th order derivatives, which is called the $k$-th order sectional variation norm of the target function. This generalizes the zero order spline representation of cadlag functions with bounded sectional variation norm to higher order smoothness classes. We use this $k$-th order spline representation of a function to define the $k$-th order spline sieve minimum loss estimator (MLE), Highly Adaptive Lasso (HAL) MLE, and Relax HAL-MLE. For first and higher order smoothness classes, in this article we analyze these three classes of estimators and establish pointwise asymptotic normality and uniform convergence at dimension free rate $n^{-k^*/(2k^*+1)}$ up till a power of $\log n$ depending on the dimension, where $k^*=k+1$, assuming appropriate undersmoothing is used in selecting the $L_1$-norm. We also establish asymptotic linearity of plug-in estimators of pathwise differentiable features of the target function.
翻译:我们考虑根据 i. d. 观察来估算数据分布的函数。 我们假设的目标函数可以被定义为 $d$- d. d. d. 观察 类中损失函数的最小值最小值。 对于所有 $= 0. 1,\ ldots, 我们定义了单位立方元顺序的顺差函数等级为 $- variet 立方元, 其中每个单位都依次定义了 $-th 顺序 的 Radon- Nikdym 衍生工具 w.r.。 Lebesgue 计量是 cadladal 和 受约束的 level 。 对于这个 $drod 平整级中的目标函数,我们提供目标函数的无限线性组合是 $leq k- tnordsion 维度, 我们的更低( $k) 命令的螺旋基函数用来在 $lood- slodeal- drequireal lad.