The nearly nonstationary first-order autoregression is a sequence of autoregressive processes $y_n(k + 1) = \phi_ny_n(k) + \varepsilon(k + 1), 0 \leq k \leq n$, where the $\varepsilon(k)$'s are iid mean zero shocks and the autoregressive coefficient $\phi_n = 1 - \beta/n$ for some $\beta > 0$, so that $\phi_n \rightarrow 1$ as $n \rightarrow \infty$. We consider a class of maximum likelihood type estimators called $M$ estimators, which are not necessarily robust. The estimates are obtained as the solution $\hat{\phi}_n$ of an equation of the form $\sum^{n - 1}_{k = 0}y_n(k)\psi(y_n(k + 1) - \phi y_n(k)) = 0,$ where $\psi$ is a given "score" function. Assuming the shocks have $2 + \delta$ moments and that $\psi$ satisfies some regularity conditions, it is shown that the limiting distribution of $n(\hat{\phi}_n - \phi_n)$ is given by the ratio of two stochastic integrals. For a given shock density $f$ satisfying regularity conditions, it is shown that the optimal $\psi$ function for minimizing asymptotic mean squared error is not the maximum likelihood score in general, but a linear combination of the maximum likelihood score and least squares score. However, numerical calculations under the constraint $y_n(0) = 0$ show that the maximum likelihood score has asymptotic efficiency no lower than 40${\tt\%}$.

An asymptotic analysis is presented for estimation in the three-parameter first-order autoregressive model, where the parameters are the mean, autoregressive coefficient and variance of the shocks. The nearly nonstationary asymptotic model is considered wherein the autoregressive coefficient tends to 1 as sample size tends to $\infty$. Three different estimators are considered: the exact Gaussian maximum likelihood estimator, the conditional maximum likelihood or least squares estimator and some "naive" estimators. It is shown that the estimators converge in distribution to analogous estimators for a continuous-time Ornstein-Uhlenbeck process. Simulation results show that the MLE has smaller asymptotic mean squared error then the other two, and that the conditional maximum likelihood estimator gives a very poor estimator of the process mean.

Least squares estimators of the parameters of a linear time series model, where the parameters are constrained by a set of nonlinear restrictions, are studied. The model may contain lags of the dependent variable as regressors and the sums of squares of the explanatory variables may grow at different rates as the sample size increases. The estimation procedures can be applied to a regression model with an error process that satisfies either a stationary or a nonstationary autoregression.

Given a realization of $T$ consecutive observations of a stationary autoregressive process of unknown, possibly infinite, order $m$, it is assumed that a process of arbitrary finite order $p$ is fitted by least squares. Under appropriate conditions it is known that the estimators of the autoregressive coefficients are asymptotically normal. The question considered here is whether the moments of the (scaled) estimators converge, as $T \rightarrow \infty$, to the moments of their asymptotic distribution. We establish a general result for stationary processes (valid, in particular, in the Gaussian case) which is sufficient to imply this convergence.

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.

Certain orthogonal polynomials are employed to estimate the prior distribution of the parameter of natural exponential families with quadratic variance functions in an approach which combines Bayesian and nonparametric empirical Bayesian methods. These estimates are based on samples from the marginal distribution rather than the conditional distribution.

If $X \sim P_\theta, \theta \in \Omega$ and $\theta \sim G \ll \mu$, where $dG/d\mu$ belongs to the convex family $\Gamma_{L, U} = \{g: L \leq \operatorname{cg} \leq U$, for some $c > 0\}$, then the sets minimizing $\lambda(S)$ subject to $\inf_{G \in \Gamma_{L,U}} P_G(S\mid X) \geq p$ are derived, where $P_G(S\mid X)$ is the posterior probability of $S$ under the prior $G$, and $\lambda$ is any nonnegative measure on $\Omega$ such that $\mu \ll \lambda \ll \mu$. Applications are shown to several multiparameter problems and connectedness (or disconnectedness) of these sets is considered. The problem of minimizing the diameter is also considered in a general probabilistic framework. It is proved that if $\mathscr{X}$ is any finite-dimensional Banach space with a convex norm, and $\{P_\alpha\}$ is a tight family of probability measures on the Borel $\sigma$-algebra of $\mathscr{X}$, then there always exists a closed connected set minimizing the diameter under the restriction $\inf_\alpha P_\alpha(S) \geq p$. It is also proved that if $P$ is a spherical unimodal measure on $\mathbb{R}^m$, then volume (Lebesgue measure) and diameter minimizing sets are the same. A result of Borell is then used to conclude that diameter minimizing sets are spheres whenever the underlying distribution $P$ is symmetric absolutely continuous and the density $f$ is such that $f^{-1/m}$ is convex. All standard symmetric multivariate densities satisfy this condition. Applications are made to several Bayes and classical problems and admissibility implications of these results are discussed.

