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International audience; When calculating independently the false alarm rate of the eight usual runs rules used in SPC control chart, it appears that the proposed rule designed to detect mixture patterns corresponds to a Type-I error strongly lower than the seven other rules. This discrepancy is underlined and the mixture rule is showed to be useless both for in-control and out-of-control processes. Thus a modification of the mixture detection rule is proposed and the impact of this new mixture rule is then illustrated and discussed using Monte Carlo calculations.

We point out some pitfalls related to the concept of an oracle property as used in Fan and Li (2001, 2002, 2004) which are reminiscent of the well-known pitfalls related to Hodges' estimator. The oracle property is often a consequence of sparsity of an estimator. We show that any estimator satisfying a sparsity property has maximal risk that converges to the supremum of the loss function; in particular, the maximal risk diverges to infinity whenever the loss function is unbounded. For ease of presentation the result is set in the framework of a linear regression model, but generalizes far beyond that setting. In a Monte Carlo study we also assess the extent of the problem in finite samples for the smoothly clipped absolute deviation (SCAD) estimator introduced in Fan and Li (2001). We find that this estimator can perform rather poorly in finite samples and that its worst-case performance relative to maximum likelihood deteriorates with increasing sample size when the estimator is tuned to sparsity., Comment: 18 pages, 5 figures

Local asymptotic minimax risk bounds in a locally asymptotically mixture of normal family of distributions have been investigated under asymmetric loss functions and the asymptotic distribution of the optimal estimator that attains the bound has been obtained., Comment: Published at http://dx.doi.org/10.1214/074921706000000527 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)

In this review of estimation problems in restricted parameter spaces, we focus through a series of illustrations on a number of methods that have proven to be successful. These methods relate to the decision-theoretic aspects of admissibility and minimaxity, as well as to the determination of dominating estimators of inadmissible procedures obtained for instance from the criteria of unbiasedness, maximum likelihood, or minimum risk equivariance. Finally, we accompany the presentation of these methods with various related historical developments.

A Blackwell $\epsilon$-optimal strategy in a Markov Decision Process is a strategy that is $\epsilon$-optimal for every discount factor sufficiently close to 1. ¶ We prove the existence of Blackwell $\epsilon$-optimal strategies in finite Markov Decision Processes with partial observation.

A first general principle and nowadays state of the art for the construction of powerful multiple test procedures controlling a multiple level $\alpha$ is the so-called closure principle. In this article we introduce another powerful tool for the construction of multiple decision procedures, especially for the construction of multiple test procedures and selection procedures. This tool is based on a partition of the parameter space and will be called partitioning principle (PP). In the first part of the paper we review basic concepts of multiple hypotheses testing and discuss a slight generalization of the current theory. In the second part we present various variants of the PP for the construction of multiple test procedures, these are a general PP (GPP), a weak PP (WPP) and a strong PP (SPP). It will be shown that, depending on the underlying decision problem, a PP may lead to more powerful test procedures than a formal application of the closure principle (FCP). Moreover, the more complex SPP may be more powerful than the WPP. Based on a duality between testing and selecting PPs can also be applied for the construction of more powerful selection procedures. In the third part of the paper FCP, WPP and SPP are applied and compared in some examples.

One of the most difficult problems occurring with stepwise multiple test procedures for a set of two-sided hypotheses is the control of direc-tional errors if rejection of a hypothesis is accomplished with a directional decision. In this paper we generalize a result for so-called step-down procedures derived by Shaffer to a large class of stepwise or closed multiple test procedures. In a unifying way we obtain results for a large class of order statistics procedures including step-down as well as step-up procedures (Hochberg, Rom), but also a procedure of Hommel based on critical values derived by Simes. Our method of proof is also applicable in situations where directional decisions are mainly based on conditionally independent $t$-statistics. A closed $F$-test procedure applicable in regression models with orthogonal design, the modified $S$-method of Scheffé applicable in the Analysis of Variance and Fisher’s LSD-test for the comparison of three means will be considered in more detail.

