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非参数回归函数的自适应置信带AI 解读

A new formulation for the construction of adaptive confidence bands in non-parametric function estimation problems is proposed. Confidence bands are constructed which have size that adapts to the smoothness of the function while guaranteeing that both the relative excess mass of the function lying outside the band and the measure of the set of points where the function lies outside the band are small. It is shown that the bands adapt over a maximum range of Lipschitz…

二次泛函的最优自适应估计AI 解读

Minimax mean-squared error estimates of quadratic functionals of smooth functions have been constructed for a variety of smoothness classes. In contrast to many nonparametric function estimation problems there are both regular and irregular cases. In the regular cases the minimax mean-squared error converges at a rate proportional to the inverse of the sample size, whereas in the irregular case much slower rates are the rule. We investigate the problem of adaptive estimation…

关于密度导数平方积分自适应估计的 Bickel–Ritov 猜想AI 解读

Bickel and Ritov suggested an optimal estimator for the integral of the square of the kth derivative of a density when the unknown density belongs to a Lipschitz class of a given order $\beta$. In this context optimality means that the estimate is asymptotically efficient, that is, it has the best constant and rate of risk convergence, whenever $\beta > 2k + 1/4$, and it is rate optimal otherwise. The suggested optimal estimator crucially depends on the value of $\beta$ which…