Modern scientific technology has provided a new class of large-scale simultaneous inference problems, with thousands of hypothesis tests to consider at the same time. Microarrays epitomize this type of technology, but similar situations arise in proteomics, spectroscopy, imaging, and social science surveys. This paper uses false discovery rate methods to carry out both size and power calculations on large-scale problems. A simple empirical Bayes approach allows the false discovery rate (fdr) analysis to proceed with a minimum of frequentist or Bayesian modeling assumptions. Closed-form accuracy formulas are derived for estimated false discovery rates, and used to compare different methodologies: local or tail-area fdr’s, theoretical, permutation, or empirical null hypothesis estimates. Two microarray data sets as well as simulations are used to evaluate the methodology, the power diagnostics showing why nonnull cases might easily fail to appear on a list of “significant” discoveries.

Consider the problem of testing $s$ hypotheses simultaneously. The usual approach restricts attention to procedures that control the probability of even one false rejection, the familywise error rate (FWER). If $s$ is large, one might be willing to tolerate more than one false rejection, thereby increasing the ability of the procedure to correctly reject false null hypotheses. One possibility is to replace control of the FWER by control of the probability of $k$ or more false rejections, which is called the $k$-FWER. We derive both single-step and step-down procedures that control the $k$-FWER in finite samples or asymptotically, depending on the situation. We also consider the false discovery proportion (FDP) defined as the number of false rejections divided by the total number of rejections (and defined to be 0 if there are no rejections). The false discovery rate proposed by Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289–300] controls $E$(FDP). Here, the goal is to construct methods which satisfy, for a given $γ$ and $α$, $P\{\mathrm{FDP} \gt γ\} ≤ α$, at least asymptotically. In contrast to the proposals of Lehmann and Romano [Ann. Statist. 33 (2005) 1138–1154], we construct methods that implicitly take into account the dependence structure of the individual test statistics in order to further increase the ability to detect false null hypotheses. This feature is also shared by related work of van der Laan, Dudoit and Pollard [Stat. Appl. Genet. Mol. Biol. 3 (2004) article 15], but our methodology is quite different. Like the work of Pollard and van der Laan [Proc. 2003 International Multi-Conference in Computer Science and Engineering, METMBS’03 Conference (2003) 3–9] and Dudoit, van der Laan and Pollard [Stat. Appl. Genet. Mol. Biol. 3 (2004) article 13], we employ resampling methods to achieve our goals. Some simulations compare finite sample performance to currently available methods.

The False Discovery Rate (FDR) paradigm aims to attain certain control on Type I errors with relatively high power for multiple hypothesis testing. The Benjamini–Hochberg (BH) procedure is a well-known FDR controlling procedure. Under a random effects model, we show that, in general, unlike the FDR, the positive FDR (pFDR) of the BH procedure cannot be controlled at an arbitrarily low level due to the limited evidence provided by the observations to separate false and true nulls. This results in a criticality phenomenon, which is characterized by a transition of the procedure’s power from being positive to asymptotically 0 without any reduction in the pFDR, once the target FDR control level is below a positive critical value. To address the constraints on the power and pFDR control imposed by the criticality phenomenon, we propose a procedure which applies BH-type procedures at multiple locations in the domain of $p$-values. Both analysis and simulations show that the proposed procedure can attain substantially improved power and pFDR control.

Some effort has been undertaken over the last decade to provide conditions for the control of the false discovery rate by the linear step-up procedure (LSU) for testing $n$ hypotheses when test statistics are dependent. In this paper we investigate the expected error rate (EER) and the false discovery rate (FDR) in some extreme parameter configurations when $n$ tends to infinity for test statistics being exchangeable under null hypotheses. All results are derived in terms of $p$-values. In a general setup we present a series of results concerning the interrelation of Simes’ rejection curve and the (limiting) empirical distribution function of the $p$-values. Main objects under investigation are largest (limiting) crossing points between these functions, which play a key role in deriving explicit formulas for EER and FDR. As specific examples we investigate equi-correlated normal and $t$-variables in more detail and compute the limiting EER and FDR theoretically and numerically. A surprising limit behavior occurs if these models tend to independence.

