A new multivariate concept of quantile, based on a directional version of Koenker and Bassett’s traditional regression quantiles, is introduced for multivariate location and multiple-output regression problems. In their empirical version, those quantiles can be computed efficiently via linear programming techniques. Consistency, Bahadur representation and asymptotic normality results are established. Most importantly, the contours generated by those quantiles are shown to coincide with the classical halfspace depth contours associated with the name of Tukey. This relation does not only allow for efficient depth contour computations by means of parametric linear programming, but also for transferring from the quantile to the depth universe such asymptotic results as Bahadur representations. Finally, linear programming duality opens the way to promising developments in depth-related multivariate rank-based inference.

Let (Y, (X_i)_1≤i≤p) be a real zero mean Gaussian vector and V be a subset of {1, …, p}. Suppose we are given n i.i.d. replications of this vector. We propose a new test for testing that Y is independent of (X_i)_{i∈{1, …, p}∖V} conditionally to (X_i)_i∈V against the general alternative that it is not. This procedure does not depend on any prior information on the covariance of X or the variance of Y and applies in a high-dimensional setting. It straightforwardly extends to test the neighborhood of a Gaussian graphical model. The procedure is based on a model of Gaussian regression with random Gaussian covariates. We give nonasymptotic properties of the test and we prove that it is rate optimal [up to a possible log(n) factor] over various classes of alternatives under some additional assumptions. Moreover, it allows us to derive nonasymptotic minimax rates of testing in this random design setting. Finally, we carry out a simulation study in order to evaluate the performance of our procedure.

In this paper, the maximum Lq-likelihood estimator (MLqE), a new parameter estimator based on nonextensive entropy [Kibernetika 3 (1967) 30–35] is introduced. The properties of the MLqE are studied via asymptotic analysis and computer simulations. The behavior of the MLqE is characterized by the degree of distortion q applied to the assumed model. When q is properly chosen for small and moderate sample sizes, the MLqE can successfully trade bias for precision, resulting in a substantial reduction of the mean squared error. When the sample size is large and q tends to 1, a necessary and sufficient condition to ensure a proper asymptotic normality and efficiency of MLqE is established.

We address the problem of parameter estimation for diffusion driven stochastic volatility models through Markov chain Monte Carlo (MCMC). To avoid degeneracy issues we introduce an innovative reparametrization defined through transformations that operate on the time scale of the diffusion. A novel MCMC scheme which overcomes the inherent difficulties of time change transformations is also presented. The algorithm is fast to implement and applies to models with stochastic volatility. The methodology is tested through simulation based experiments and illustrated on data consisting of US treasury bill rates.

We propose a two-sample test for the means of high-dimensional data when the data dimension is much larger than the sample size. Hotelling’s classical T² test does not work for this “large p, small n” situation. The proposed test does not require explicit conditions in the relationship between the data dimension and sample size. This offers much flexibility in analyzing high-dimensional data. An application of the proposed test is in testing significance for sets of genes which we demonstrate in an empirical study on a leukemia data set.

In this paper, we study inference for high-dimensional data characterized by small sample sizes relative to the dimension of the data. In particular, we provide an infinite-dimensional framework to study statistical models that involve situations in which (i) the number of parameters increase with the sample size (that is, allowed to be random) and (ii) there is a possibility of missing data. Under a variety of tail conditions on the components of the data, we provide precise conditions for the joint consistency of the estimators of the mean. In the process, we clarify and improve some of the recent consistency results that appeared in the literature. An important aspect of the work presented is the development of asymptotic normality results for these models. As a consequence, we construct different test statistics for one-sample and two-sample problems concerning the mean vector and obtain their asymptotic distributions as a corollary of the infinite-dimensional results. Finally, we use these theoretical results to develop an asymptotically justifiable methodology for data analyses. Simulation results presented here describe situations where the methodology can be successfully applied. They also evaluate its robustness under a variety of conditions, some of which are substantially different from the technical conditions. Comparisons to other methods used in the literature are provided. Analyses of real-life data is also included.

We present fixed domain asymptotic results that establish consistent estimates of the variance and scale parameters for a Gaussian random field with a geometric anisotropic Matérn autocovariance in dimension d>4. When d<4 this is impossible due to the mutual absolute continuity of Matérn Gaussian random fields with different scale and variance (see Zhang [J. Amer. Statist. Assoc. 99 (2004) 250–261]). Informally, when d>4, we show that one can estimate the coefficient on the principle irregular term accurately enough to get a consistent estimate of the coefficient on the second irregular term. These two coefficients can then be used to separate the scale and variance. We extend our results to the general problem of estimating a variance and geometric anisotropy for more general autocovariance functions. Our results illustrate the interaction between the accuracy of estimation, the smoothness of the random field, the dimension of the observation space and the number of increments used for estimation. As a corollary, our results establish the orthogonality of Matérn Gaussian random fields with different parameters when d>4. The case d=4 is still open.