We address the problem of finding a design that minimizes the Bayes risk with respect to a fixed prior subject to being robust with respect to misspecification of the prior. Uncertainty in the prior is formulated in terms of having a family of priors instead of one single prior. Two different classes of priors are considered: $\Gamma_1$ is a family of conjugate priors, and a second family of priors $\Gamma_2$ is induced by a metric on the space of nonnegative measures. The family $\Gamma_1$ has earlier been suggested by Leamer and Polasek, while $\Gamma_2$ was considered by DeRobertis and Hartigan and Berger. The setup assumed is that of a canonical normal linear model with independent homoscedastic errors. Optimal robust designs are considered for the problem of estimating the vector of regression coefficients or a linear combination of the regression coefficients and also for testing and set estimation problems. Concrete examples are given for polynomial regression and completely randomized designs. A very surprising finding is that for $\Gamma_2$, the same design is optimal for a variety of different problems with different loss structures. In general, the results for $\Gamma_2$ are significantly more substantive. Our results are applicable to group decision making and reconciliation of opinions among experts with different priors.

Deconvolution problems arise in a variety of situations in statistics. An interesting problem is to estimate the density $f$ of a random variable $X$ based on $n$ i.i.d. observations from $Y = X + \varepsilon$, where $\varepsilon$ is a measurement error with a known distribution. In this paper, the effect of errors in variables of nonparametric deconvolution is examined. Insights are gained by showing that the difficulty of deconvolution depends on the smoothness of error distributions: the smoother, the harder. In fact, there are two types of optimal rates of convergence according to whether the error distribution is ordinary smooth or supersmooth. It is shown that optimal rates of convergence can be achieved by deconvolution kernel density estimators.

We discuss the difficulties of estimating quadratic functionals based on observations $Y(t)$ from the white noise model $Y(t) = \int^t_0 f(u) du + \sigma W(t),\quad t \in \lbrack 0, 1\rbrack,$ where $W(t)$ is a standard Wiener process on $\lbrack 0, 1\rbrack$. The optimal rates of convergence (as $\sigma \rightarrow 0$) for estimating quadratic functionals under certain geometric constraints are found. Specifically, the optimal rates of estimating $\int^1_0\lbrack f^{(k)}(x)\rbrack^2 dx$ under hyperrectangular constraints $\sum = \{f: |x_j(f)| \leq Cj^{-\alpha}\}$ and weighted $l_p$-body constraints $\sum_p = \{f: \sum^\infty_1 j^r|x_j(f)|^p \leq C\}$ are computed explicitly, where $x_j(f)$ is the $j$th Fourier-Bessel coefficient of the unknown function $f$. We develop lower bounds based on testing two highly composite hypercubes and address their advantages. The attainable lower bounds are found by applying the hardest one-dimensional approach as well as the hypercube method. We demonstrate that for estimating regular quadratic functionals [i.e., the functionals which can be estimated at rate $O(\sigma^2)$], the difficulties of the estimation are captured by the hardest one-dimensional subproblems, and for estimating nonregular quadratic functionals [i.e., no $O(\sigma^2)$-consistent estimator exists], the difficulties are captured at certain finite-dimensional (the dimension goes to infinity as $\sigma \rightarrow 0$) hypercube subproblems.

Many important statistics can be written as functions of sample means of vector variables. A fundamental contribution to the Edgeworth expansion for functions of sample means was made by Bhattacharya and Ghosh. In their work the crucial Cramer $c$-condition is assumed on the joint distribution of all the components of the vector variable. However, in many practical situations, only one or a few of the components satisfy (conditionally) this condition while the rest do not (such a case is referred to as satisfying the partial Cramer $c$-condition). The purpose of this paper is to establish Edgeworth expansions for functions of sample means when only the partial Cramer $c$-condition is satisfied.