The “large $p$, small $n$” paradigm arises in microarray studies, image analysis, high throughput molecular screening, astronomy, and in many other high dimensional applications. False discovery rate (FDR) methods are useful for resolving the accompanying multiple testing problems. In cDNA microarray studies, for example, $p$-values may be computed for each of $p$ genes using data from $n$ arrays, where typically $p$ is in the thousands and $n$ is less than 30. For FDR methods to be valid in identifying differentially expressed genes, the $p$-values for the nondifferentially expressed genes must simultaneously have uniform distributions marginally. While feasible for permutation $p$-values, this uniformity is problematic for asymptotic based $p$-values since the number of $p$-values involved goes to infinity and intuition suggests that at least some of the $p$-values should behave erratically. We examine this neglected issue when $n$ is moderately large but $p$ is almost exponentially large relative to $n$. We show the somewhat surprising result that, under very general dependence structures and for both mean and median tests, the $p$-values are simultaneously valid. A small simulation study and data analysis are used for illustration.

Bayesian variable selection has gained much empirical success recently in a variety of applications when the number K of explanatory variables $(x_1,\ldots,x_K)$ is possibly much larger than the sample size $n$. For generalized linear models, if most of the $x_j$’s have very small effects on the response $y$, we show that it is possible to use Bayesian variable selection to reduce overfitting caused by the curse of dimensionality $K\gg n$. In this approach a suitable prior can be used to choose a few out of the many $x_j$’s to model $y$, so that the posterior will propose probability densities $p$ that are “often close” to the true density $p^*$ in some sense. The closeness can be described by a Hellinger distance between $p$ and $p^*$ that scales at a power very close to $n^{-1/2}$, which is the “finite-dimensional rate” corresponding to a low-dimensional situation. These findings extend some recent work of Jiang [Technical Report 05-02 (2005) Dept. Statistics, Northwestern Univ.] on consistency of Bayesian variable selection for binary classification.

The removal of blur from a signal, in the presence of noise, is readily accomplished if the blur can be described in precise mathematical terms. However, there is growing interest in problems where the extent of blur is known only approximately, for example in terms of a blur function which depends on unknown parameters that must be computed from data. More challenging still is the case where no parametric assumptions are made about the blur function. There has been a limited amount of work in this setting, but it invariably relies on iterative methods, sometimes under assumptions that are mathematically convenient but physically unrealistic (e.g., that the operator defined by the blur function has an integrable inverse). In this paper we suggest a direct, noniterative approach to nonparametric, blind restoration of a signal. Our method is based on a new, ridge-based method for deconvolution, and requires only mild restrictions on the blur function. We show that the convergence rate of the method is close to optimal, from some viewpoints, and demonstrate its practical performance by applying it to real images.

Kernel methods for deconvolution have attractive features, and prevail in the literature. However, they have disadvantages, which include the fact that they are usually suitable only for cases where the error distribution is infinitely supported and its characteristic function does not ever vanish. Even in these settings, optimal convergence rates are achieved by kernel estimators only when the kernel is chosen to adapt to the unknown smoothness of the target distribution. In this paper we suggest alternative ridge methods, not involving kernels in any way. We show that ridge methods (a) do not require the assumption that the error-distribution characteristic function is nonvanishing; (b) adapt themselves remarkably well to the smoothness of the target density, with the result that the degree of smoothness does not need to be directly estimated; and (c) give optimal convergence rates in a broad range of settings.

We extend the isotonic analysis for Wicksell’s problem to estimate a regression function, which is motivated by the problem of estimating dark matter distribution in astronomy. The main result is a version of the Kiefer–Wolfowitz theorem comparing the empirical distribution to its least concave majorant, but with a convergence rate $n^{-1}\log n$ faster than $n^{-2/3}\log n$. The main result is useful in obtaining asymptotic distributions for estimators, such as isotonic and smooth estimators.

