Consider a multiple linear regression in which $Y_i, i = 1, \cdots, n$, are independent normal variables with variance $\sigma^2$ and $E(Y_i) = \alpha + V'_i\beta$, where $V_i \in \mathbb{R}^r$ and $\beta \in \mathbb{R}^r.$ Let $\hat{\alpha}$ denote the usual least squares estimator of $\alpha$. Suppose that $V_i$ are themselves observations of independent multivariate normal random variables with mean 0 and known, nonsingular covariance matrix $\theta$. Then $\hat{\alpha}$ is admissible under squared error loss if $r \geq 2$. Several estimators dominating $\hat{\alpha}$ when $r \geq 3$ are presented. Analogous results are presented for the case where $\sigma^2$ or $\theta$ are unknown and some other generalizations are also considered. It is noted that some of these results for $r \geq 3$ appear in earlier papers of Baranchik and of Takada. $\{V_i\}$ are ancillary statistics in the above setting. Hence admissibility of $\hat{\alpha}$ depends on the distribution of the ancillary statistics, since if $\{V_i\}$ is fixed instead of random, then $\hat{\alpha}$ is admissible. This fact contradicts a widely held notion about ancillary statistics; some interpretations and consequences of this paradox are briefly discussed.

We study approximations to the distribution of counts of matches in the best matching segment of specified length when comparing two long sequences of i.i.d. letters. The key tools used are large-deviation inequalities and the Chen-Stein method of Poisson approximation. The origin of the problem in molecular biology is indicated.

Random left truncation is modelled by the conditional distribution of the random variable $X$ of interest, given that it is larger than the truncating random variable $Y$; usually $X$ and $Y$ are assumed independent. The present paper is based on a simple reparametrization of the left truncation model as a three-state Markov process. The derivation of a nonparametric estimator is a distribution function under random truncation is then a special case of results on the statistical theory of counting processes by Aalen and Johansen. This framework also clarifies the status of the estimator as a nonparametric maximum likelihood estimator, and consistency, asymptotic normality and efficiency may be derived directly as special cases of Aalen and Johansen's general theorems and later work. Although we do not carry through these here, we note that the present framework also allows several generalizations: censoring may be incorporated; the independence hypothesis underlying the truncation models may be tested; ties (occurring when the distributions of $F$ and $G$ have discrete components) may be handled.

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.

In shrinkage estimation of a multivariate normal mean, the two dominant approaches to construction of estimators have been the hierarchical or empirical Bayes approach and the minimax approach. The first has been most extensively used in practice, because of its greater flexibility in adapting to varying situations, while the second has seen the most extensive theoretical development. In this paper we consider several topics on the interface of these approaches, concentrating, in particular, on the interface between hierarchical Bayes and frequentist shrinkage estimation. The hierarchical Bayes setup considered is quite general, allowing (and encouraging) utilization of subjective second stage prior distributions to represent knowledge about the actual location of the normal means. (The first stage of the prior is used, as usual, to model suspected relationships among the means.) We begin by providing convenient representations for the hierarchical Bayes estimators to be considered, as well as formulas for their associated posterior covariance matrices and unbiased estimators of matrical mean square error; these are typically proposed by Bayesians and frequentists, respectively, as possible "error matrices" for use in evaluating the accuracy of the estimators. These two measures of accuracy are extensively compared in a special case, to highlight some general features of their differences. Risks and various estimated risks or losses (with respect to quadratic loss) of the hierarchical Bayes estimators are also considered. Some rather surprising minimax results are established (such as one in which minimaxity holds for any subjective second stage prior of the mean), and the various risks and estimated risks are extensively compared. Finally, a conceptually trivial (but often calculationally difficult) method of verifying minimaxity is illustrated, based on numerical maximization of the unbiased estimator of risk (using certain convenient calculational formulas for hierarchical Bayes estimators), and is applied to an illustrative example.

We illustrate with contemporary examples Hotelling's geometric approach to simultaneous probability calculations. Hotelling reduces the evaluation of certain normal theory significance probabilities to finding the volume of a tube about a curve in a hypersphere, and shows that this volume is often exactly given by length times cross-sectional area. We review Hotelling's result together with some recent complements, and then use the approach to set simultaneous prediction regions for some data from gait analysis, to study Andrews' plots in multivariate data analysis, and to construct significance tests for projection pursuit regression. A by-product is a numerical criterion for tube self-overlap, relevant, for example, to uniqueness of certain nonlinear least squares estimates.