Estimation of $p, p \geq 3$, location parameters of a distribution of a p-dimensional random vector $\mathsf{X}$ is considered under quadratic loss. Explicit estimators which are better than the best invariant one are given for a sign-invariantly distributed random vector $\mathsf{X}$. The results depend only on the second and the third moments of $|| \mathsf{X} - \theta ||$. The generalizations to concave loss functions and to n observations are also considered. Additionally, if the scale is unknown, we investigate the estimators of the location parameters when the observation contains a residual vector.

LetX1,...,Xn be pairwise independent random variables of known (but not necessarily identical) distribution; we wish to select a subset of these whose sum will be as close as possible to some known target value T. Conditions described below force the selections to be made by a primitive distributed system (similar to one considered by Papadimitriou and Yannakakis [2] in PODC '91); here we are able to obtain a surprising amount of information about optimal solutions.The conditions are that each variable must be “programmed” in advance, joining the selected set according to its own value. Thus, for example, one variable might be programmed to join just if its value lies between a and b, while another is told to join regardless of its value. Our object is to find a strategy, that is, a collection of programs, which minimizes the mean square error in approximating T. Typical applications involve producing a steady flow of some commodity when supply is controlled at a multiplicity of random sources.It turns out that there is always an optimal strategy in which each Xi is programmed to join if its value is between 0 and ti, for appropriate choice of thresholds ti. When the variables are identically distributed, we examine conditions under which the ti's must be equal. The case of uniform distributions on [0,1], for which the above conditions are not satisfied, is analyzed in detail, showing the rather bizarre behavior of the ti's which may take place in general as the target value is gradually changed.Next, we analyze the problem in which the variables are permitted to contribute any part of themselves to the sum; here it turns out that in an optimal strategy each program will be of the form “contribute the minimum of Xi and ei” with all the ei's equal in the i.i.d. case.Finally, we show how the original target shooting problem can be generalized to a kind of load balancing, where variables are assigned to different buckets, each with its own target, and the penalty is a weighted sum of squared errors. The surprising result here is that when the weights are equal, an optimal solution assigns variables only according to their signs.

This paper is concerned with estimating the loss of a point estimator when sampling from a spherically symmetric distribution. We examine the canonical setting of a general linear model where the dimension of the parameter space is greater than 4 and less than the dimension of the sampling space. We consider two location estimators ― the least squares estimator and a shrinkage estimator ― and we compare their unbiased loss estimator with an improved loss estimator. The domination results are valid for a large class of spherically symmetric distributions and, in so far as the sampling distribution does not need to be precisely specified, the estimates have desirable robustness properties.

A new criterion is proposed for the evaluation of variable selection procedures in multiple regression. This criterion, which we call the risk inflation, is based on an adjustment to the risk. Essentially, the risk inflation is the maximum increase in risk due to selecting rather than knowing the "correct" predictors. A new variable selection procedure is obtained which, in the case of orthogonal predictors, substantially improves on AIC, $C_p$ and BIC and is close to optimal. In contrast to AIC, $C_p$ and BIC which use dimensionality penalties of 2, 2 and $\log n$, respectively, this new procedure uses a penalty $2 \log p$, where $p$ is the number of available predictors. For the case of nonorthogonal predictors, bounds for the optimal penalty are obtained.

The purpose of this paper is to give an explicit estimator dominating the positive-part James-Stein rule. The James-Stein estimator improves on the "usual" estimator $X$ of a multivariate normal mean vector $\theta$ if the dimension $p$ of the problem is at least 3. It has been known since at least 1964 that the positive-part version of this estimator improves on the James-Stein estimator. Brown's 1971 results imply that the positive-part version is itself inadmissible although this result was assumed to be true much earlier. Explicit improvements, however, have not previously been found; indeed, 1988 results of Bock and of Brown imply that no estimator dominating the positive-part estimator exists whose unbiased estimator of risk is uniformly smaller than that of the positive-part estimator.

In the point and interval estimation of the variance of a normal distribution with an unknown mean, the best affine equivariant estimators are dominated by Stein's truncated and Brewster and Zidek's smooth procedures, which are separately derived. This paper gives a unified approach to this problem by using a simple definite integral and provides a class of improved procedures in both point and interval estimation of powers of the scale parameter of normal, lognormal, exponential and Pareto distributions. Finally, the same method is applied to the improvement on the James-Stein rule in the simultaneous estimation of a multinormal mean.

