In the class of normal regression models with a finite number of regressors, and for a wide class of prior distributions, a Bayesian model selection procedure based on the Bayes factor is consistent [Casella and Moreno J. Amer. Statist. Assoc. 104 (2009) 1261–1271]. However, in models where the number of parameters increases as the sample size increases, properties of the Bayes factor are not totally understood. Here we study consistency of the Bayes factors for nested normal linear models when the number of regressors increases with the sample size. We pay attention to two successful tools for model selection [Schwarz Ann. Statist. 6 (1978) 461–464] approximation to the Bayes factor, and the Bayes factor for intrinsic priors [Berger and Pericchi J. Amer. Statist. Assoc. 91 (1996) 109–122, Moreno, Bertolino and Racugno J. Amer. Statist. Assoc. 93 (1998) 1451–1460].

We find that the the Schwarz approximation and the Bayes factor for intrinsic priors are consistent when the rate of growth of the dimension of the bigger model is O(n^b) for b < 1. When b = 1 the Schwarz approximation is always inconsistent under the alternative while the Bayes factor for intrinsic priors is consistent except for a small set of alternative models which is characterized.

In this paper, we investigate the (in)-consistency of different bootstrap methods for constructing confidence intervals in the class of estimators that converge at rate n^1/3. The Grenander estimator, the nonparametric maximum likelihood estimator of an unknown nonincreasing density function f on [0, ∞), is a prototypical example. We focus on this example and explore different approaches to constructing bootstrap confidence intervals for f(t₀), where t₀ ∈ (0, ∞) is an interior point. We find that the bootstrap estimate, when generating bootstrap samples from the empirical distribution function $\mathbb{F}_{n}$ or its least concave majorant F̃_n, does not have any weak limit in probability. We provide a set of sufficient conditions for the consistency of any bootstrap method in this example and show that bootstrapping from a smoothed version of F̃_n leads to strongly consistent estimators. The m out of n bootstrap method is also shown to be consistent while generating samples from $\mathbb{F}_{n}$ and F̃_n.

This paper develops a theory for group Lasso using a concept called strong group sparsity. Our result shows that group Lasso is superior to standard Lasso for strongly group-sparse signals. This provides a convincing theoretical justification for using group sparse regularization when the underlying group structure is consistent with the data. Moreover, the theory predicts some limitations of the group Lasso formulation that are confirmed by simulation studies.

Most results in nonparametric regression theory are developed only for the case of additive noise. In such a setting many smoothing techniques including wavelet thresholding methods have been developed and shown to be highly adaptive. In this paper we consider nonparametric regression in exponential families with the main focus on the natural exponential families with a quadratic variance function, which include, for example, Poisson regression, binomial regression and gamma regression. We propose a unified approach of using a mean-matching variance stabilizing transformation to turn the relatively complicated problem of nonparametric regression in exponential families into a standard homoscedastic Gaussian regression problem. Then in principle any good nonparametric Gaussian regression procedure can be applied to the transformed data. To illustrate our general methodology, in this paper we use wavelet block thresholding to construct the final estimators of the regression function. The procedures are easily implementable. Both theoretical and numerical properties of the estimators are investigated. The estimators are shown to enjoy a high degree of adaptivity and spatial adaptivity with near-optimal asymptotic performance over a wide range of Besov spaces. The estimators also perform well numerically.

In this paper, the maximal nonlinear conditional correlation of two random vectors X and Y given another random vector Z, denoted by ρ₁(X, Y|Z), is defined as a measure of conditional association, which satisfies certain desirable properties. When Z is continuous, a test for testing the conditional independence of X and Y given Z is constructed based on the estimator of a weighted average of the form ∑_k=1^n_Zf_Z(z_k)ρ₁²(X, Y|Z = z_k), where f_Z is the probability density function of Z and the z_k’s are some points in the range of Z. Under some conditions, it is shown that the test statistic is asymptotically normal under conditional independence, and the test is consistent.

