We introduce a new sampling algorithm, the equi-energy sampler, for efficient statistical sampling and estimation. Complementary to the widely used temperature-domain methods, the equi-energy sampler, utilizing the temperature–energy duality, targets the energy directly. The focus on the energy function not only facilitates efficient sampling, but also provides a powerful means for statistical estimation, for example, the calculation of the density of states and microcanonical averages in statistical mechanics. The equi-energy sampler is applied to a variety of problems, including exponential regression in statistics, motif sampling in computational biology and protein folding in biophysics.

We congratulate Samuel Kou, Qing Zhou and Wing Wong (referred to subsequently as KZW) for this beautifully written paper, which opens a new direction in Monte Carlo computation. This discussion has two parts. First, we describe a very closely related method, multicanonical sampling (MCS), and report a simulation example that compares the equi-energy (EE) sampler with MCS. Overall, we found the two algorithms to be of comparable efficiency for the simulation problem considered. In the second part, we develop some additional convergence results for the EE sampler.

Novel sampling algorithms can significantly impact open questions in computational biology, most notably the in silico protein folding problem. By using computational methods, protein folding aims to find the three-dimensional structure of a protein chain given the sequence of its amino acid building blocks. The complexity of the problem strongly depends on the protein representation and its energy function. The more detailed the model, the more complex its corresponding energy function and the more challenge it sets for sampling algorithms. Kou, Zhou and Wong have introduced a novel sampling method, which could contribute significantly to the field of structural prediction.

We study a standard method of regularization by projections of the linear inverse problem Y=Af+ε, where ε is a white Gaussian noise, and A is a known compact operator with singular values converging to zero with polynomial decay. The unknown function f is recovered by a projection method using the singular value decomposition of A. The bandwidth choice of this projection regularization is governed by a data-driven procedure which is based on the principle of risk hull minimization. We provide nonasymptotic upper bounds for the mean square risk of this method and we show, in particular, that in numerical simulations this approach may substantially improve the classical method of unbiased risk estimation.

We study large sample properties of Bayesian analysis of the proportional hazard model with neutral to the right process priors on the baseline hazard function. We show that the posterior distribution of the baseline cumulative hazard function and regression coefficients centered at the maximum likelihood estimator is jointly asymptotically equivalent to the sampling distribution of the maximum likelihood estimator.

It is proved that nonparametric autoregression is asymptotically equivalent in the sense of Le Cam’s deficiency distance to nonparametric regression with random design as well as with regular nonrandom design.

Nested-error regression models are widely used for analyzing clustered data. For example, they are often applied to two-stage sample surveys, and in biology and econometrics. Prediction is usually the main goal of such analyses, and mean-squared prediction error is the main way in which prediction performance is measured. In this paper we suggest a new approach to estimating mean-squared prediction error. We introduce a matched-moment, double-bootstrap algorithm, enabling the notorious underestimation of the naive mean-squared error estimator to be substantially reduced. Our approach does not require specific assumptions about the distributions of errors. Additionally, it is simple and easy to apply. This is achieved through using Monte Carlo simulation to implicitly develop formulae which, in a more conventional approach, would be derived laboriously by mathematical arguments.

This paper introduces a nonparametric algorithm for bootstrapping a stationary random field and proves certain consistency properties of the algorithm for the case of mixing random fields. The motivation for this paper comes from relating a heuristic texture synthesis algorithm popular in computer vision to general nonparametric bootstrapping of stationary random fields. We give a formal resampling scheme for the heuristic texture algorithm and prove that it produces a consistent estimate of the joint distribution of pixels in a window of certain size under mixing and regularity conditions on the random field. The joint distribution of pixels is the quantity of interest here because theories of human perception of texture suggest that two textures with the same joint distribution of pixel values in a suitably chosen window will appear similar to a human. Thus we provide theoretical justification for an algorithm that has already been very successful in practice, and suggest an explanation for its perceptually good results.

In this paper we consider the problem of bootstrapping a class of spatial regression models when the sampling sites are generated by a (possibly nonuniform) stochastic design and are irregularly spaced. It is shown that the natural extension of the existing block bootstrap methods for grid spatial data does not work for irregularly spaced spatial data under nonuniform stochastic designs. A variant of the blocking mechanism is proposed. It is shown that the proposed block bootstrap method provides a valid approximation to the distribution of a class of M-estimators of the spatial regression parameters. Finite sample properties of the method are investigated through a moderately large simulation study and a real data example is given to illustrate the methodology.

In observational studies designed to estimate the effects of interventions or exposures, such as cigarette smoking, it is desirable to try to control background differences between the treated group (e.g., current smokers) and the control group (e.g., never smokers) on covariates X (e.g., age, education). Matched sampling attempts to effect this control by selecting subsets of the treated and control groups with similar distributions of such covariates. This paper examines the consequences of matching using affinely invariant methods when the covariate distributions are “discriminant mixtures of proportional ellipsoidally symmetric” (DMPES) distributions, a class herein defined, which generalizes the ellipsoidal symmetry class of Rubin and Thomas [Ann. Statist. 20 (1992) 1079–1093]. The resulting generalized results help indicate why earlier results hold quite well even when the simple assumption of ellipsoidal symmetry is not met [e.g., Biometrics 52 (1996) 249–264]. Extensions to conditionally affinely invariant matching with conditionally DMPES distributions are also discussed.