The usual confidence interval, based on Student's t distribution, has conditional confidence that is larger than the nominal confidence level. Although this fact is known, along with the fact that increased conditional confidence can be used to improve a confidence assertion, the confidence assertion of Student's t interval has never been critically examined. We do so here, and construct a confidence estimator that allows uniformly higher confidence in the interval and is closer (than 1 - a) to the indicator of coverage. 1. Introduction and summary. The usual confidence interval for a normal mean, when the population variance is unknown, is based on Student's

A strong adaptive criteria is defined for density estimation problems. In a particular case it is shown that there is no strongly adaptive sequence of estimators. In contrast Woodroofe has shown that a weakly adaptive result holds.

Confidence intervals for the variance of a normal distribution with unknown mean are constructed which improve upon the usual shortest interval based on the sample variance alone. These intervals have guaranteed coverage probability uniformly greater than a predetermined value $1-\alpha$ and have uniformly shorter length. Using information relating the size of the samples mean to that of the sample variance, we smoothly shift the usual minimum length interval closer to zero, simultaneously bringing the endpoints closer to each other. The gains in coverage probability and expected length are also investigated numerically. Lastly, we examine the posterior probabilities of the intervals, quantities which can be used as post-data confidence reports.

This paper is primarily concerned with extending the results of Stein to spherically symmetric distributions. Specifically, when $X \sim f(\|X - \theta\|^2)$, we investigate conditions under which estimators of the form $X + ag(X)$ dominate $X$ for loss functions $\|\delta - \theta\|^2$ and loss functions which are concave in $\|\delta - \theta\|^2$. Additionally, if the scale is unknown we investigate estimators of the location parameter of the form $X + aVg(X)$ in two different settings. In the first, an estimator $V$ of the scale is independent of $X$. In the second, $V$ is the sum of squared residuals in the usual canonical setting of a generalized linear model when sampling from a spherically symmetric distribution. These results are also generalized to concave loss. The conditions for domination of $X + ag(X)$ are typically (a) $\|g\|^2 + 2\nabla \circ g \leq 0$, (b) $\nabla \circ g$ is superharmonic and (c) $0 < a < 1/pE_0(1/\|X\|^2)$, plus technical conditions.

A general representation is obtained for the formal Bayes estimator of a parameter matrix. We assume that the prior distribution is symmetric in some sense, but it is not specified otherwise. The formal Bayes risk is minimized subject to order constraints by a variational technique; hence our representation is called "the variational form of the Bayes estimator" (VFBE). The VFBE is used to obtain estimators that have good frequency properties relative to the usual estimators. Such estimators are obtained for the mean vector and covariance matrix of a multivariate normal distribution. Also, for possibly nonnormal data, we give the VFBE of several Pearson means. A certain emphasis is placed on the problem of estimating the covariance matrix. For that problem, our constrained optimization provides an estimator with very good properties: Its eigenvalues are in the proper order, and they are not as distorted as those in the sample covariance matrix. The VFBE for the covariance matrix is related to an estimator of Stein. Of the two, the VFBE deals with order relations in a more natural way; that is, it is more criterion dependent. In addition, it is easier to compute than Stein's estimator, and a brief Monte Carlo simulation indicates that it has better risk properties as well.

Consider the problem of estimating the $p \times 1$ mean vector $\theta$ under expected squared error loss, based on the observation of two independent multivariate normal vectors $Y_1 \sim N_p(\theta, \sigma^2I)$ and $Y_2 \sim N_p(\theta, \lambda\sigma^2I)$ when $\lambda$ and $\sigma^2$ are unknown. For $p \geq 3$, estimators of the form $\delta_\eta = \eta Y_1 + (1 - \eta)Y_2$ where $\eta$ is a fixed number in (0, 1), are shown to be uniformly dominated in risk by Stein estimators in spite of the fact that independent estimates of scale are unavailable. A consequence of this result is that when $\lambda$ is assumed known, shrinkage domination is robust to incorrect specification of $\lambda$.

