We study the minimax rate of estimation in nonparametric exponential family regression under star-shaped constraints. Specifically, the parameter space $K$ is a star-shaped set contained within a bounded box $[-M, M]^n$, where $M$ is a known positive constant. Moreover, we assume that the exponential family is nonsingular and that its cumulant function is twice continuously differentiable. Our main result shows that the minimax rate for this problem is $\varepsilon^{*2} \wedge \operatorname{diam}(K)^2$, up to absolute constants, where $\varepsilon^*$ is defined as \[ \varepsilon^* = \sup \{\varepsilon: \varepsilon^2 \kappa(M) \leq \log N^{\operatorname{loc}}(\varepsilon)\}, \] with $N^{\operatorname{loc}}(\varepsilon)$ denoting the local entropy and $\kappa(M)$ is an absolute constant allowed to depend on $M$. We also provide an example and derive its corresponding minimax optimal rate.

翻译：本文研究了星形约束下非参数指数族回归的极小极大估计速率。具体而言，参数空间 $K$ 是一个包含在有界盒子 $[-M, M]^n$ 内的星形集，其中 $M$ 是一个已知的正常数。此外，我们假设该指数族是非奇异的，且其累积量函数是二次连续可微的。我们的主要结果表明，该问题的极小极大速率（在绝对常数范围内）为 $\varepsilon^{*2} \wedge \operatorname{diam}(K)^2$，其中 $\varepsilon^*$ 定义为 \[ \varepsilon^* = \sup \{\varepsilon: \varepsilon^2 \kappa(M) \leq \log N^{\operatorname{loc}}(\varepsilon)\}, \] 这里 $N^{\operatorname{loc}}(\varepsilon)$ 表示局部熵，$\kappa(M)$ 是一个允许依赖于 $M$ 的绝对常数。我们还提供了一个例子，并推导出其相应的极小极大最优速率。