Local asymptotic minimax theory for block-decreasing densities

In this paper, we study Lebesgue densities on (0,∞)d(0,∞)d that are non-increasing in each coordinate, while keeping all other coordinates fixed, from the perspective of local asymptotic minimax lower bound theory. In particular, we establish a local optimal rate of convergence of the order n−1/(d+2)n−1/(d+2).