Let $S_1$ and $S_2$ be two independent $p \times p$ Wishart matrices with $S_1 \sim W_p(\sum_1, n_1)$ and $S_2 \sim W_p(\sum_2, n_2)$. We wish to estimate $(\sum_1, \sum_2)$ under the loss function $L(\hat{\sum}_1, \hat{\sum}_2; \sum_1, \sum_2) = \sum_i\{\operatorname{tr}(\sum^{-1}_i \hat{\sum}_i) - \log|\sum^{-1}_i\hat{\sum}_i| - p\}$. Our approach is to first utilize the principle of invariance to narrow the class of estimators under consideration to the equivariant ones. The unbiased estimates of risk of these estimators are then computed and promising estimators are derived from them. A Monte Carlo study is also conducted to evaluate the risk performances of these estimators. The results of this paper extend those of Stein, Haff, Dey and Srinivasan from the one sample problem to the two sample one.

This paper compares three methods for producing lower bounds on the minimax risk under quadratic loss. The first uses the bounds from Brown and Gajek. The second method also uses the information inequality and results in bounds which are always at least as good as those from the first method. The third method is the hardest-linear-family method described by Donoho and Liu. These methods are applied in four examples, the last of which relates to a frequently considered problem in nonparametric regression.

This paper presents lower bounds, derived from the information inequality, for the Bayes risk under scaled quadratic loss. Some numerical results are also presented which give some idea concerning the precision of these bounds. An appendix contains a proof of the information inequality without conditions on the estimator. This result is a direct extension of an earlier result of Fabian and Hannan.

A procedure for Bayes nonparametric estimation from a Markov renewal process is developed. It is based on a conjugate class of a priori distributions on the parameter space of semi-Markov transition distributions. The class is characterized by a Dirichlet family of distributions for random Markov matrices and a Beta family of Levy processes for random cumulative hazard functions. The main result is the derivation of the posterior law from an observation of the Markov renewal process over a period of time.

The usual confidence interval for the variance o,2 of a normal distribution is a function of the sample variance alone. In this paper, we construct confidence intervals for o-2 that also depend on the sample mean. These intervals have the same length as the shortest interval depending only on the sample variance and have uniformly higher probability of coverage. The coverage probabilities of these intervals and others are compared. 1. Background. Stein (1964) constructed a point estimator for the nor

Rating scales are used to elicit data about qualitative entities (e.g., research collaboration). This study presents an innovative method for reducing the number of rating scale items without the predictability loss. The "area under the receiver operator curve method" (AUC ROC) is used. The presented method has reduced the number of rating scale items (variables) to 28.57\% (from 21 to 6) making over 70\% of collected data unnecessary. Results have been verified by two methods of analysis: Graded Response Model (GRM) and Confirmatory Factor Analysis (CFA). GRM revealed that the new method differentiates observations of high and middle scores. CFA proved that the reliability of the rating scale has not deteriorated by the scale item reduction. Both statistical analysis evidenced usefulness of the AUC ROC reduction method., 14 pages, 5 figures

Let $S_{p \times p}$ have a nonsingular Wishart distribution with unknown matrix $\Sigma$ and $k$ degrees of freedom. For two different loss functions, estimators of $\Sigma^{-1}$ are given which dominate the obvious estimators $aS^{-1}, 0 < a \leqslant k - p - 1$. Our class of estimators $\mathscr{C}$ includes random mixtures of $S^{-1}$ and $I$. A subclass $\mathscr{C}_0 \subset \mathscr{C}$ was given by Haff. Here, we show that any member of $\mathscr{C}_0$ is dominated in $\mathscr{C}$. Some troublesome aspects of the estimation problem are discussed, and the theory is supplemented by simulation results.

For $p \geqslant 4$ and one observation $X$ on a $p$-dimensional spherically symmetric distribution, minimax estimators of $\theta$ whose risks are smaller than the risk of $X$ (the best invariant estimator) are found when the loss is a nondecreasing concave function of quadratic loss. For $n$ observations $X_1, X_2, \cdots, X_n$, we have classes of minimax estimators which are better than the usual procedures, such as the best invariant estimator, $\bar{X}$, or a maximum likelihood estimator.

