Local asymptotic minimax theory for block-decreasing densities

In this paper, we study Lebesgue densities on ( 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 ) .