Confidence intervals are widely used in statistical practice as indicators of precision for related point estimators or as estimators in their own right. In the present paper it is shown that for some models, including most linear and nonlinear errors-in-variables regression models, and for certain estimation problems arising in the context of classical linear models, such as the inverse regression problem, it is impossible to construct confidence intervals for key parameters which have both positive confidence and finite expected length. The results are generalized to cover general confidence sets for both scalar and vector parameters.

The usual confidence sphere for a multivariate normal mean can be uniformly improved upon, in terms of coverage probability, by recentering it at a Stein-type estimator. However, these improved sets can have poor conditional performance. Using the theory of relevant betting procedures, which provides an objective means for assessing the conditional performance of a statistical procedure, a criterion for conditional acceptability can be established. A method of constructing such sets is outlined and applied to some recentered confidence sets. In particular, recentering at the positive-part James-Stein estimator yields a conditionally acceptable confidence set.

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).

Estimating the mean of a $p$-variate normal distribution is considered when the loss is squared error plus a complexity cost. The complexity of estimates is defined using a partition of the parameter space into sets corresponding to models of different complexity. The model implied by the use of an estimate determines the estimate's complexity cost. Complete classes of estimators are developed which consist of preliminary-test estimators. As is the case when loss is just squared error, the maximum-likelihood estimator is minimax. However, unlike the no-complexity-cost case, the maximum-likelihood estimator is inadmissible even in the case when $p = 1$ or 2.

General results on the limiting equivalence of local and nonlocal measures of efficiency are obtained. Why equivalence occurs in so many testing and estimation problems is clarified. Uniformity of the convergence is a key point. The concepts of Frechet- and Hadamard-type differentiability, which imply uniformity, play an important role. The theory is applied to tests based on linear rank statistics, showing equivalence of the local limit of exact Bahadur efficiency and Pitman efficiency. As a second application, the relation between the inaccuracy rate and the asymptotic variance of $L$-estimators is investigated.

This paper considers Cramer-Rao type bounds for the estimation error of a parameter in a Bayesian setup. This class of bounds, introduced by Van Trees, proved useful in various stochastic communications and control problems. Two issues are considered in this paper. The first deals with a comparison of the tightness of several different versions of the bound in the multivariate case. The second introduces several useful generalizations of the original version of the bound.

In Glenn Shafer's theory of parametric statistical inference, observational evidence and prior evidence are separately represented by belief or commonality functions $Q$ and $R$, which are then combined by Dempster's rule. We characterise, for finite parameter spaces, the functionals $Q$ and $R$ for which statistically independent observations may be combined by Dempster's rule, and those for which Dempster's rule is consistent with Bayes' rule. The functionals are determined up to an arbitrary partition of the parameter space and an arbitrary scale parameter, which might be chosen to reflect aspects of the evidence on which the statistical model is based. Our results suggest that Dempster's rule is not generally suitable for combining evidence from independent observations nor for combining prior beliefs with observational evidence.

A two parameter family of distributions on partitions is obtained by mixing a multinomial with a symmetric Dirichlet distribution. Estimates for the parameters are proposed and studied in various asymptotic limits.

Let $\mathbf{Y} = (Y_1, \cdots, Y_n)$ be random variables satisfying the weak negative dependence condition: $P(Y_i < a_i\mid Y_1 < a_1, \cdots, Y_{i-1}) \leq P(Y_i < a_i)$ for $i = 2, \cdots, n$ and all constants $a_1, \cdots, a_n$. Let $\mathbf{X} = (X_1, \cdots, X_n)$ have independent components, where $X_i$ and $Y_i$ have the same marginal distribution, $i = 1, \cdots, n$. It is shown that $V(\mathbf{X}) \leq V(\mathbf{Y})$, where $V(\mathbf{Y}) = \sup \{EY_t: t \text{is a stopping rule for} Y_1,\cdots, Y_n\}$. Also, the classical inequality which for nonnegative variables compares the expected return of a prophet $E\{Y_1 \vee \cdots \vee Y_n\}$ with that of the statistician $V(\mathbf{Y})$, i.e., $E\{Y_1 \vee \cdots \vee Y_n\} < 2V(\mathbf{Y})$, holds for nonnegative $\mathbf{Y}$ satisfying the negative dependence condition. Moreover, this inequality can be obtained by an explicitly described threshold rule $t(b)$, i.e., $E\{Y_1 \vee \cdots \vee Y_n\} < 2EY_{t(b)}$. Generalizations of this prophet inequality are given. Extensions of the results to infinite sequences are obtained.

"Discrimination information," or Kullback-Leibler loss, is an appropriate measure of distance in problems of discrimination. We examine it in the context of nonparametric kernel density estimation and show that its asymptotic properties are profoundly influenced by tail properties of the kernel and of the unknown density. We suggest ways of choosing the kernel so as to reduce loss, and describe the extent to which likelihood cross-validation asymptotically minimises loss. Likelihood cross-validation generally leads to selection of a window width of the correct order of magnitude, but not necessarily to a window with the correct first-order properties. However, if the kernel is chosen appropriately, then likelihood cross-validation does result in asymptotic minimisation of Kullback-Leibler loss.