Assume $X = (X_1,\cdots, X_p)^t$ has a $p$-variate density, with respect to Lebesgue measure, of the form $f((x - \theta)^t\not\sum^{-1}(x - \theta))$. Here $\not\sum$ is a known positive definite $p \times p$ matrix and $p \geqq 3$. Assume either (i) $f$ is completely monotonic, or (ii) there exist $\alpha > 0$ and $K > 0$ for which $h(s) = f(s)e^{\alpha s}$ is nondecreasing and nonzero if $s > K$. Then for estimating $\theta$ under a known quadratic loss, classes of minimax estimators are found.

The problem is considered of testing symmetry of a bivariate distribution $\mathscr{L}(X, Y)$ against "asymmetry towards high $X$-values," subject to the restriction of invariance under the transformations $(x_i, y_i) \mapsto (g(x_i), g(y_i)) (1 \leq i \leq n)$ for increasing bijections $g$. This invariance restriction prohibits the common reduction to the differences $x_i - y_i$. The intuitive concept of "asymmetry towards high $X$-values" is approached in several ways, and a mathematical formulation for this concept is proposed. Most powerful and locally most powerful invariant similar tests against certain subalternatives are characterized by means of a Hoeffding formula. Asymptotic normality and consistency results are obtained for appropriate linear rank tests.

The problem of selecting a random nonempty subset from $k$ populatinos, characterized by $\theta_1, \cdots, \theta_k$ with possible nuisance parameters $\sigma$, is considered using a decision-theoretic approach. The concept of asymptotic consistency is defined as the property that the risk of a procedure at $(\theta, \sigma)$ tends to the minimum loss at $(\theta, \sigma)$. Necessary and sufficient conditions for both pointwise and uniform (on compact sets) consistency for permutation-invariant procedures are derived with general loss functions. Various loss functions when the goal is to select populations with $\theta_i$ close to $\max \theta_j$ are considered. Applications are made to normal populations. It is shown that Gupta's procedure is the only procedure in Seal's class that can be consistent. Other Bayes and admissible procedures are also considered.

Let $X_1,X_2,...$ be a Bernoulli sequence with parameter p. An algorithm $T_{n=1}=\\f(T_n,X_n,n)$; $d_n = d(T_n); \\f:\{1,2,\1dots,8\} \times \{0,1\} \times \{0,1, \1dots}\rightarrow \{1, \1dots, 8\}; d:\{1,2,\dots,8\} \rigtharrow \{H_0,H_1\}$; is found such that $d(T_n)= H_0$ all but a finite number of times with probability one if p is rational, and $d(T_n)= H_1$ all but a finite number of times with probability one if p is irrational (and not in a given null set of irrationals). Thus, an 8-state memory with a time-varying algorithm makes only a finite number of mistakes with probability one on determining the rationality of the parameter of a coin. Thus, determining the rationality of the Bernoulli parameter p does not depend on infinite memory of the data.

Families of minimax estimators are found for the location parameter of a $p$-variate $(p \geqq 3)$ spherically symmetric unimodal distribution with respect to general quadratic loss. The estimators of James and Stein, Baranchik, Bock and Strawderman are all considered for this general problem. Specifically, when the loss is general quadratic loss given by $L(\delta, \theta) = (\delta - \theta)'D(\delta - \theta)$ where $D$ is a known $p \times p$ positive definite matrix, one main result, for one observation, $X$, on a multivariate s.s.u. distribution about $\theta$, presents a class of minimax estimators whose risk dominate the risk of $X$, provided $p \geqq 3$ and trace $D \geqq 2d_L$ where $d_L$ is the maximum eigenvalue of $D$. This class is given by $\delta_{a,r}(X) = (1 - a(r(\|X\|^2)/\|X\|^2)) X$ where $0 \leqq r(\bullet) \leqq 1, r(\|X\|^2)$ is nondecreasing, $r(\|X\|^2)/\|X\|^2$ is nonincreasing, and $0 \leqq a \leqq (c_0/E_0(\|X\|^{-2}))((\operatorname{trace} D/d_L) - 2)$, where $c_0 = 2p/((p + 2)(p - 2))$ when $p \geqq 4$ and $c_0 \approx .96$ when $p = 3$.