Let $v$ be a vector field in a bounded open set $G\subset {\mathbb{R}}^d$. Suppose that $v$ is observed with a random noise at random points $X_i,\ i=1,\dots, n$, that are independent and uniformly distributed in $G$. The problem is to estimate the integral curve of the differential equation \[\frac{dx(t)}{dt}=v(x(t)),\qquad t\geq 0,x(0)=x_{0}\in G,\] starting at a given point $x(0)=x_0\in G$ and to develop statistical tests for the hypothesis that the integral curve reaches a specified set $\Gamma\subset G$. We develop an estimation procedure based on a Nadaraya–Watson type kernel regression estimator, show the asymptotic normality of the estimated integral curve and derive differential and integral equations for the mean and covariance function of the limit Gaussian process. This provides a method of tracking not only the integral curve, but also the covariance matrix of its estimate. We also study the asymptotic distribution of the squared minimal distance from the integral curve to a smooth enough surface $\Gamma\subset G$. Building upon this, we develop testing procedures for the hypothesis that the integral curve reaches $\Gamma$.

The problems of this nature are of interest in diffusion tensor imaging, a brain imaging technique based on measuring the diffusion tensor at discrete locations in the cerebral white matter, where the diffusion of water molecules is typically anisotropic. The diffusion tensor data is used to estimate the dominant orientations of the diffusion and to track white matter fibers from the initial location following these orientations. Our approach brings more rigorous statistical tools to the analysis of this problem providing, in particular, hypothesis testing procedures that might be useful in the study of axonal connectivity of the white matter.

In longitudinal and spatial studies, observations often demonstrate strong correlations that are stationary in time or distance lags, and the times or locations of these data being sampled may not be homogeneous. We propose a nonparametric estimator of the correlation function in such data, using kernel methods. We develop a pointwise asymptotic normal distribution for the proposed estimator, when the number of subjects is fixed and the number of vectors or functions within each subject goes to infinity. Based on the asymptotic theory, we propose a weighted block bootstrapping method for making inferences about the correlation function, where the weights account for the inhomogeneity of the distribution of the times or locations. The method is applied to a data set from a colon carcinogenesis study, in which colonic crypts were sampled from a piece of colon segment from each of the 12 rats in the experiment and the expression level of p27, an important cell cycle protein, was then measured for each cell within the sampled crypts. A simulation study is also provided to illustrate the numerical performance of the proposed method.

Asymptotic equivalence results for nonparametric regression experiments have always assumed that the variances of the observations are known. In practice, however the variance of each observation is generally considered to be an unknown nuisance parameter. We establish an asymptotic approximation to the nonparametric regression experiment when the value of the variance is an additional parameter to be estimated or tested. This asymptotically equivalent experiment has two components: the first contains all the information about the variance and the second has all the information about the mean. The result can be extended to regression problems where the variance varies slowly from observation to observation.

This paper studies statistical aggregation procedures in the regression setting. A motivating factor is the existence of many different methods of estimation, leading to possibly competing estimators. We consider here three different types of aggregation: model selection (MS) aggregation, convex (C) aggregation and linear (L) aggregation. The objective of (MS) is to select the optimal single estimator from the list; that of (C) is to select the optimal convex combination of the given estimators; and that of (L) is to select the optimal linear combination of the given estimators. We are interested in evaluating the rates of convergence of the excess risks of the estimators obtained by these procedures. Our approach is motivated by recently published minimax results [Nemirovski, A. (2000). Topics in non-parametric statistics. Lectures on Probability Theory and Statistics (Saint-Flour, 1998). Lecture Notes in Math. 1738 85–277. Springer, Berlin; Tsybakov, A. B. (2003). Optimal rates of aggregation. Learning Theory and Kernel Machines. Lecture Notes in Artificial Intelligence 2777 303–313. Springer, Heidelberg]. There exist competing aggregation procedures achieving optimal convergence rates for each of the (MS), (C) and (L) cases separately. Since these procedures are not directly comparable with each other, we suggest an alternative solution. We prove that all three optimal rates, as well as those for the newly introduced (S) aggregation (subset selection), are nearly achieved via a single “universal” aggregation procedure. The procedure consists of mixing the initial estimators with weights obtained by penalized least squares. Two different penalties are considered: one of them is of the BIC type, the second one is a data-dependent $\ell_1$-type penalty.

