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非参数响应面设计的渐近贝叶斯准则AI 解读

This paper deals with Bayesian design for response surface prediction when the prior may be finite or infinite dimensional, the design space arbitrary. In order that the resulting problems be manageable, we resort to asymptotic versions of D-, G- and A-optimality. Here the asymptotics stem from allowing the error variance to be large. The problems thus elicited have strong game-like characteristics. Examples of theoretical solutions are brought forward, especially when the…

回归函数的置信带AI 解读

A method for obtaining (conservative) confidence bands for regression functions in low dimensions is given. Of particular importance is the applicability of the method to nonparametric regression. Computations, examples, and comparisons indicate the effectiveness of the method.

回归中的若干模型稳健设计AI 解读

Theory for finding designs in estimating a linear functional of a regression function is developed for classes of regression functions which are infinite dimensional. These classes can be viewed as representing possible departures from an "ideal" simple model and thus describe a model robust setting. The estimates are restricted to be linear and the design (and estimate) sought is minimax for mean square error. The structure of the design is obtained in a variety of cases;…

L 估计、R 估计与极小极大性质AI 解读

Let $\{X_i\}$ be a sample from $F(x - \theta)$ where $F$ is in a class $\mathscr{F}$ of symmetric distributions on the line and $\theta$ is the location parameter to be estimated. Huber has shown that maximum likelihood estimation has a minimax property over a convex $\mathscr{F}$. Here a simple convex $\mathscr{F}$ is given for which neither $L$- nor $R$-estimation has the minimax property. In particular, this example shows that a recent assertion concerning $L$-estimation…

点态密度估计的渐近最优核AI 解读

Kernel estimation of $f(0)$ is considered where $f$ is a density in some class $\mathscr{F}$ of $d$-dimensional densities, described in terms of a Taylor series expansion. A sequence of kernels which asymptotically minimizes the maximum mean square error of estimation over $\mathscr{F}$ is given. The shape of the kernel is fixed, the size of the window depends on $f(0)$, and an easily computed estimate is obtained to efficiently adapt the sequence to the unknown value of $f(0)$…

指数族的共轭先验AI 解读

Let $X$ be a random vector distributed according to an exponential family with natural parameter $\theta \in \Theta$. We characterize conjugate prior measures on $\Theta$ through the property of linear posterior expectation of the mean parameter of $X : E\{E(X|\theta)|X = x\} = ax + b$. We also delineate which hyperparameters permit such conjugate priors to be proper.