This paper is concerned with rates of convergence in the sequence-compound decision problem when the component problem is the squared-distance loss estimation of the mean of $m$-variate normal distribution with covariance matrix I. Section 1 introduces some notation and discusses the earlier work related to this problem. In Section 2, we prove two lemmas which are required in later sections. Sections 3, 4 and 5 exhibit sequence-compound decision procedures whose modified regrets are $O(n^{-1/m+4})$, near $O(n^{-\frac{1}{4}})$ and near $O(n^{-\frac{1}{2}})$ respectively, all the orders being uniform in parameter sequences concerned. In Section 6, comparisons have been made between the procedures given in Sections 3, 4 and 5. We conclude the paper with a few remarks.

If $X$ is a $N(\theta, 1)$ random variable, let $\rho (m)$ be the minimax risk for estimation with quadratic loss subject to $|\theta| \leq m$. Then $\rho (m) = 1 - \pi^2/m^2 + o(m^{-2})$. We exhibit estimates which are asymptotically minimax to this order as well as approximations to the least favorable prior distributions. The approximate least favorable distributions (correct to order $m^{-2}$) have density $m^{-1} \cos^2 \big(\frac{\pi}{2m} s\big), |s| \leq m$ rather than the naively expected uniform density on $\lbrack -m, m \rbrack$. We also show how our results extend to estimation of a vector mean and give some explicit solutions.

Inference about the correlation coefficient $\rho$ in a bivariate normal distribution is considered when observations from the distribution are available only in the form of a broken random sample. In other words, a random sample of $n$ pairs is drawn from the distribution but the observed data are only the first components of the $n$ pairs and, separately, some unknown permutation of the second components of the $n$ pairs. Under these conditions, the estimation of $\rho$ is, as Samuel Johnson put it, "like a dog's walking on his hinder legs. It is not done well; but you are surprised to find it done at all." We study the maximum likelihood estimation of $\rho$ and present some effective procedures for estimating the sign of $\rho$.

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.

In this paper a necessary and sufficient condition for a predictor to be the best invariant predictor in location and scale families is given. Using this condition, it is shown that the best invariant predictor is expressed by a linear combination of the best unbiased predictor and the best unbiased estimator of the scale parameter.

Stein's general technique for improving upon the best invariant unbiased and minimax estimators of the normal covariance matrix is described. The technique is to obtain solutions to a certain differential inequality involving the eigenvalues of the sample covariance matrix. Several improved estimators are obtained by solving the differential inequality. These estimators shrink or expand the sample eigenvalues depending on their magnitude. A scale invariant, adaptive minimax estimator is also obtained.

This paper provides an empirical Bayes approach to the problem of nonparametric estimation of a distribution (or survival) function when the observations are censored on the right. The results use the notion of a Dirichlet process prior (Ferguson, 1973, Ann. Stat., 2, 209-230). The paper presents a generalization to the case of right censored observations of the rate result of an empirical Bayes nonparametric estimator of a distribution function of Korwar and Hollander ((1974, Tech. Rept. No. 288, Dept. of Statistics, Florida State University) in the uncensored case. The rate of asymptotic convergence to optimality is shown to be the best obtainable for the problem considered.

Minimax interval estimators of location parameters analogous to the Pitman point estimators are found by employing invariance in decision theory. The half normal distribution is used as an illustrative example.

Consider the problem of estimating the ordered means $\mu_1 \leq \mu_2 \leq \cdots \leq \mu_k$ of independent normal random variables, $Y_1, Y_2, \cdots, Y_k$. It is shown that the absolute error of each component $\hat{\mu}_i$ of the isotonic regression estimator is stochastically smaller than that of the usual estimator $Y_i$. Thus $\hat{\mu}_i$ is superior to $Y_i$ under any nonconstant loss which is a nondecreasing function of absolute error.

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.

Let $X$ be an observation from a $p$-variate normal distribution $(p \geqq 3)$ with mean vector $\theta$ and unknown positive definite covariance matrix $\not\Sigma$. It is desired to estimate $\theta$ under the quadratic loss $L(\delta, \theta, \not\Sigma) = (\delta - \theta)^tQ(\delta - \theta)/\operatorname{tr} (Q\not\Sigma)$, where $Q$ is a known positive definite matrix. Estimators of the following form are considered: $\delta^c(X, W) = (I - c\alpha Q^{-1}W^{-1}/(X^tW^{-1}X))X,$ where $W$ is a $p \times p$ random matrix with a Wishart $(\not\Sigma, n)$ distribution (independent of $X$), $\alpha$ is the minimum characteristic root of $(QW)/(n - p - 1)$ and $c$ is a positive constant. For appropriate values of $c, \delta^c$ is shown to be minimax and better than the usual estimator $\delta^0(X) = X$.

