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Non-parametric density estimation

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Let X_1, X_2… X_n be data from an unknown density f on a fixed closed compact interval which contains a point x_0. For some m fixed, we assume that the first m derivatives of f are known to be bounded. We will then show that we have an estimate f_n such that the expected value of |f_n(x_0) – f(x_0)| is less than some constant (which does not depend on n) multiplied by n^{-m/(2m+1)}. We will also show that this is minimax optimal over all densities where the first m derivatives of f are bounded.

This talk is part of the Statistical Laboratory Graduate Seminars series.

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