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纵向研究中变化率的经验贝叶斯估计AI 解读

The usually irregular follow-up intervals in epidemiologic studies preclude the use of classical growth curve analysis. For short follow-up intervals, it is suggested that individual rates of change are useful for exploratory analysis. Empirical Bayes estimates of these rates of change are recommended and developed. An example of bone loss with age in women is also given.

控制问题中估计量可容许性与不可容许性的一般结果AI 解读

Let $\mathbf{X} = (X_1, \cdots, X_p)^t$ be an observation from a $p$-variate normal distribution with unknown mean $\mathbf{\theta} = (\theta_1, \cdots, \theta_p)^t$ and identity covariance matrix. We consider a control problem which, in canonical form, is the problem of estimating $\mathbf{\theta}$ under the loss $L(\mathbf{\theta, \delta}) = (\mathbf{\theta}^t \mathbf{\delta} - 1)^2$, where $\mathbf{\delta(x)} = (\delta_1(\mathbf{x}), \cdots, \delta_p(\mathbf{x}))^t$ is…

多元正态均值最小最大估计量的选择AI 解读

The problem of estimating a $p$-variate normal mean under arbitrary quadratic loss when $p \geq 3$ is considered. Any estimator having uniformly smaller risk than the maximum likelihood estimator $\delta^0$ will have significantly smaller risk only in a fairly small region of the parameter space. A relatively simple minimax estimator is developed which allows the user to select the region in which significant improvement over $\delta^0$ is to be achieved. Since the desired…

多元问题中的广义贝叶斯估计量AI 解读

Several problems involving multivariate generalized Bayes estimators are investigated. First, a characterization of admissible estimators as generalized Bayes estimators is developed for certain multivariate exponential families and quadratic loss. The problem of verifying whether or not an estimator is generalized Bayes is also considered. Next, an important class of estimators for a multivariate normal mean is considered. (The class includes many minimax, empirical Bayes,…

位置向量两个坐标的最佳不变估计量不可容许性AI 解读

Let $X = (X_1, X_2, X_3)$ be a random vector with density $f(x - \theta)$, where $\theta = (\theta_1, \theta_2, \theta_3)$ is unknown. It is desired to estimate $(\theta_1, \theta_2)$ using an estimator $(\delta_1(X), \delta_2(X))$, and under a loss function $L(\delta_1 - \theta_1, \delta_2 - \theta_2)$. (Note that $\theta_3$ is a nuisance parameter.) Under certain conditions on $f$ and $L$, it is shown that the best invariant estimator of $(\theta_1, \theta_2)$ is…

未知方差下独立正态均值估计问题的合并AI 解读

Let $X = (X_1, \cdots, X_p)^t$ be a $p$-variate normal random vector with unknown mean $\theta = (\theta_1, \cdots, \theta_p)^t$ and unknown positive definite diagonal covariance matrix $A$. Assume that estimates $V_i$ of the variances $A_i$ are available, and that $V_i/A_i$ is $\chi^2_{n_i}$. Assume also that all $X_i$ and $V_i$ are independent. It is desired to estimate $\theta$ under the quadratic loss $\lbrack\sum^p_{i=1} q_i(\delta_i - \theta_i)^2\rbrack/\lbrack\sum^p_{i=1} q_i A_i\rbrack,\quad\text{where} q_i > 0, i = 1, \cdots, p.$…

位置向量坐标的广义贝叶斯估计可容许性AI 解读

Let $X$ be an $n$-dimensional random vector with density $f(x - \theta)$. It is desired to estimate $\theta_1$, under a strictly convex loss $L(\delta - \theta_1)$. If $F$ is a generalized Bayes prior density, the admissibility of the corresponding generalized Bayes estimator, $\delta_F$, is considered. An asymptotic approximation to $\delta_F$ is found. Using this approximation, it is shown that if (i) $f$ has enough moments, (ii) $L$ and $F$ are smooth enough, and (iii) $F(\theta) \leqq K(|\theta_1| + \sum^n_{i=2} \theta_i^2)^{(3-n)/2}$…

位置向量坐标的广义贝叶斯估计量不可容许性结果AI 解读

Let $X = (X_1, \cdots, X_n)$ be an $n$-dimensional random vector with density $f(x - \theta)$. Assume the loss incurred in estimating $\theta$ by $d$ is $L(d - \theta)$ and is convex. Let $F$ be a generalized prior density. The question of inadmissibility of generalized Bayes estimators of $\theta$ is considered. As in Brown (1974c), the crucial role played by the "moment structure" in determining inadmissibility results is indicated (moments defined as the quantities $m_{i,j(1), \cdots, j(l)} = \int \prod^l_{k=1} x_{j(k)} \cdot \lbrack(\partial/\partial x_i)L(x)\rbrack f(x) dx)$…

消除位置向量 Stein 型估计量的奇点AI 解读

Summary In estimating a multivariate location vector in three or more dimensions, it is well known that Stein-type estimators can offer considerable improvement in risk over the best invariant estimator. A common problem arising in the use of such estimators is that simple versions often have a singularity at zero. Methods of eliminating such singularities are discussed in this paper. The resulting estimators are usually considerably better than the uncorrected versions, and…

任意二次损失下多元正态均值的可容许极小极大估计AI 解读

The problem of estimating the mean of a $p$-variate $(p \geqq 3)$ normal distribution is considered. It is assumed that the covariance matrix $\not\sum$ is known and that the loss function is quadratic. A class of minimax estimators is given, out of which admissible minimax estimators are developed.

位置向量问题中的尾部极小极大性及其应用AI 解读

Let $X = (X_1,\cdots, X_p)^t, p \geqq 3$, have density $f(x - \theta)$ with respect to Lebesgue measure. It is desired to estimate $\theta = (\theta_1,\cdots, \theta_p)^t$ under the loss $L(\delta - \theta)$. Assuming the problem has a minimax risk $R_0$, an estimator is defined to be tail minimax if its risk is no larger than $R_0$ outside some compact set. Under quite general conditions on $f$ and $L$, sufficient conditions for an estimator to be tail minimax are given. A…