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惩罚最大似然与半参数二阶效率AI 解读

We consider the problem of estimation of a shift parameter of an unknown symmetric function in Gaussian white noise. We introduce a notion of semiparametric second-order efficiency and propose estimators that are semiparametrically efficient and second-order efficient in our model. These estimators are of a penalized maximum likelihood type with an appropriately chosen penalty. We argue that second-order efficiency is crucial in semiparametric problems since only the…

逆问题的 Oracle 不等式AI 解读

We consider a sequence space model of statistical linear inverse problems where we need to estimate a function $f$ from indirect noisy observations. Let a finite set $\Lambda$ of linear estimators be given. Our aim is to mimic the estimator in $\Lambda$ that has the smallest risk on the true $f$. Under general conditions, we show that this can be achieved by simple minimization of an unbiased risk estimator, provided the singular values of the operator of the inverse problem…

Wicksell 问题中的渐近有效估计AI 解读

We consider the classical Wicksell problem of estimating an unknown distribution function $G$of the radii of balls, based on their observed crosssections. It is assumed that the underlying distribution function$G$ belongs to a Hölder class of smoothness $\gamma >1/2$. We prove that, for a suitable choice of the smoothing parameters, kernel-type estimators are asymptotically efficient for a large class of symmetric bowl-shaped loss functions.