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密度及其导数估计量的精确渐近行为AI 解读

For an integer $p \geq 0$, Singh has proposed a class of kernel estimators $\hat{f}^{(p)}$ of the $p$th order derivative $f^{(p)}$ of a density $f$. This paper examines the detailed asymptotic behavior of these estimators. In particular, asymptotically equivalent expressions for the bias $(E\hat{f}^{(p)} - f^{(p)})$, the mean squared error $E(\hat{f}^{(p)} - f^{(p)})^2$ and the error $(\hat{f}^{(p)} - f^{(p)})$ are obtained, which in turn give exact rates of convergence of…

Lebesgue 指数族中接近最优速率的经验贝叶斯估计AI 解读

Asymptotically optimal (a.o.) empirical Bayes (EB) estimators are proposed. Speeds and the best possible speed at which these estimators are a.o. are investigated. The underlying component problem is the squared error loss estimation of $\theta$ based on an observation $X$ whose conditional (on $\theta$) $\operatorname{pdf}$ is of the form $u(x)C(\theta)\exp(\theta x)$. The function $u$ could have infinitely many discontinuities; $\theta$ is distributed according to an…

密度及其导数估计量的均方误差AI 解读

The behaviour of the mean squared errors of estimators of a density and its derivatives proposed by Singh (1977a) is examined. Asymptotic values of mean squared errors at some points, for the normal population model in particular, are compared with different choices of estimators leading to estimators giving minimum asymptotic mean squared errors.

密度及其导数估计量的均方误差AI 解读

The behaviour of the mean squared errors of estimators of a density and its derivatives proposed by Singh (1977a) is examined. Asymptotic values of mean squared errors at some points, for the normal population model in particular, are compared with different choices of estimators leading to estimators giving minimum asymptotic mean squared errors.

密度及其导数估计在若干统计问题中的应用AI 解读

Summary Several statistical problems are considered. By reducing them to problems of estimating functionals of a density, derivatives of a density, or both, a method of finding partial solutions is obtained.

改进若干已知的密度导数非参数一致估计量AI 解读

Based on a random sample from a univariate distribution with density $f$, this note exhibits a class of kernel estimators of the $p$th order derivative $f^{(p)}$ of $f, p \geqq 0$ fixed. These estimators improve some known estimators of $f^{(p)}$ by weakening the conditions, sharpening the rates of convergence, or both for the properties of strong consistency, asymptotic unbiasedness and mean square consistency, each uniform on the real line.

Empirical Bayes estimators, asymptotically optimal with rates, are proposed. In the component problem there is a pair $(X, \omega)$ of real valued random variables. The Lebesgue density of $X$, conditional on $\omega$, is of the form $u(x)C(\omega)e^{\omega x}$. Based on a realization of $X$, the problem is squared error loss estimation of $\omega$. Let $G$ be a prior distribution on $\omega$, and $R(G)$ be the Bayes optimal risk wrt $G$. Using $(X_1, \cdots, X_n)$, the…