For statistical procedures including Scheffe-type simultaneous confidence bounds for response surfaces and likelihood ratio tests for an additional regressor with unspecified parameters in a regression model, the confidence level or size can be expressed in terms of probabilities of the form $P\lbrack U \in D(\Gamma, \theta)\rbrack$, where $\Gamma$ is a subset of $S^m$ (the unit sphere in $R^{m + 1}), U$ is uniformly distributed in $S^m$ and $D(\Gamma, \theta)$ denotes the tubular neighborhood of $\Gamma$ of angular radius $\theta$, the set of points in $S^m$ whose angular distance from $\Gamma$ is at most $\theta$. Consequently, determining critical points involves the calculation of the volumes of tubes. For the case when $\Gamma$ is the diffeomorphic image of an $r$-dimensional convex polytope, an upper bound is given for the volume of its tubular neighborhood when the tube radius is sufficiently small, and which is exact in some special cases. Even if the tubular radius is moderate in size, the expression can be used to approximate the volume. The volume expression is a sum of $r$-fold integrals, one corresponding to each face of the polytope, and is derived using a result of Weyl (1939), which gives the volume of a tubular neighborhood of a $k$-dimensional submanifold of the unit sphere. Use of the expression leads to conservative statistical procedures when the desired error probability is sufficiently small and to asymptotically valid procedures as the error probability goes to zero.

Let $f$ be a continuous and positive unknown density on a known compact interval $\mathscr{Y}$. Let $F$ denote the distribution function of $f$ and let $Q = F^{-1}$ denote its quantile function. A finite-parameter exponential family model based on $B$-splines is constructed. Maximum-likelihood estimation of the parameters of the model based on a random sample of size $n$ from $f$ yields estimates $\hat{f, F}$ and $\hat{Q}$ of $f, F$ and $Q$, respectively. Under mild conditions, if the number of parameters tends to infinity in a suitable manner as $n \rightarrow \infty$, these estimates achieve the optimal rate of convergence. The asymptotic behavior of the corresponding confidence bounds is also investigated. In particular, it is shown that the standard errors of $\hat{F}$ and $\hat{Q}$ are asymptotically equal to those of the usual empirical distribution function and empirical quantile function.

The ACE (alternating conditional expectations) algorithm developed by Breiman and Friedman is an iterative method for finding optimal transformations of variables in multiple regression. Recently, several authors have extended ACE to discriminant analysis, time series and principal components. The central idea of ACE and of each of these extensions is that an optimal transformation $\phi^\ast$ minimizes a squared error-related functional over a Hilbert space, subject to nonlinear functional constraints. An estimate $\hat{\phi}^{(N)}$ is obtained by minimizing an estimate of the functional, subject to estimates of the constraints, over a smoothness restricted class of transformations. Using the method of sieves, conditions are established for consistency of $\hat{\phi}^{(N)}$ in the $\mathbf{L}^2$ sense.

For nonparametric regression estimation, when the unknown function belongs to a Sobolev smoothness class, sharp risk bounds for integrated mean square error have been found recently which improve on optimal rates of convergence results. The key to these has been the fact that under normality of the errors, the minimax linear estimator is asymptotically minimax in the class of all estimators. We extend this result to the nonnormal case, when the noise distribution is unknown. The pertaining lower asymptotic risk bound is established, based on an analogy with a location model in the independent identically distributed case. Attainment of the bound and its relation to adaptive optimal smoothing are discussed.

Conditions for the existence of the density of a minimum contrast estimator in a parametric statistical family are given together with a formula for this density. The formula is exact if multiple local minima cannot occur; otherwise the formula is an exact expression for the point process of local minima of the contrast function. Although it is not in general feasible to compute the expression for the density, the formula can be used as a basis for further expansion of the large deviation type. When the estimate is sufficient, either in the original model or after conditioning on an approximate or exact ancillary, the formula simplifies drastically. In particular, it is shown how Barndorff-Nielsen's formula for the density of the maximum likelihood estimator given an ancillary statistic is derived from the formula given here. In this way the nature of Barndorff-Nielsen's formula as an asymptotic approximation and its appearance as an exact formula for certain cases are demonstrated.

The behaviour of a graphical interaction model under marginalization is discussed. A graphical interaction model is called collapsible onto a set of variables if the class of marginal distributions is the same as that implied by the related subgraph. The necessary and sufficient condition for collapsibility is found and it is shown that collapsibility is equivalent to a range of other important statistical properties of the model.

Let $X_1, X_2, \cdots$ be iid observations from a mixture density $f(x) = \int f(x \mid \theta)dG(\theta)$, where $f(x \mid \theta)$ is a known parametric family of density functions and $G$ is an unknown distribution function. This paper concerns estimating the mixing density $g = G'$ and the mixing distribution $G$. Fourier methods are used to derive kernel estimators, upper bounds for their rates of convergence and lower bounds for the optimal rate of convergence. Sufficient conditions are given under which the kernel estimators are asymptotically normal. Our estimators achieve the optimal rate of convergence $(\log n)^{-1/2}$ for the normal family and $(\log n)^{-1}$ for the Cauchy family.

Supposing a given collection $y_1, \cdots, y_N$ of i.i.d. random points on a Riemannian manifold, we discuss how to estimate the underlying distribution from a differential geometric viewpoint. The main hypothesis is that the manifold is closed and that the distribution is (sufficiently) smooth. Under such a hypothesis a convergence arbitrarily close to the $N^{-1/2}$ rate is possible, both in the $L_2$ and the $L_\infty$ senses.

