Minimax boundaries in estimation of the intensity of a three-dimensional Poisson process (Q1842706)

We study nonparametric estimation of the intensity (measure) \(\Lambda(\cdot)\) of an inhomogeneous Poisson process defined on a bounded set in \(R^ d\). Without significant limitations, we can consider this set to be the cube \([0,1]^ d\). We assume that a sequence of independent realizations of this process is given. Theorem 1 provides an asymptotic expansion of the risk for the case where the estimate of \(\Lambda (\cdot)\) is taken to be the arithmetic mean \(\widehat \Lambda_ n (\cdot)\). We make use of the general theory of the construction of asymptotically minimax boundaries. To resort to this theory, Theorem 2 proves convergence of experiments corresponding to the initial problem of a Gaussian experiment with shift. We then directly apply the Le Cam - Millar theorem. The proof of the reachability of this boundary by the estimate \(\widehat \Lambda_ n\) rests on a respective functional limit theorem in the space of functions continuous on \([0,1]^ d\).