This paper makes two important contributions to the theory of bandwidth selection for kernel density estimators under right censorship. First, an asymptotic representation of the integrated squared error into easily understood variance and squared bias components is given. Second, it is shown that if the bandwidth is chosen by the data-based method of least-squares cross-validation, then it is asymptotically optimal in a compelling sense. A by-product of the first part is an interesting comparison of the two most popular kernel estimators.

A double censoring mechanism is such that each variable $X$ in the sample is observable if and only if $X$ is within the observation interval $\lbrack Z, Y \rbrack$. Otherwise, we can only determine whether $X$ is less than $Z$ or greater than $Y$ and observe $Z$ or $Y$ correspondingly. This kind of censoring occurs often in collecting lifetime data. Our problem is to estimate the survival function of $X, S_X(t) = P \lbrack X > t \rbrack$, from a doubly censored sample, where $X$ is assumed to be independent of the random interval $\lbrack Z, Y \rbrack$. We establish sufficient conditions for which $S_X(t)$ is identifiable and then prove the strong consistency of the self-consistent estimator $\hat{S}_X(t)$ for $S_X(t)$. This investigation generalizes the results available for the right censored data.

Consider estimators which behave locally asymptotically like an average of some function taken at the observations. This function is called the influence function and one calls such estimators locally asymptotically linear. It is shown that the influence function of a locally asymptotically linear estimator can be estimated consistently and conversely, that, given a consistent estimator of the influence function, estimators can be constructed which are locally asymptotically linear in that influence function. With the help of these results an adaptive estimator is constructed for a partially irregular model.

The asymptotic unbiasedness and consistency of three types of jackknife variance estimators in the presence of error variance heteroscedasticity in linear models are studied. The results are given in terms of the number of observations deleted and measures of imbalance of the model. The consistency of a class of Wu's weighted jackknife variance estimators for nonlinear parameters is also studied. A necessary and sufficient condition is given for the asymptotic unbiasedness and consistency of the unweighted delete-1 jackknife variance estimator and Hinkley's weighted delete-1 jackknife variance estimator. This condition is more stringent than those required for Wu's weighted jackknife. Comparison of the three delete-1 jackknife variance estimators in terms of their biases also favors the latter method.

In a general univariate linear model, $M$-estimation of a subset of parameters is considered when the complementary subset is plausibly redundant. Along with the classical versions, both the preliminary test and shrinkage versions of the usual $M$-estimators are considered and, in the light of their asymptotic distributional risks, the relative asymptotic risk-efficiency results are studied in detail. Though the shrinkage $M$-estimators may dominate their classical versions, they do not, in general, dominate the preliminary test versions.

Let $\Omega = \{\mathbf{x} = (x_1, \cdots, x_n)^T: 0 \leq x_i \leq 1\}$ be the unit cube in $R^n$. For any probability measure $\xi$ on $\Omega$, let $\mathbf{M}(\xi) = \int_\Omega\mathbf{xx}^T \xi(d\mathbf{x})$. Harwit and Sloane (1976) conjectured that if $\mathbf{X}^\ast$ is the incidence matrix of a balanced incomplete block design (BIBD) with $n$ treatments and $n$ blocks of size $(n + 1)/2$, then $\mathbf{X}^\ast$ minimizes $\operatorname{tr}(\mathbf{X}^T\mathbf{X})^{-1}$ over the $n \times n$ matrices with entries $0 \leq x_{ij} \leq 1$. This arises from a problem in spectroscopy. In order to solve the conjecture, we consider the more general problem of maximizing $j_a(\mathbf{M}(\xi))$ over the probability measures on $\Omega$ for $-\infty \leq a \leq 1$, where $j_0 (\mathbf{M}(\xi)) = \{\det \mathbf{M}(\xi))\}^{1/n}, j_{-\infty} (\mathbf{M}(\xi)) =$ the minimum eigenvalue of $\mathbf{M}(\xi)$ and $j_a(\mathbf{M}(\xi)) = \{n^{-1} \operatorname{tr}\lbrack \mathbf{M}(\xi) \rbrack^a\}^{1/a}$ for other $a$'s. A complete solution is obtained by using the equivalence theorem in optimal design theory. Let $\xi_k$ be the uniform measure on the vertices of $\Omega$ with $k$ coordinates equal to 1. Then depending on the value of $a$, optimality is attained by $\xi_k$ or a mixture of $\xi_k$ and $\xi_{k+1}$ with $k \geq \lbrack (n + 1)/2\rbrack$. Optimal $\xi$'s with a smaller support can be found by using BIBDs. It follows that if $n$ is odd and $\mathbf{X}^\ast$ is the block-treatment incidence matrix of a BIBD with $n$ treatments and $N$ blocks of size $(n + 1)/2$, then $\mathbf{X}^\ast$ minimizes $\operatorname{tr}(\mathbf{X}^T\mathbf{X})^a$ for all $a < 0$ and maximizes $\det(\mathbf{X}^T\mathbf{X})^a$ and $\operatorname{tr}(\mathbf{X}^T\mathbf{X})^a$ for all $0 < a \leq 1 - \ln(n/2 + 1)/ \ln(n/2 + 1)/ \ln( n + 1)$ over the $N \times n$ matrices with entries $0 \leq x_{ij} \leq 1$. Similar results are derived for the even case and the incidence matrices of BIBDs of larger block sizes.