Suppose that $P$ is the distribution of a pair of random variables $(X, Y)$ on a product space $\mathbb{X} \times \mathbb{Y}$ with known marginal distributions $P_X$ and $P_Y$. We study efficient estimation of functions $\theta(h) = \int h dP$ for fixed $h: \mathbb{X} \times \mathbb{Y} \rightarrow R$ under iid sampling of $(X, Y)$ pairs from $P$ and a regularity condition on $P$. Our proposed estimator is based on partitions of both $\mathbb{X}$ and $\mathbb{Y}$ and the modified minimum chi-square estimates of Deming and Stephan (1940). The asymptotic behavior of our estimator is governed by the projection on a certain sum subspace of $L_2(P)$, or equivalently by a pair of equations which we call the "ACE equations."

Probability density functions are estimated by the method of maximum likelihood in sequences of regular exponential families. This method is also familiar as entropy maximization subject to empirical constraints. The approximating families of log-densities that we consider are polynomials, splines and trigonometric series. Bounds on the relative entropy (Kullback-Leibler distance) between the true density and the estimator are obtained and rates of convergence are established for log-density functions assumed to have square integrable derivatives.

Buckley and James proposed an extension of the classical least squares estimator to the censored regression model. It has been found in some empirical and Monte Carlo studies that their approach provides satisfactory results and seems to be superior to other extensions of the least squares estimator in the literature. To develop a complete asymptotic theory for this approach, we introduce herein a slight modification of the Buckley-James estimator to get around the difficulties caused by the instability at the upper tail of the associated Kaplan-Meier estimate of the underlying error distribution and show that the modified Buckley-James estimator is consistent and asymptotically normal under certain regularity conditions. A simple formula for the asymptotic variance of the modified Buckley-James estimator is also derived and is used to study the asymptotic efficiency of the estimator. Extensions of these results to the multiple regression model are also given.

A general weak convergence theory is developed for time-sequential censored rank statistics in the two-sample problem of comparing time to failure between two treatment groups, such as in the case of a clinical trial in which patients enter serially and, after being randomly allocated to one of two treatments, are followed until they fail or withdraw from the study or until the study is terminated. Applications of the theory to time-sequential tests based on these censored rank statistics are also discussed.

The present paper derives various survival tests and their optimality results for randomly censored lifetime data by an extension of familiar rank tests arguments. The approach is based on local asymptotic normal models which have a natural interpretation in terms of hazard rates. In particular, the description of classical rank tests by hazard rates may be of separate interest. As an application of the new methods, a justification of conditional survival tests is given also under unequal censoring distributions. It turns out that censoring is a nuisance phenomenon which asymptotically drops out. Conditional tests are exact permutation tests which are shown to be equivalent to their unconditional counterparts. They have all kinds of optimality properties and can be recommended for applications at least in those cases when a model with equal censorship cannot be excluded under the null hypothesis.

This paper deals with censored data estimation of a general class of von Mises-type functionals of the survival time distribution $F$. Conditions are given under which an almost sure asymptotic representation holds for the estimator, obtained by applying the same functional to $\hat{F}_n$, the product-limit estimator of Kaplan and Meier.

Consider a sequence $X_1, X_2,\ldots, X_n$ of independent random variables, where $X_1, X_2,\ldots, X_{n\theta}$ have distribution $P,$ and $X_{n\theta + 1}, X_{n\theta + 2},\ldots, X_n$ have distribution $Q$. The change-point $\theta \in (0,1)$ is an unknown parameter to be estimated, and $P$ and $Q$ are two unknown probability distributions. The nonparametric estimators of Darkhovskh and Carlstein are imbedded in a more general framework, where random seminorms are applied to empirical measures for making inference about $\theta$. Carlstein's and Darkhovskh's results about consistency are improved, and the limiting distributions of some particular estimators are derived in various models. Further we propose asymptotically valid confidence regions for the change point $\theta$ by inverting bootstrap tests. As an example this method is applied to the Nile data.