We consider the problem of adaptation to the margin and to complexity in binary classification. We suggest an exponential weighting aggregation scheme. We use this aggregation procedure to construct classifiers which adapt automatically to margin and complexity. Two main examples are worked out in which adaptivity is achieved in frameworks proposed by Steinwart and Scovel [Learning Theory. Lecture Notes in Comput. Sci. 3559 (2005) 279–294. Springer, Berlin; Ann. Statist. 35 (2007) 575–607] and Tsybakov [Ann. Statist. 32 (2004) 135–166]. Adaptive schemes, like ERM or penalized ERM, usually involve a minimization step. This is not the case for our procedure.

Consider the following multi-phase project management problem. Each project is divided into several phases. All projects enter the next phase at the same point chosen by the decision maker based on observations up to that point. Within each phase, one can pursue the projects in any order. When pursuing the project with one unit of resource, the project state changes according to a Markov chain. The probability distribution of the Markov chain is known up to an unknown parameter. When pursued, the project generates a random reward depending on the phase and the state of the project and the unknown parameter. The decision maker faces two problems: (a) how to allocate resources to projects within each phase, and (b) when to enter the next phase, so that the total expected reward is as large as possible. In this paper we formulate the preceding problem as a stochastic scheduling problem and propose asymptotic optimal strategies, which minimize the shortfall from perfect information payoff. Concrete examples are given to illustrate our method.

A stochastic algorithm for the recursive approximation of the location $θ$ of a maximum of a regression function was introduced by Kiefer and Wolfowitz [Ann. Math. Statist. 23 (1952) 462–466] in the univariate framework, and by Blum [Ann. Math. Statist. 25 (1954) 737–744] in the multivariate case. The aim of this paper is to provide a companion algorithm to the Kiefer–Wolfowitz–Blum algorithm, which allows one to simultaneously recursively approximate the size $μ$ of the maximum of the regression function. A precise study of the joint weak convergence rate of both algorithms is given; it turns out that, unlike the location of the maximum, the size of the maximum can be approximated by an algorithm which converges at the parametric rate. Moreover, averaging leads to an asymptotically efficient algorithm for the approximation of the couple $(\theta,\mu)$.

We consider asymptotic problems in spectral analysis of stationary causal processes. Limiting distributions of periodograms and smoothed periodogram spectral density estimates are obtained and applications to the spectral domain bootstrap are given. Instead of the commonly used strong mixing conditions, in our asymptotic spectral theory we impose conditions only involving (conditional) moments, which are easily verifiable for a variety of nonlinear time series.

Let $( X_{i}) _{i=1,\ldots,n}$ be a possibly nonstationary sequence such that $\mathscr{L} (X_{i})=P_{n}$ if $i\leq n\theta $ and $\mathscr{L}(X_{i})=Q_{n}$ if $i \gt n\theta $, where $0 \lt \theta \lt 1$ is the location of the change-point to be estimated. We construct a class of estimators based on the empirical measures and a seminorm on the space of measures defined through a family of functions $\mathcal{F}$. We prove the consistency of the estimator and give rates of convergence under very general conditions. In particular, the $1/n$ rate is achieved for a wide class of processes including long-range dependent sequences and even nonstationary ones. The approach unifies, generalizes and improves on the existing results for both parametric and nonparametric change-point estimation, applied to independent, short-range dependent and as well long-range dependent sequences.

A general rate estimation method is proposed that is based on studying the in-sample evolution of appropriately chosen diverging/converging statistics. The proposed rate estimators are based on simple least squares arguments, and are shown to be accurate in a very general setting without requiring the choice of a tuning parameter. The notion of scanning is introduced with the purpose of extracting useful subsamples of the data series; the proposed rate estimation method is applied to different scans, and the resulting estimators are then combined to improve accuracy. Applications to heavy tail index estimation as well as to the problem of estimating the long memory parameter are discussed; a small simulation study complements our theoretical results.