We study the Cox models with semiparametric relative risk, which can be partially linear with one nonparametric component, or multiple additive or nonadditive nonparametric components. A penalized partial likelihood procedure is proposed to simultaneously estimate the parameters and select variables for both the parametric and the nonparametric parts. Two penalties are applied sequentially. The first penalty, governing the smoothness of the multivariate nonlinear covariate effect function, provides a smoothing spline ANOVA framework that is exploited to derive an empirical model selection tool for the nonparametric part. The second penalty, either the smoothly-clipped-absolute-deviation (SCAD) penalty or the adaptive LASSO penalty, achieves variable selection in the parametric part. We show that the resulting estimator of the parametric part possesses the oracle property, and that the estimator of the nonparametric part achieves the optimal rate of convergence. The proposed procedures are shown to work well in simulation experiments, and then applied to a real data example on sexually transmitted diseases.

Covariance matrix plays a central role in multivariate statistical analysis. Significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices. However, a minimax theory has yet been developed. In this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and Frobenius norm. It is shown that optimal procedures under the two norms are different and consequently matrix estimation under the operator norm is fundamentally different from vector estimation. The minimax upper bound is obtained by constructing a special class of tapering estimators and by studying their risk properties. A key step in obtaining the optimal rate of convergence is the derivation of the minimax lower bound. The technical analysis requires new ideas that are quite different from those used in the more conventional function/sequence estimation problems.

Let (X_n,i)_{1≤i≤n,n∈ℕ} be a triangular array of row-wise stationary ℝ^d-valued random variables. We use a “blocks method” to define clusters of extreme values: the rows of (X_n,i) are divided into m_n blocks (Y_n,j), and if a block contains at least one extreme value, the block is considered to contain a cluster. The cluster starts at the first extreme value in the block and ends at the last one. The main results are uniform central limit theorems for empirical processes $Z_{n}(f):=\frac{1}{\sqrt{nv_{n}}}\sum_{j=1}^{m_{n}}(f(Y_{n,j})-Ef(Y_{n,j}))$, for v_n = P{X_n,i ≠ 0} and f belonging to classes of cluster functionals, that is, functions of the blocks Y_n,j which only depend on the cluster values and which are equal to 0 if Y_n,j does not contain a cluster. Conditions for finite-dimensional convergence include β-mixing, suitable Lindeberg conditions and convergence of covariances. To obtain full uniform convergence, we use either “bracketing entropy” or bounds on covering numbers with respect to a random semi-metric. The latter makes it possible to bring the powerful Vapnik–Červonenkis theory to bear. Applications include multivariate tail empirical processes and empirical processes of cluster values and of order statistics in clusters. Although our main field of applications is the analysis of extreme values, the theory can be applied more generally to rare events occurring, for example, in nonparametric curve estimation.

The paper considers nonparametric specification tests of quantile curves for a general class of nonstationary processes. Using Bahadur representation and Gaussian approximation results for nonstationary time series, simultaneous confidence bands and integrated squared difference tests are proposed to test various parametric forms of the quantile curves with asymptotically correct type I error rates. A wild bootstrap procedure is implemented to alleviate the problem of slow convergence of the asymptotic results. In particular, our results can be used to test the trends of extremes of climate variables, an important problem in understanding climate change. Our methodology is applied to the analysis of the maximum speed of tropical cyclone winds. It was found that an inhomogeneous upward trend for cyclone wind speeds is pronounced at high quantile values. However, there is no trend in the mean lifetime-maximum wind speed. This example shows the effectiveness of the quantile regression technique.

Clinical trials are complex and usually involve multiple objectives such as controlling type I error rate, increasing power to detect treatment difference, assigning more patients to better treatment, and more. In literature, both response-adaptive randomization (RAR) procedures (by changing randomization procedure sequentially) and sequential monitoring (by changing analysis procedure sequentially) have been proposed to achieve these objectives to some degree. In this paper, we propose to sequentially monitor response-adaptive randomized clinical trial and study it’s properties. We prove that the sequential test statistics of the new procedure converge to a Brownian motion in distribution. Further, we show that the sequential test statistics asymptotically satisfy the canonical joint distribution defined in Jennison and Turnbull (2000). Therefore, type I error and other objectives can be achieved theoretically by selecting appropriate boundaries. These results open a door to sequentially monitor response-adaptive randomized clinical trials in practice. We can also observe from the simulation studies that, the proposed procedure brings together the advantages of both techniques, in dealing with power, total sample size and total failure numbers, while keeps the type I error. In addition, we illustrate the characteristics of the proposed procedure by redesigning a well-known clinical trial of maternal-infant HIV transmission.

