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从 Stein 无偏风险估计到广义交叉验证AI 解读

This paper concerns the method of generalized cross validation (GCV), a promising way of choosing between linear estimates. Based on Stein estimates and the associated unbiased risk estimates (Stein, 1981), a new approach to GCV is developed. Many consistency results are obtained for the cross-validated (Steinized) estimates in the contexts of nearest-neighbor nonparametric regression, model selection, ridge regression, and smoothing splines. Moreover, the associated Stein's…

Stein估计的数据平滑性质AI 解读

The data smoothing aspect of Stein estimates is explored in the nonparametric regression settings. We show that appropriately shrinking the raw data towards any linear smoother will provide a robust "smoother" (which dominates the raw data and hence has a bounded maximum risk when the average squared error loss is concerned).

无限参数回归模型中有界线性泛函的一致估计AI 解读

Consider a linear model with infinitely many parameters given by $y = \sum^\infty_{i = 1} x_i\theta_i + \varepsilon$ where $\mathbf{x} = (x_1, x_2, \cdots)'$ and $\theta = (\theta_1, \theta_2, \cdots)'$ are infinite dimensional vectors such that $\sum^\infty_{i = 1}x^2_i < \infty$ and $\sum^\infty_{i = 1} \theta^2_i < \infty$. Suppose independent observations $y_1, y_2, \cdots, y_n$ are observed at levels $\mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_n$. Under suitable…

We consider the nearly linear regression problem when the assumed first degree model is contaminated by some small constant, and the design space consists of finitely many points, symmetrically distributed on the interval $\lbrack -1/2, 1/2\rbrack$. Under the usual squared loss function for the estimation of the slope and the intercept, and with the use of the least squares estimators, the problem is to find the designs which are optimal in the sense of minimizing the…

非参数回归中交叉验证近邻估计的一致性AI 解读

Under suitable conditions, we show that the cross-validated nearest neighbor estimates for the unknown smooth regression function in $R^p$ is asymptotically consistent.

抽样调查的极小极大方法AI 解读

Suppose that there is a population of $N$ identifiable units each with two associated values $x_i$ and $y_i$. All $N$ values of $x$ are given but $y_i$ is determined only after the $i$th unit is selected and observed. The objective is to estimate the population total $\sum^N_{i=1} y_i$. It is assumed that $y_i = \theta x_i + \delta_ig(x_i), 1 \leq i \leq N$, where $(\delta_1, \cdots, \delta_N)$ is in some bounded neighborhood of $(0, \cdots 0)$. The Rao-Hartley-Cochran and…

随机化设计的极小极大性：若干一般结果AI 解读

In many design settings where model violations are present, a "stochastic" minimaxity for many standard randomization procedures is demonstrated. This result requires no special analytic properties of the loss function and estimators. Next, under the squared loss and with the restriction to the use of linear estimators, a recipe for finding a randomized strategy is given. As a special case, randomizing an $A$-optimal design in the standard manner and using the least squares…

随机过程正则化方法的极小极大性AI 解读

The idea of smoothing-spline interpolation is generalized to propose an estimator for the mean function of a stochastic process. A minimax property of the proposed estimator is then demonstrated under the usual squared loss function.