In this paper, we consider the type II(a) designs of Williams. It was shown, essentially, by Kiefer that the type II(a) designs are asymptotically universally optimum for a first order autoregression with parameter $\lambda > 0$. We concentrate on the stationary first order autoregression with $\lambda > 0$ and the extra plot version of the II(a) designs. Our main results are that the design is $D$- and $A$-optimal then, but is not necessarily $E$-optimal when $\lambda$ is small.

We consider the problem of comparing test treatments with a control in a proper block design. We give conditions on the parameters of both R-type and S-type designs that guarantee their $A$-optimality, and demonstrate how these conditions can be used to obtain families of $A$-optimal designs. We give an example for the construction of the desired S-type designs. A table with optimal R-type designs $(3 \leq k \leq 10, k \leq v \leq 30)$ is also given.

Every positive definite matrix $\Sigma$ has a unique Cholesky decomposition $\Sigma = \theta\theta'$, where $\theta$ is lower triangular with positive diagonal elements. Suppose that $S$ has a Wishart distribution with mean $n\Sigma$ and that $S$ has the Cholesky decomposition $S = XX'$. We show, for a variety of loss functions, that $XD$, where $D$ is diagonal, is a best equivariant estimator of $\theta$. Explicit expressions for $D$ are provided.

We consider the problem of estimating parameter matrices which occur in the noncentral Wishart, noncentral multivariate $F$ and canonical correlations distributions. A decision-theoretic approach is taken with squared error as the loss function. In these three settings the eigenvalues of the parameter matrices are of primary interest. Sensible estimates of these are obtained by restricting attention to orthogonally invariant estimates of the parameter matrices, whose eigenvalues are functions only of sample eigenvalues.

Herein we consider the asymptotic performance of the least squares predictors $\hat{y}_n$ of the stochastic regression model $y_n = \beta_1 x_{n1} + \cdots + \beta_p x_{np} + \varepsilon_n$. In particular, the accumulated cost function $\sum^n_{k=1} (y_k - \hat{y}_k - \varepsilon_k)^2$ is studied. The results are then applied to nonstationary autoregressive time series. A statistic is also constructed to show how many times one should difference a nonstationary time series in order to obtain a stationary series.

A class of procedures is proposed for testing the composite hypothesis that a stationary stochastic process is Gaussian. Requiring very limited prior knowledge about the structure of the process, the tests rely on quadratic forms in deviations of certain sample statistics from their population counterparts, minimized with respect to the unknown parameters. A specific test is developed, which employs differences between components of the sample and Gaussian characteristic functions, evaluated at certain points on the real line. By demonstrating that, under $H_0$, the normalized empirical characteristic function converges weakly to a continuous Gaussian process, it is shown that the test remains valid when arguments of the characteristic functions are in certain ways data dependent.

It is shown that the convergence in law of estimators entails convergence uniformly over all Borel sets whenever the estimators are asymptotically equivariant in a suitable sense and the likelihood ratios of the sample are appropriately smooth. This result generalizes a recent result of Boos in many directions.

Let $\{S_n = \sum^n_1 X_i\}_{n\geq 0}$ be a random walk with positive drift $\mu = EX_1 > 0$ and finite variance $\sigma^2 = \operatorname{Var} X_1$. Let $\tau(b) = \inf\{n \geq 1: S_n > b\}, R_b = S_{\tau(b)} - b, M = \min_{n\geq 0} S_n, \tau^+ = \tau(0)$ and $H = S_\tau +$. Lai and Siegmund show that $\operatorname{Var} \tau(b) = b\sigma^2/\mu^3 + K/\mu^2 + o(1)$ as $b \rightarrow \infty$, but give an unpleasant expression for the constant $K$. Using the identity $\int Eh(R_{-y}) dP(M \leq y) = E^+_\tau h(H)/E\tau^+$, the expression for $K$ can be simplified to a form that depends only on moments of ladder variables.

The phenomenon of partial $n^{-1/2}$-term correction by the bootstrap in the estimation of sampling distributions of nonstandardized statistics is explained and studied in this note.