The dynamic properties and independence structure of stochastic kinetic models (SKMs) are analyzed. An SKM is a highly multivariate jump process used to model chemical reaction networks, particularly those in biochemical and cellular systems. We identify SKM subprocesses with the corresponding counting processes and propose a directed, cyclic graph (the kinetic independence graph or KIG) that encodes the local independence structure of their conditional intensities. Given a partition [A, D, B] of the vertices, the graphical separation A ⊥ B|D in the undirected KIG has an intuitive chemical interpretation and implies that A is locally independent of B given A ∪ D. It is proved that this separation also results in global independence of the internal histories of A and B conditional on a history of the jumps in D which, under conditions we derive, corresponds to the internal history of D. The results enable mathematical definition of a modularization of an SKM using its implied dynamics. Graphical decomposition methods are developed for the identification and efficient computation of nested modularizations. Application to an SKM of the red blood cell advances understanding of this biochemical system.

We consider a nonparametric additive model of a conditional mean function in which the number of variables and additive components may be larger than the sample size but the number of nonzero additive components is “small” relative to the sample size. The statistical problem is to determine which additive components are nonzero. The additive components are approximated by truncated series expansions with B-spline bases. With this approximation, the problem of component selection becomes that of selecting the groups of coefficients in the expansion. We apply the adaptive group Lasso to select nonzero components, using the group Lasso to obtain an initial estimator and reduce the dimension of the problem. We give conditions under which the group Lasso selects a model whose number of components is comparable with the underlying model, and the adaptive group Lasso selects the nonzero components correctly with probability approaching one as the sample size increases and achieves the optimal rate of convergence. The results of Monte Carlo experiments show that the adaptive group Lasso procedure works well with samples of moderate size. A data example is used to illustrate the application of the proposed method.

A new thresholding method, based on L-statistics and called order thresholding, is proposed as a technique for improving the power when testing against high-dimensional alternatives. The new method allows great flexibility in the choice of the threshold parameter. This results in improved power over the soft and hard thresholding methods. Moreover, order thresholding is not restricted to the normal distribution. An extension of the basic order threshold statistic to high-dimensional ANOVA is presented. The performance of the basic order threshold statistic and its extension is evaluated with extensive simulations.

This article considers estimation of constant and time-varying coefficients in nonlinear ordinary differential equation (ODE) models where analytic closed-form solutions are not available. The numerical solution-based nonlinear least squares (NLS) estimator is investigated in this study. A numerical algorithm such as the Runge–Kutta method is used to approximate the ODE solution. The asymptotic properties are established for the proposed estimators considering both numerical error and measurement error. The B-spline is used to approximate the time-varying coefficients, and the corresponding asymptotic theories in this case are investigated under the framework of the sieve approach. Our results show that if the maximum step size of the p-order numerical algorithm goes to zero at a rate faster than n^−1/(p∧4), the numerical error is negligible compared to the measurement error. This result provides a theoretical guidance in selection of the step size for numerical evaluations of ODEs. Moreover, we have shown that the numerical solution-based NLS estimator and the sieve NLS estimator are strongly consistent. The sieve estimator of constant parameters is asymptotically normal with the same asymptotic co-variance as that of the case where the true ODE solution is exactly known, while the estimator of the time-varying parameter has the optimal convergence rate under some regularity conditions. The theoretical results are also developed for the case when the step size of the ODE numerical solver does not go to zero fast enough or the numerical error is comparable to the measurement error. We illustrate our approach with both simulation studies and clinical data on HIV viral dynamics.

We consider kernel estimation of marginal densities and regression functions of stationary processes. It is shown that for a wide class of time series, with proper centering and scaling, the maximum deviations of kernel density and regression estimates are asymptotically Gumbel. Our results substantially generalize earlier ones which were obtained under independence or beta mixing assumptions. The asymptotic results can be applied to assess patterns of marginal densities or regression functions via the construction of simultaneous confidence bands for which one can perform goodness-of-fit tests. As an application, we construct simultaneous confidence bands for drift and volatility functions in a dynamic short-term rate model for the U.S. Treasury yield curve rates data.

