Given a Markov chain sampling scheme, does the standard empirical estimator make best use of the data? We show that this is not so and construct better estimators. We restrict attention to nearest-neighbor random fields and to Gibbs samplers with deterministic sweep, but our approach applies to any sampler that uses reversible variable-at-a-time updating with deterministic sweep. The structure of the transition distribution of the sampler is exploited to construct further empirical estimators that are combined with the standard empirical estimator to reduce asymptotic variance. The extra computational cost is negligible. When the random field is spatially homogeneous, symmetrizations of our estimator lead to further variance reduction. The performance of the estimators is evaluated in a simulation study of the Ising model.

The coordinatewise median of a multivariate data set is a highly robust location estimator, but it depends on the choice of coordinates. A popular alternative which avoids this drawback is the spatial median, defined as the value that minimizes the sum of distances to the individual data points. In this paper we introduce and discuss another orthogonal equivariant version of the multivariate median, obtained by averaging the coordinatewise median over all orthogonal transformations. We investigate the asymptotic behavior of this estimator and compare it to the spatial median.

In this paper we discuss a new approach to solve calibration problems in a nonparametric setting. This approach is appealing because it yields estimates of the required quantities directly. The method combines kernel and robust estimation techniques. It relies on strong approximations of the estimating process and the extreme value theorem of Bickel and Rosenblatt. Using these results, we first obtain robust pointwise estimates of the parameters of interest. Second, we set up asymptotic simultaneous tolerance regions for many unknown values of the quantity to be calibrated. The technique is illustrated on a radiocarbon dating problem. The nonparametric calibration procedure proves to be of practical, as well as theoretical interest; moreover, it is quick and simple to implement.

This paper describes a method of robust estimation of shift and scale parameters in a mixed unbalanced interlaboratory model. Estimators presented result from "easily computable" Fréchet differentiable functionals which enjoy some optimal properties in a small neighborhood of the model. A rigorous treatment of their asymptotic behaviour under departures from the model assumptions and a simulation study are given.

Principal curves were introduced to formalize the notion of "a curve passing through the middle of a dataset." Vaguely speaking, a curve is said to pass through the middle of a dataset if every point on the curve is the average of the observations projecting onto it. This idea can be made precise by defining principal curves for probability densities. In this paper we study principal curves in the plane. Like linear principal components, principal curves are critical points of the expected squared distance from the data. However, the largest and smallest principal components are extrema of the distance, whereas all principal curves are saddle points. This explains why cross-validation does not appear to be a viable method for choosing the complexity of principal curve estimates.

In a fascinating article on models for particulate microstructure, Hermann and Ohser argue that the superposition of stochastically independent Boolean models may be thought of as producing sets whose boundaries have unusual properties of dimension. Hermann and Ohser employ such superposition models to analyse the particulate structure of rust. In the present paper we provide a theoretical foundation for their work, with respect to both the definition of dimension and its statistical estimation. We argue that the dimension of set boundaries is not well defined in either the Minkowski or capacity senses. It can, however, be properly defined using a mathematical formalization of the practical notion of analysing a spatial pattern at different resolution levels. Adopting this as a basis, we consider properties of estimators of dimension.

The problem of constructing confidence sets for the structural errors-in-variables model is considered under the assumption that the variance of the error associated with the covariate is known. Previously proposed confidence sets for this model suffer from the problem that they all have zero confidence levels for any sample size, where the confidence level of a confidence set is defined to be the infimum of coverage probability over the parameter space. In this paper we construct some asymptotically honest confidence sets; that is, the limiting values of their confidence levels are at least as large as the nominal probabilities when the sample size goes to $\infty$. A desirable property of the proposed confidence set for the slope is also established.

In a general normal regression model, this paper first derives the least upper bound (LUB) for the covariance matrix of a generalized least squares estimator (GLSE) relative to the covariance matrix of the Gauss-Markov estimator. Second the result is applied to the (unrestricted) Zellner estimator in an N-equation seemingly unrelated regression (SUR) model and to the GLSE in a heteroscedastic model.