Let $X$ be an observation from a $p$-variate normal distribution $(p \geqslant 3)$ with mean vector $\theta$ and unknown positive definite covariance matrix $\sum$. We wish to estimate $\theta$ under the quadratic loss $L(\delta; \theta, \sum) = \lbrack\mathrm{tr}(Q\sum)\rbrack^{-1}(\delta - \theta)'Q(\delta - \theta)$, where $Q$ is a known positive definite matrix. Estimators of the following form are considered: $\delta_{k, h}(X, W) = \lbrack I - kh(X'W^{-1}X)\lambda_1(QW/n^\ast)Q^{-1}W^{-1}\rbrack X,$ where $W : p \times p$ is observed independently of $X$ and has a Wishart distribution with $n$ degrees of freedom and parameter $\sum, \lambda_1(A)$ denotes the minimum characteristic root of $A$, and $h(t): \lbrack 0, \infty) \rightarrow \lbrack 0, \infty)$ is absolutely continuous with respect to Lebesgue measure, is nonincreasing, and satisfies the additional requirements that $th(t)$ is nondecreasing and $\sup_{t \geqslant 0}th(t) = 1$. With $h(t) = t^{-1}$, the class $\delta_{k, h}$ specializes to that considered by Berger, Bock, Brown, Casella and Gleser (1977). For the more general class considered in the present paper, it is shown that there is an interval $\lbrack 0, k_{n, p}\rbrack$ of values of $k$ (which may be degenerate for small values of $n - p)$ for which $\delta_{k, h}$ is minimax and dominates the usual estimator $\delta_0 \equiv X$ in risk.

Both one-sided and two-sided interval estimators are examined using conditional criteria and we find that most common intervals have acceptable conditional properties. In the two-sided case we further examine three well known intervals and find the shortest-unbiased (Neyman-shortest) interval possessing the strongest conditional properties, with the minimum-length interval a close second. In the one-sided case we have the somewhat surprising result that the lower confidence interval (which results from inverting the UMP test of $H_0: \sigma \leq \sigma_0$) has weaker conditional properties than the upper interval (where a UMP test does not exist).

The problem of estimating a normal mean has received much attention in recent years. If one assumes, however, that the true mean lies in a bounded interval, the problem changes drastically. In this paper we show that if the interval is small (approximately two standard deviations wide) then the Bayes rule against a two point prior is the unique minimax estimator under squared error loss. For somewhat wider intervals we also derive sufficient conditions for minimaxity of the Bayes rule against a three point prior.

The problem of selecting a random subset of good populations out of $k$ populations is considered. The populations $\Pi_1, \cdots, \Pi_k$ are characterized by the location parameters $\theta_1, \cdots, \theta_k$ and $\Pi_i$ is said to be a good population if $\theta_i > \max_{1 \leq j\leq k}\theta_j - \Delta$, and a bad population if $\theta_i \leq \max_{1 \leq j \leq k} \theta_j - \Delta$, where $\Delta$ is a specified positive constant. A theory for a special class of procedures, called Schur procedures, is developed, and applied to certain minimax problems. Subject to controlling the minimum expected number of good populations selected or the probability that the best population is in the selected subset, procedures are derived which minimize the expected number of bad populations selected or some similar criterion. For normal populations it is known that the classical "maximum-type" procedures has certain minimax properties. In this paper, two other procedures are shown to have several minimax properties. One is the "average-type" procedure. The other procedure has not previously been considered as a serious contender.

This paper establishes the asymptotic optimality (in the sense of Robbins) of the empirical Bayes distribution function created from the Bayes rule relative to the Dirichlet process prior with unknown parameter $\alpha(.)$. It will follow that the same result applies to the estimation of the mean of a distribution function.