We study limit distributions for the tuning-free max-min block estimator originally proposed in [FLN17] in the problem of multiple isotonic regression, under both fixed lattice design and random design settings. We show that, if the regression function $f_0$ admits vanishing derivatives up to order $\alpha_k$ along the $k$-th dimension ($k=1,\ldots,d$) at a fixed point $x_0 \in (0,1)^d$, and the errors have variance $\sigma^2$, then the max-min block estimator $\hat{f}_n$ satisfies \begin{align*} (n_\ast/\sigma^2)^{\frac{1}{2+\sum_{k \in \mathcal{D}_\ast} \alpha_k^{-1}}}\big(\hat{f}_n(x_0)-f_0(x_0)\big)\rightsquigarrow \mathbb{C}(f_0,x_0). \end{align*} Here $\mathcal{D}_\ast, n_\ast$, depending on $\{\alpha_k\}$ and the design points, are the set of all `effective dimensions' and the size of `effective samples' that drive the asymptotic limiting distribution, respectively. If furthermore either $\{\alpha_k\}$ are relative primes to each other or all mixed derivatives of $f_0$ of certain critical order vanish at $x_0$, then the limiting distribution can be represented as $\mathbb{C}(f_0,x_0) =_d K(f_0,x_0) \cdot \mathbb{D}_{\alpha}$, where $K(f_0,x_0)$ is a constant depending on the local structure of the regression function $f_0$ at $x_0$, and $\mathbb{D}_{\alpha}$ is a non-standard limiting distribution generalizing the well-known Chernoff distribution in univariate problems. The above limit theorem is also shown to be optimal both in terms of the local rate of convergence and the dependence on the unknown regression function whenever such dependence is explicit (i.e. $K(f_0,x_0)$), for the full range of $\{\alpha_k\}$ in a local asymptotic minimax sense.

中文翻译：

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多元等渗回归中块估计量的极限分布理论

我们在固定晶格设计和随机设计设置下研究了最初在 [FLN17] 中在多重等渗回归问题中提出的无调谐最大-最小块估计器的极限分布。我们证明，如果回归函数 $f_0$ 允许消失的导数直到 $\alpha_k$ 沿着 $k$-th 维（$k=1,\ldots,d$）在一个固定点 $x_0\in（ 0,1)^d$，并且误差有方差 $\sigma^2$，那么最大最小块估计器 $\hat{f}_n$ 满足 \begin{align*} (n_\ast/\sigma^ 2)^{\frac{1}{2+\sum_{k \in \mathcal{D}_\ast} \alpha_k^{-1}}}\big(\hat{f}_n(x_0)-f_0 (x_0)\big)\rightsquigarrow \mathbb{C}(f_0,x_0)。\end{align*} 这里 $\mathcal{D}_\ast, n_\ast$，取决于 $\{\alpha_k\}$ 和设计点，是所有“有效尺寸”的集合和“有效样本” 分别驱动渐近极限分布。此外，如果 $\{\alpha_k\}$ 是彼此的相对素数，或者某个临界顺序的 $f_0$ 的所有混合导数都在 $x_0$ 处消失，则极限分布可以表示为 $\mathbb{C}( f_0,x_0) =_d K(f_0,x_0) \cdot \mathbb{D}_{\alpha}$，其中 $K(f_0,x_0)$ 是一个常数，取决于回归函数 $f_0$ 的局部结构在 $x_0$ 和 $\mathbb{D}_{\alpha}$ 是一个非标准的极限分布，概括了单变量问题中著名的 Chernoff 分布。上述极限定理在局部收敛速度和对未知回归函数的依赖性方面也被证明是最优的，只要这种依赖性是明确的（即 $K(f_0,x_0)$），