This paper considers the problem of adaptive estimation of a mean pattern in a randomly shifted curve model. We show that this problem can be transformed into a linear inverse problem, where the density of the random shifts plays the role of a convolution operator. An adaptive estimator of the mean pattern, based on wavelet thresholding is proposed. We study its consistency for the quadratic risk as the number of observed curves tends to infinity, and this estimator is shown to achieve a near-minimax rate of convergence over a large class of Besov balls. This rate depends both on the smoothness of the common shape of the curves and on the decay of the Fourier coefficients of the density of the random shifts. Hence, this paper makes a connection between mean pattern estimation and the statistical analysis of linear inverse problems, which is a new point of view on curve registration and image warping problems. We also provide a new method to estimate the unknown random shifts between curves. Some numerical experiments are given to illustrate the performances of our approach and to compare them with another algorithm existing in the literature.

In this paper we consider a novel statistical inverse problem on the Poincaré, or Lobachevsky, upper (complex) half plane. Here the Riemannian structure is hyperbolic and a transitive group action comes from the space of 2 × 2 real matrices of determinant one via Möbius transformations. Our approach is based on a deconvolution technique which relies on the Helgason–Fourier calculus adapted to this hyperbolic space. This gives a minimax nonparametric density estimator of a hyperbolic density that is corrupted by a random Möbius transform. A motivation for this work comes from the reconstruction of impedances of capacitors where the above scenario on the Poincaré plane exactly describes the physical system that is of statistical interest.

Deriving optimal designs for nonlinear models is, in general, challenging. One crucial step is to determine the number of support points needed. Current tools handle this on a case-by-case basis. Each combination of model, optimality criterion and objective requires its own proof. The celebrated de la Garza Phenomenon states that under a (p−1)th-degree polynomial regression model, any optimal design can be based on at most p design points, the minimum number of support points such that all parameters are estimable. Does this conclusion also hold for nonlinear models? If the answer is yes, it would be relatively easy to derive any optimal design, analytically or numerically. In this paper, a novel approach is developed to address this question. Using this new approach, it can be easily shown that the de la Garza phenomenon exists for many commonly studied nonlinear models, such as the Emax model, exponential model, three- and four-parameter log-linear models, Emax-PK1 model, as well as many classical polynomial regression models. The proposed approach unifies and extends many well-known results in the optimal design literature. It has four advantages over current tools: (i) it can be applied to many forms of nonlinear models; to continuous or discrete data; to data with homogeneous or nonhomogeneous errors; (ii) it can be applied to any design region; (iii) it can be applied to multiple-stage optimal design and (iv) it can be easily implemented.

This paper studies sparse density estimation via ℓ₁ penalization (SPADES). We focus on estimation in high-dimensional mixture models and nonparametric adaptive density estimation. We show, respectively, that SPADES can recover, with high probability, the unknown components of a mixture of probability densities and that it yields minimax adaptive density estimates. These results are based on a general sparsity oracle inequality that the SPADES estimates satisfy. We offer a data driven method for the choice of the tuning parameter used in the construction of SPADES. The method uses the generalized bisection method first introduced in [10]. The suggested procedure bypasses the need for a grid search and offers substantial computational savings. We complement our theoretical results with a simulation study that employs this method for approximations of one and two-dimensional densities with mixtures. The numerical results strongly support our theoretical findings.

This paper develops a point impact linear regression model in which the trajectory of a continuous stochastic process, when evaluated at a sensitive time point, is associated with a scalar response. The proposed model complements and is more interpretable than the functional linear regression approach that has become popular in recent years. The trajectories are assumed to have fractal (self-similar) properties in common with a fractional Brownian motion with an unknown Hurst exponent. Bootstrap confidence intervals based on the least-squares estimator of the sensitive time point are developed. Misspecification of the point impact model by a functional linear model is also investigated. Non-Gaussian limit distributions and rates of convergence determined by the Hurst exponent play an important role.