Showing papers by "James O. Berger published in 1976"

Admissible Minimax Estimation of a Multivariate Normal Mean with Arbitrary Quadratic Loss

Abstract: The problem of estimating the mean of a $p$-variate $(p \geqq 3)$ normal distribution is considered. It is assumed that the covariance matrix $ ot\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.

Combining Independent Normal Mean Estimation Problems with Unknown Variances

Abstract: 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 $\lbrack\sum^p_{i=1} q_i(\delta_i - \theta_i)^2\rbrack/\lbrack\sum^p_{i=1} q_i A_i\rbrack,\quad\text{where} q_i > 0, i = 1, \cdots, p.$ Defining $W_i = V_i/(n_i - 2), W = (W_1, \cdots, W_p)^t$, and $\|X\|_{W^2} = \sum^p_{j=1} \lbrack X_{j^2}/(q_jW_j^2)\rbrack$, it is shown that under certain conditions on $r(X, W)$, the estimator given componentwise by $\delta_i(X, W) = (1 - r(X, W)/\lbrack\|X\|_{W^2}q_i W_i\rbrack)X_i$ is a minimax estimator of $\theta$. (The conditions on $r$ require $p \geqq 3$.) A good practical version of this estimator is also given.

Tail Minimaxity in Location Vector Problems and Its Applications

Abstract: 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 class of good tail minimax estimators is then developed and compared with the best invariant estimator

Admissibility Results for Generalized Bayes Estimators of Coordinates of a Location Vector

Abstract: 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) $F(\theta) \leqq K(|\theta_1| + \sum^n_{i=2} \theta_i^2)^{(3-n)/2}$, then $\delta_F$ is admissible for estimating $\theta_1$. For example, assume that $F(\theta) \equiv 1$ and that $L$ is squared error loss. Under appropriate conditions it can be shown that $\delta_F(x) = x_1$, and that $\delta_F$ is the best invariant estimator. If, in addition, $f$ has 7 absolute moments and $n \leqq 3$, it can be concluded that $\delta_F$ is admissible.

Inadmissibility Results for the Best Invariant Estimator of Two Coordinates of a Location Vector

Abstract: 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 inadmissible.