The Second-Order Perspectives of Minimax Parameter Estimation in Restricted Space with Weighted Squared Error Loss

In this paper, we investigate the parameter estimation problems, where the unknown parameter is assumed to be within certain restricted parameter space. To this end, we adopt the conventional minimax formulation and apply the weighted mean-squared error as the loss function. In particular, we focus on analyzing the minimax risk in the asymptotic regime, where the second-order convergence rate, and the ordinary differential equation for computing the second-order terms are presented. Moreover, our results are applied to some widely considered probability distribution models, where the analytical expressions of the second-order terms and the corresponding estimators are explicitly provided. Finally, some numerical simulations are also presented when the second-order term cannot be explicitly expressed, which further supports our theoretical results.