In this paper we consider the problem of identifiability and estimation for the scale parameter $\theta$ in the location mixture model $\theta (X + Y)$, where X has a known distribution independent of the Y, whose distribution is unknown. Identification of $\theta$ is ensured by constraining Y based on the tail behavior of the distribution for X. Rates for estimation are described for those X which can be written as a square summable series of exponential variables. As a special case, our analysis shows that the structural parameters in the Weibull semiparametric mixture (Heckman and Singer) are not estimable at the usual parametric $O_p(1/ \sqrt{n})$. The exact relationship between identifying constraints and achievable rates is explained.

This paper presents a uniform estimator for a finite-dimensional parameter in the semiparametric Weibull mixture model. The rates achieved by the estimator hold uniformly over shrinking sequences of models much more general than traditional sequences that are required to satisfy a Hellinger differentiable property. We show that these rates are optimal in a class of identified models constrained by a moment condition on the nonparametric mixing distribution.

In semiparametric ANOCOVA (mixed-effects) models, the role of regression rank scores in robust estimation of fixed-effects parameters as well as covariate regression functionals is critically appraised, and the relevant asymptotic theory is presented.

Local likelihood was introduced by Tibshirani and Hastie as a method of smoothing by local polynomials in non-Gaussian regression models. In this paper an extension of these methods to density estimation is discussed, and comparison with other methods of density estimation presented. The local likelihood method has particularly strong advantages over kernel methods when estimating tails of densities and in multivariate settings. Suppose constraints are incorporated in a simple manner. Asymptotic properties of the estimate are discussed. A method for computing the estimate is outlined.

C code to implement the estimation procedure described in this paper, together with S interfaces for graphical display of results, are available at:

http://cm.bell-labs.com/stat/project/locfit/index.html

This paper develops a nonparametric density estimator with parametric overtones. Suppose $f(x, \theta)$ is some family of densities, indexed by a vector of parameters $\theta$. We define a local kernel-smoothed likelihood function which, for each x, can be used to estimate the best local parametric approximant to the true density. This leads to a new density estimator of the form $f(x, \hat{\theta}(x))$, thus inserting the best local parameter estimate for each new value of x. When the bandwidth used is large, this amounts to ordinary full likelihood parametric density estimation, while for moderate and small bandwidths the method is essentially nonparametric, using only local properties of data and the model. Alternative ways more general than via the local likelihood are also described. The methods can be seen as ways of nonparametrically smoothing the parameter within a parametric class.

Properties of this new semiparametric estimator are investigated. Our preferred version has approximately the same variance as the ordinary kernel method but potentially a smaller bias. The new method is seen to perform better than the traditional kernel method in a broad nonparametric vicinity of the parametric model employed, while at the same time being capable of not losing much in precision to full likelihood methods when the model is correct. Other versions of the method are approximately equivalent to using particular higher order kernels in a semiparametric framework. The methodology we develop can be seen as the density estimation parallel to local likelihood and local weighted least squares theory in nonparametric regression.

For d-dimensional images and regression functions the true object is estimated by median smoothing. The mean square error of the median smoother is calculated using the framework of M-estimation, and an expression for the asymptotic rate of convergence of the mean square error is given. It is shown that the median smoother performs asymptotically as well as the local mean. The optimal window size and the bandwidth of the median smoother are given in terms of the sample size and the dimension of the problem. The rate of convergence is found to decrease as the dimension increases, and its functional dependence on the dimension changes when the dimension reaches 4.

We consider a regression model in which the mean function may have a discontinuity at an unknown point. We propose an estimate of the location of the discontinuity based on one-side nonparametric regression estimates of the mean function. The change point estimate is shown to converge in probability at rate $O(n^{-1})$ and to have the same asymptotic distribution as maximum likelihood estimates considered by other authors under parametric regression models. Confidence regions for the location and size of the change are also discussed.

