MINIMAX ESTIMATION OF A CONSTRAINED POISSON VECTOR

and minimax risk p(T), we give analytical and numerical results for small intervals when p = 1. Usually, however, approximations are needed. If T is "rectangulary convex" at 0, there exist linear estimators with risk at most 1.26p(T). For general T, p(T) 2 p2/(p + A(fQ)), where A(Q) is the principal eigenvalue of the Laplace operator on the polydisc transform Q = Ql(T), a domain in twice-p-dimensional space. The bound is asymptotically sharp: p(mT) = p - A(l)/m + o(m'-). Explicit forms are given for T a simplex or a hyperrectangle. We explore the curious parallel of the results for T with those for a Gaussian vector of double the dimension lying in fl.

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