The Kolmogorov distance between the empirical $\operatorname{cdf} F_n$ and its symmetrization $sF_n$ with respect to an adequate estimator of the center of symmetry of $P$ is a natural statistic for testing symmetry. However, its limiting distribution depends on $P$. Using critical values from the symmetrically bootstrapped statistic (where the resampling is made from $sF_n$) produces tests that can be easily implemented and have asymptotically the correct levels as well as good consistency properties. This article deals with the asymptotic theory that justifies this procedure in particular for a test proposed by Schuster and Barker. Because of lack of smoothness (in some cases implying non-Gaussianness of the limiting processes), these tests do not seem to fall into existing general frameworks.

Some relationships have been established between unbiased linear predictors of processes, in signal and noise models, minimizing the predictive mean square error and some smoothing spline functions. We construct a new family of multidimensional splines adapted to the prediction of locally homogeneous random fields, whose "$m$-spectral measure" (to be defined) is absolutely continuous with respect to Lebesgue measure and satisfies some minor assumptions. By considering partial splines, one may include an arbitrary drift in the signal. This type of correspondence underlines the potentialities of cross-fertilization between statistics and the numerical techniques in approximation theory.

The problem of bandwidth selection for nonparametric kernel regression is considered. It is well recognized that the classical bandwidth selectors are subject to large sample variation. Due to the large variation, these selectors might not be very useful in practice. Based on the frequency-domain representation of the residual sum of squares (RSS), the source of the variation is pointed out. The observation leads to consideration of a procedure which stabilizes the RSS by modifying the periodogram of the observations. The stabilized bandwidth selectors are obtained by substituting the stabilized RSS for the RSS in the classical selectors. The strong consistency of the stabilized bandwidth estimate is established. For sufficiently smooth regression functions, it is shown that the stabilized bandwidth is asymptotically normal, and the relative convergence rate of the stabilized bandwidth estimate is $T^{-1/2}$ instead of the rate $T^{-1/10}$ of the classical estimates. In a simulation study, it is confirmed that the stabilized selectors perform much better than the classical selectors. The simulation results are consistent with the theoretic results. The article contains the important message that the commonly used cross-validation can be improved substantially. The procedure and the theoretic results are developed for a rather restrictive case. Further studies are required for more general situations.

Let $X_1, X_2,\ldots$ be possibly dependent random variables having the same marginal distribution. Consider the situation where $\bar{F}(x) := P\lbrack X_1 > x\rbrack$ is regularly varying at $\infty$ with an unknown index $- \alpha < 0$ which is to be estimated. In the i.i.d. setting, it is well known that Hill's estimator is consistent for $\alpha^{-1}$, and is asymptotically normally distributed. It is the purpose of this paper to demonstrate that such properties of Hill's estimator extend considerably beyond the independent setting. In addition to some basic results derived under very general conditions, the case where the observations are strictly stationary and satisfy a certain mixing condition is considered in detail. Also a finite moving average sequence is studied to illustrate the results.

Amari's dual geometries are used to study measures of influence in exponential family regression. The dual geometries are presented as a natural extension of the Euclidean geometry used for the normal regression model. These geometries are then used to extend Cook's distance to generalized linear models and exponential family regression. Some of these extensions lead to measures already considered while other extensions lead to new measures of influence. The advantages of one of these new measures are discussed.

Some general results on the existence of trend-free and partially trend-free designs are given for both varietal and factorial experiments. In particular, trend-free properties of cyclic and generalized cyclic designs are investigated. It is shown that, for factorial experiments, certain designs which are not completely trend-free are nevertheless trend-free for estimating a subset of the main effect and interaction contrasts.

A total of $nb$ judges rank $t$ objects $k$ at a time according to $n$ replications of a BIBD with $b$ blocks. The Durbin statistic is commonly used in this context and is equivalent to the usual analysis of variance on the rankings. The approach considered here is to introduce the notion of compatibility so as to define distances between incomplete rankings based on metrics on the space of complete rankings. Through this device we define a class of test statistics which includes the Durbin statistic as a special case, and derive their asymptotic distributions. This analysis also yields a new interpretation of the Durbin statistic.

An explicit formula is derived to compute the $A$-optimal design weights on linearly independent regression vectors, for the mean parameters in a linear model with homoscedastic variances. The formula emerges as a special case of a general result which holds for a wide class of optimality criteria. There are close links to iterative algorithms for computing optimal weights.