We consider two problems in nonparametric survival analysis under the restriction of stochastic ordering. The first problem is that of estimating a survival function $\overline{F}(t)$ under the restriction $\overline{F}(t) \geq $\overline{F}_0 (t)$, all t, where $\overline{F}_0 (t)$ is known. The second problem consists of estimating two unknown survival functions $\overline{F}^{(1)}(t)$ and $\overline{F}^{(2)}(t)$ when it is known that $\overline{F}^{(1)}(t) \geq \overline{F}^{(2)}(t)$, all t. The nonparametric maximum likelihood estimators in these problems were derived by Brunk, Franck, Hansen and Hogg and Dykstra. In the present paper we derive their large-sample distributions. We present two sets of proofs depending on whether or not the data are right-censored. When centered and scaled by $n^{1/2}$, the estimators converge in distribution to limiting processes related to the concave majorant of Brownian motion. The limiting distributions are not known in closed form, but can be simulated for the purpose of forming asymptotic pointwise confidence limits.

Yang and Hall and Wellner initiated investigations of the asymptotic uniform behaviour of mean residual life (MRL) processes. They obtained results holding true over fixed and expanding compact subintervals of $[0, \infty)$.

In this exposition we study MRL processes over the whole positive half-line $[0, \infty)$. We describe classes of weight functions which enable us to establish the (a) strong uniform-over-$[0, \infty)$ consistency and (b)weak uniform-over-$[0, \infty)$ approximation of MRL processes. We give examples which show the necessity of employing weight functions in order to have (a) and (b), and prove the optimality of the weight function classes which we make use of. Extending our results concerning (b), we discuss constructions of asymptotic confidence bands for unknown MRL functions. The width of the obtained confidence bands is regulated by weight functions depending on the available information on the underlying distribution function.

The paper investigates the structure of the self-consistent estimators (SCE) and the nonparametric maximum likelihood estimator (NPMLE) for doubly censored data. An explicit sufficient and necessary condition for an SCE to be the NPMLE is given. Based on this, algorithms for computing the SCE and the NPMLE are provided. The relation between our algorithms and the EM algorithm is studied.

We consider a sequential procedure for comparing three treatments with the goal of ultimately selecting the best treatment. This procedure starts with a sequential test to detect an overall treatment difference and eliminates the apparently inferior treatment if this test rejects the equality of the treatments. It then proceeds with a sequential test of the remaining two treatments. We base these sequential tests on the stopping boundaries popularized by O'Brien and Fleming. Our procedure is similar in structure to that used by Siegmund in conjunction with modified repeated significance tests. We compare the performances of the two procedures via a simulation experiment. We derive analytic approximations for an error probability, the power and the expected sample size of our procedure, which we compare to simulated values. Furthermore, we propose a modification of the procedure for the comparison of a standard treatment with experimental treatments.

We propose a sequential procedure for estimating with prescribed proportional accuracy one parameter in a class of two-parameter exponential family of distributions. We study the properties of the resulting stopping time and provide second-order analysis of the coverage probability associated with it by using Edgeworth expansion.

We consider, in a Bayesian framework, the model $W_t = B_t + \theta (t - \nu)^+$, where B is a standard Brownian motion, $\theta$ is arbitrary but known and $\nu$ is the unknown change-point. We transfer the construction of Ritov to this continuous time setup and show that the corresponding Bayes problems can be reduced to generalized parking problems.

If $\mu$ is a positive measure on $\mathbb{R}^n$ with Laplace transform $L_{\mu}$ , we show that there exists a positive measure $\mu$ on $\mathbb{R}^n$ such that det $L_{\mu}^'' = L_{\nu}$. We deduce various corollaries from this result and, in particular, we obtain the Rao-Blackwell estimator of the determinant of the variance of a natural exponential family on $\mathbb{R}^n$ based on $(n + 1)$ observations. A new proof and extensions of Lindsay's results on the determinants of moment matrices are also given. Finally we give a characterization of the Gaussian law in $\mathbb{R}^n$.

The present paper describes all the natural exponential families on $\mathbb{R}^d$ whose variance function is of the form $V(m) = am \otimes m + B(m) + C$, with $m \otimes m(\theta) = \langle \theta, m \rangle m$ and B linear in m. There are $2d + 4$ types of such families, which are built from particular mixtures of families of Normal, Poisson, gamma, hyperbolic on $\mathbb{R}^d$ and negative-multinomial distributions. The proof of this result relies mainly on techniques used in the elementary theory of Lie algebras.