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Modern data-driven approaches to modeling make extensive use of covariate/model selection. Such selection incurs a cost: it invalidates classical statistical inference. A conservative remedy to the problem was proposed by Berk et al. (Ann. Statist. 41 (2013) 802–837) and further extended by Bachoc, Preinerstorfer and Steinberger (2016). These proposals, labeled “PoSI methods,” provide valid inference after arbitrary model selection. They are computationally NP-hard and have…

自由结点多项式样条及其置信区间AI 解读

Summary We construct approximate confidence intervals for a nonparametric regression function, using polynomial splines with free-knot locations. The number of knots is determined by generalized cross-validation. The estimates of knot locations and coefficients are obtained through a non-linear least squares solution that corresponds to the maximum likelihood estimate. Confidence intervals are then constructed based on the asymptotic distribution of the maximum likelihood…

若干非参数问题的贝叶斯视角AI 解读

We study the Bayesian approach to nonparametric function estimation problems such as nonparametric regression and signal estimation. We consider the asymptotic properties of Bayes procedures for conjugate (= Gaussian) priors. We show that so long as the prior puts nonzero measure on the very large parameter set of interest then the Bayes estimators are not satisfactory. More specifically, we show that these estimators do not achieve the correct minimax rate over norm bounded…

非参数回归问题中的改进估计量AI 解读

Linear estimators of multivariate means are considered. Generalizations of some well-known theorems about admissibility of linear estimators are given. The results then are applied to show that commonly used kernel-type estimators in nonparametric regression problems can be constructively improved in a simple way. An asymptotic result is described that gives a quantitative measure of the maximum improvement to be gained in certain situations. A theoretical bound shows that…

非参数回归问题中的改进估计量AI 解读

Linear estimators of multivariate means are considered. Generalizations of some well-known theorems about admissibility of linear estimators are given. The results then are applied to show that commonly used kernel-type estimators in nonparametric regression problems can be constructively improved in a simple way. An asymptotic result is described that gives a quantitative measure of the maximum improvement to be gained in certain situations. A theoretical bound shows that…

非参数函数估计中的超效率AI 解读

Fixed parameter asymptotic statements are often used in the context of nonparametric curve estimation problems (e.g., nonparametric density or regression estimation). In this context several forms of superefficiency can occur. In contrast to what can happen in regular parametric problems, here every parameter point (e.g., unknown density or regression function) can be a point of superefficiency. We begin with an example which shows how fixed parameter asymptotic statements…

白噪声问题中的极小极大线性估计AI 解读

.Linear estimation of $f(x)$ at a point in a white noise model is considered. The exact linear minimax estimator of $f(0)$ is found for the family of $f(x)$ in which $f'(x)$ is Lip (M). The resulting estimator is then used to verify a conjecture of Sacks and Ylvisaker concerning the near optimality of the Epanechnikov kernel.