We investigate the properties of the Benjamini–Hochberg method for multiple testing and of a variant of Storey’s generalization of it, extending and complementing the asymptotic and exact results available in the literature. Results are obtained under two different sets of assumptions and include asymptotic and exact expressions and bounds for the proportion of rejections, the proportion of incorrect rejections out of all rejections and two other proportions used to quantify the efficacy of the method.

Consider the multiple testing problem of testing null hypotheses H₁,…, H_s. A classical approach to dealing with the multiplicity problem is to restrict attention to procedures that control the familywise error rate (FWER), the probability of even one false rejection. But if s is large, control of the FWER is so stringent that the ability of a procedure that controls the FWER to detect false null hypotheses is limited. It is therefore desirable to consider other measures of error control. This article considers two generalizations of the FWER. The first is the k-FWER, in which one is willing to tolerate k or more false rejections for some fixed k≥1. The second is based on the false discovery proportion (FDP), defined to be the number of false rejections divided by the total number of rejections (and defined to be 0 if there are no rejections). Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289–300] proposed control of the false discovery rate (FDR), by which they meant that, for fixed α, E(FDP)≤α. Here, we consider control of the FDP in the sense that, for fixed γ and α, P{FDP>γ}≤α. Beginning with any nondecreasing sequence of constants and p-values for the individual tests, we derive stepup procedures that control each of these two measures of error control without imposing any assumptions on the dependence structure of the p-values. We use our results to point out a few interesting connections with some closely related stepdown procedures. We then compare and contrast two FDP-controlling procedures obtained using our results with the stepup procedure for control of the FDR of Benjamini and Yekutieli [Ann. Statist. 29 (2001) 1165–1188].

Limit distributions for the greatest convex minorant and its derivative are considered for a general class of stochastic processes including partial sum processes and empirical processes, for independent, weakly dependent and long range dependent data. The results are applied to isotonic regression, isotonic regression after kernel smoothing, estimation of convex regression functions, and estimation of monotone and convex density functions. Various pointwise limit distributions are obtained, and the rate of convergence depends on the self similarity properties and on the rate of convergence of the processes considered.

Itô processes are the most common form of continuous semimartingales, and include diffusion processes. This paper is concerned with the nonparametric regression relationship between two such Itô processes. We are interested in the quadratic variation (integrated volatility) of the residual in this regression, over a unit of time (such as a day). A main conceptual finding is that this quadratic variation can be estimated almost as if the residual process were observed, the difference being that there is also a bias which is of the same asymptotic order as the mixed normal error term.

The proposed methodology, “ANOVA for diffusions and Itô processes,” can be used to measure the statistical quality of a parametric model and, nonparametrically, the appropriateness of a one-regressor model in general. On the other hand, it also helps quantify and characterize the trading (hedging) error in the case of financial applications.

Estimating high quantiles plays an important role in the context of risk management. This involves extrapolation of an unknown distribution function. In this paper we propose three methods, namely, the normal approximation method, the likelihood ratio method and the data tilting method, to construct confidence regions for high quantiles of a heavy tailed distribution. A simulation study prefers the data tilting method.

Consider n i.i.d. random vectors on ℝ², with unknown, common distribution function F. Under a sharpening of the extreme value condition on F, we derive a weighted approximation of the corresponding tail copula process. Then we construct a test to check whether the extreme value condition holds by comparing two estimators of the limiting extreme value distribution, one obtained from the tail copula process and the other obtained by first estimating the spectral measure which is then used as a building block for the limiting extreme value distribution. We derive the limiting distribution of the test statistic from the aforementioned weighted approximation. This limiting distribution contains unknown functional parameters. Therefore, we show that a version with estimated parameters converges weakly to the true limiting distribution. Based on this result, the finite sample properties of our testing procedure are investigated through a simulation study. A real data application is also presented.

Linear regression models are among the models most used in practice, although the practitioners are often not sure whether their assumed linear regression model is at least approximately true. In such situations, only designs for which the linear model can be checked are accepted in practice. For important linear regression models such as polynomial regression, optimal designs do not have this property. To get practically attractive designs, we suggest the following strategy. One part of the design points is used to allow one to carry out a lack of fit test with good power for practically interesting alternatives. The rest of the design points are determined in such a way that the whole design is optimal for inference on the unknown parameter in case the lack of fit test does not reject the linear regression model.

To solve this problem, we introduce efficient lack of fit designs. Then we explicitly determine the e_k-optimal design in the class of efficient lack of fit designs for polynomial regression of degree k−1.

Motivated by studying asymptotic properties of the maximum likelihood estimator (MLE) in stochastic volatility (SV) models, in this paper we investigate likelihood estimation in state space models. We first prove, under some regularity conditions, there is a consistent sequence of roots of the likelihood equation that is asymptotically normal with the inverse of the Fisher information as its variance. With an extra assumption that the likelihood equation has a unique root for each n, then there is a consistent sequence of estimators of the unknown parameters. If, in addition, the supremum of the log likelihood function is integrable, the MLE exists and is strongly consistent. Edgeworth expansion of the approximate solution of likelihood equation is also established. Several examples, including Markov switching models, ARMA models, (G)ARCH models and stochastic volatility (SV) models, are given for illustration.