Let $\mathscr{F}$ denote the set of cdf's on $\mathbb{R}$ with density everywhere positive. Let $C_A = \{(F,G) \in \mathscr{F} \times \mathscr{F}$: there exists a unique $x^\ast \in \mathbb{R}$ such that $F(x) > G(x) \text{for} x << x^\ast \text{and} F(x) < G(x) \text{for} x > x^\ast\}, C_B - \{(F,G) \in \mathscr{F} \times \mathscr{F}: (G,F) \in C_A\}$. Based on independent random samples from $F$ and $G$ (assumed unknown), we give distribution-free tests of $H_0: F = G$ versus the alternatives that $(F,G) \in C_A, (F,G) \in C_B \text{or} (F,G) \in C_A \cup C_B$. Next, assuming that $(F,G) \in C_A$ (or in $C_B$), a point estimate of the crossing point $x^\ast$ is obtained and is shown to be strongly consistent and asymptotically normal. Finally, an asymptotically distribution-free confidence interval for $x^\ast$ is obtained. All inferences are based on a special criterion functional of $F$ and $G$, which yields $x^\ast$ when maximized (minimized) if $(F,G) \in C_A \lbrack(F,G) \in C_B\rbrack$.

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.

This paper shows that Cox's partial score function is the projection of the score function on the (locally) $E$-ancillary subspace for the nuisance parameter (Small and McLeish). This is done by adapting the concepts of (locally) $E$-ancillarity and (locally) $E$-sufficiency for inference functions (McLeish and Small) to an extended Cox's regression model, where the baseline function is allowed to be a predictable process.

Plane-fitting, for example, linear regression, principal components or projection pursuit, is treated from a general perspective. It is shown that any method of plane-fitting satisfying very mild hypotheses must have singularities, that is, data sets near which the procedure is unstable. The well-known collinearity phenomenon in least squares regression is a special case. Severity of singularities is also discussed. The results, which are applications of algebraic topology, may be viewed as putting limits on how much can be done through robustification to stabilize plane-fitting.

Let $\mathscr{D}_{\upsilon,b,k}$ be the set of all the binary equireplicate incomplete-block designs for $\upsilon$ treatments in $b$ blocks of size $k$. It is shown that if $\mathscr{D}_{\upsilon,b,k}$ contains a connected two-associate-class partially balanced design $d^\ast$ with $\lambda_2 = \lambda_1 \pm 1$ which has a singular concurrence matrix, then it is optimal over $\mathscr{D}_{\upsilon,b,k}$ with respect to a large class of criteria including the $A,D$ and $E$ criteria. The dual of $d^\ast$ is also optimal over $\mathscr{D}_{b,\upsilon,r}$ with respect to the same criteria, where $r = bk/\upsilon$. The result can be applied to many designs which were not previously known to be optimal. In another application, Bailey's (1988) conjecture on the optimality of Trojan squares over semi-Latin squares is confirmed.

The effect of a linear filter with monotone gain on the first-order autocorrelation of a weakly stationary time series is discussed. When the gain is monotone increasing, the first-order autocorrelation cannot increase. Otherwise, when the gain is monotone decreasing, the correlation cannot decrease. Further, when the gain is strictly monotone, the first-order autocorrelation is unchanged if and only if the process is a pure sinusoid with probability 1. Under the Gaussian assumption, the zero-crossing rate moves oppositely from the first-order autocorrelation.

Let $\{X_n, n \geq 1\}$ be a sequence of random variables and let $P_\theta$ be a probability measure under which $(X_1, \ldots, X_n)$ have joint pdf's $f_n(X_1, \ldots, X_n, \theta) = L_n(\theta), n \geq 1$. Suppose $u_n = u_n(X_1, \ldots, X_n), n \geq 1$, are statistics such that $(u_n - c)(L_n(\theta') - L_n(\theta)) \geq 0, \forall (X_1, \ldots, X_n), n \geq 1$, for some constant $c = c(\theta,\theta'), \theta \neq \theta'$. For any increasing function $\psi$ and stopping time $T$, it is shown that $E_\theta\phi(u_T) \leq E_{\theta'}\phi(u_T)$, provided that one of the expectations is finite and $P_\theta(T < \infty) = P_{\theta'}(T < \infty) = 1$. The given result holds for a certain monotone likelihood ratio family and an exponential family in particular. This generalizes a result of Chow and Studden and provides a sequential version of a result of Lehmann.