Nonparametric Exponential Family Regression Under Star-Shaped Constraints

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.