For a random field $z(t)$ defined for $t \in R \subseteq \mathbb{R}^d$ with specified second-order structure (mean function $m$ and covariance function $K$), optimal linear prediction based on a finite number of observations is a straightforward procedure. Suppose $(m_0, K_0)$ is the second-order structure used to produce the predictions when in fact $(m_1, K_1)$ is the correct second-order structure and $(m_0, K_0)$ and $(m_1, K_1)$ are "compatible" on $R$. For bounded $R$, as the points of observation become increasingly dense in $R$, predictions based on $(m_0, K_0)$ are shown to be uniformly asymptotically optimal relative to the predictions based on the correct $(m_1, K_1)$. Explicit bounds on this rate of convergence are obtained in some special cases in which $K_0 = K_1$. A necessary and sufficient condition for the consistency of best linear unbiased predictors is obtained, and the asymptotic optimality of these predictors is demonstrated under a compatibility condition on the mean structure.

The bandwidth selection problem in kernel density estimation is investigated in situations where the observed data are dependent. The classical leave-out technique is extended, and thereby a class of cross-validated bandwidths is defined. These bandwidths are shown to be asymptotically optimal under a strong mixing condition. The leave-one out, or ordinary, form of cross-validation remains asymptotically optimal under the dependence model considered. However, a simulation study shows that when the data are strongly enough correlated, the ordinary version of cross-validation can be improved upon in finite-sized samples.

In this paper we obtain the asymptotic distribution of robust nonparametric autoregression estimators for dependent observations. Weights based on kernel and nearest-neighbor methods are considered.

In this paper, an entropy approach is proposed to establish rates of convergence for estimators of a regression function. General regression problems are considered, with linear regression, splines and isotonic regression as special cases. The estimation methods studied are least squares, least absolute deviations and penalized least squares. Common features of these methods and various regression problems are highlighted.

We consider in this paper two widely studied examples of nonparametric and semiparametric models in which the standard information bounds are totally misleading. In fact, no estimators converge at the $n^{-\alpha}$ rate for any $\alpha > 0$, although the information is strictly positive "promising" that $n^{-1/2}$ is achievable. The examples are the estimation of $\int p^2$ and the slope in the model of Engle et al. A class of models in which the parameter of interest can be estimated efficiently is discussed.

Consider the model $X_i = \rho X_{i - 1} + \varepsilon_i, |\rho| > 1$, where $X_0, \varepsilon_1, \varepsilon_2, \cdots$ are independent random variables with $\varepsilon_1, \varepsilon_2, \cdots$ having common density $\psi$. This paper gives sufficient conditions under which the sequence of experiments induced by $\{X_0, X_1, \cdots, X_n\}$ has a weak limit in the sense of Le Cam. In general, the limiting experiment is translation invariant and neither LAN nor LAMN. The paper further shows that the sequence of Pitman-type estimators of $\rho$ at a given $\psi$ converges weakly to $T$, where $T$ is minimax for the limiting experiment under a weighted squared error loss function. Finally, for an unknown $\psi$, a sequence of Pitman-type estimators that converges weakly to $T$ is constructed. These estimators are called weakly adaptive. The class of error densities for which these results hold include some that may not have finite Fisher information.

We propose a definition of the partial autocorrelation function $\beta(\cdot)$ for multivariate stationary time series suggested by the canonical analysis of the forward and backward innovations. Here $\beta(\cdot)$ satisfies $\beta(-n) = \beta(n)', n = 0, 1, \cdots,$ where $\beta(0)$ is nonnegative definite, $\{\beta(n), n = 1, 2, \cdots\}$ is a sequence of square matrices having singular values less than or equal to 1 and such that the order of $\beta(n + 1)$ is equal to the rank of $I - \beta(n)\beta(n)',$ the order of $\beta(1)$ being equal to the rank of $\beta(0)$. We show that there exists a one-to-one correspondence between the set of matrix autocovariance functions $\Lambda(\cdot)$, with the positive definiteness property, and the set of canonical partial autocorrelation functions $\beta(\cdot)$ as described above.

The usual confidence interval for the variance $\sigma^2$ of a normal distribution is a function of the sample variance alone. In this paper, we construct confidence intervals for $\sigma^2$ 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.

This paper investigates the use of Edgeworth expansions for approximating the distribution function of the normalized sum of $n$ independent and identically distributed lattice-valued random variables. We prove that the continuity-corrected Edgeworth series, using Sheppard-adjusted cumulants, is accurate to the same order in $n$ as the usual Edgeworth approximation for continuous random variables. Finally, as a partial justification of the Sheppard adjustments, it is shown that if a continuous random variable $Y$ is rounded into a discrete part $D$ and a truncation error $U$, such that $Y = D + U$, then under suitable limiting conditions the truncation error is approximately uniformly distributed and independent of $Y$, but not independent of $D$.