We propose MC+, a fast, continuous, nearly unbiased and accurate method of penalized variable selection in high-dimensional linear regression. The LASSO is fast and continuous, but biased. The bias of the LASSO may prevent consistent variable selection. Subset selection is unbiased but computationally costly. The MC+ has two elements: a minimax concave penalty (MCP) and a penalized linear unbiased selection (PLUS) algorithm. The MCP provides the convexity of the penalized loss in sparse regions to the greatest extent given certain thresholds for variable selection and unbiasedness. The PLUS computes multiple exact local minimizers of a possibly nonconvex penalized loss function in a certain main branch of the graph of critical points of the penalized loss. Its output is a continuous piecewise linear path encompassing from the origin for infinite penalty to a least squares solution for zero penalty. We prove that at a universal penalty level, the MC+ has high probability of matching the signs of the unknowns, and thus correct selection, without assuming the strong irrepresentable condition required by the LASSO. This selection consistency applies to the case of p≫n, and is proved to hold for exactly the MC+ solution among possibly many local minimizers. We prove that the MC+ attains certain minimax convergence rates in probability for the estimation of regression coefficients in ℓ_r balls. We use the SURE method to derive degrees of freedom and C_p-type risk estimates for general penalized LSE, including the LASSO and MC+ estimators, and prove their unbiasedness. Based on the estimated degrees of freedom, we propose an estimator of the noise level for proper choice of the penalty level. For full rank designs and general sub-quadratic penalties, we provide necessary and sufficient conditions for the continuity of the penalized LSE. Simulation results overwhelmingly support our claim of superior variable selection properties and demonstrate the computational efficiency of the proposed method.

High-frequency data observed on the prices of financial assets are commonly modeled by diffusion processes with micro-structure noise, and realized volatility-based methods are often used to estimate integrated volatility. For problems involving a large number of assets, the estimation objects we face are volatility matrices of large size. The existing volatility estimators work well for a small number of assets but perform poorly when the number of assets is very large. In fact, they are inconsistent when both the number, p, of the assets and the average sample size, n, of the price data on the p assets go to infinity. This paper proposes a new type of estimators for the integrated volatility matrix and establishes asymptotic theory for the proposed estimators in the framework that allows both n and p to approach to infinity. The theory shows that the proposed estimators achieve high convergence rates under a sparsity assumption on the integrated volatility matrix. The numerical studies demonstrate that the proposed estimators perform well for large p and complex price and volatility models. The proposed method is applied to real high-frequency financial data.

We develop a new Gibbs sampler for a linear mixed model with a Dirichlet process random effect term, which is easily extended to a generalized linear mixed model with a probit link function. Our Gibbs sampler exploits the properties of the multinomial and Dirichlet distributions, and is shown to be an improvement, in terms of operator norm and efficiency, over other commonly used MCMC algorithms. We also investigate methods for the estimation of the precision parameter of the Dirichlet process, finding that maximum likelihood may not be desirable, but a posterior mode is a reasonable approach. Examples are given to show how these models perform on real data. Our results complement both the theoretical basis of the Dirichlet process nonparametric prior and the computational work that has been done to date.

We consider the detection of multivariate spatial clusters in the Bernoulli model with N locations, where the design distribution has weakly dependent marginals. The locations are scanned with a rectangular window with sides parallel to the axes and with varying sizes and aspect ratios. Multivariate scan statistics pose a statistical problem due to the multiple testing over many scan windows, as well as a computational problem because statistics have to be evaluated on many windows. This paper introduces methodology that leads to both statistically optimal inference and computationally efficient algorithms. The main difference to the traditional calibration of scan statistics is the concept of grouping scan windows according to their sizes, and then applying different critical values to different groups. It is shown that this calibration of the scan statistic results in optimal inference for spatial clusters on both small scales and on large scales, as well as in the case where the cluster lives on one of the marginals. Methodology is introduced that allows for an efficient approximation of the set of all rectangles while still guaranteeing the statistical optimality results described above. It is shown that the resulting scan statistic has a computational complexity that is almost linear in N.

Calculating a Monte Carlo standard error (MCSE) is an important step in the statistical analysis of the simulation output obtained from a Markov chain Monte Carlo experiment. An MCSE is usually based on an estimate of the variance of the asymptotic normal distribution. We consider spectral and batch means methods for estimating this variance. In particular, we establish conditions which guarantee that these estimators are strongly consistent as the simulation effort increases. In addition, for the batch means and overlapping batch means methods we establish conditions ensuring consistency in the mean-square sense which in turn allows us to calculate the optimal batch size up to a constant of proportionality. Finally, we examine the empirical finite-sample properties of spectral variance and batch means estimators and provide recommendations for practitioners.

We show that scale-adjusted versions of the centroid-based classifier enjoys optimal properties when used to discriminate between two very high-dimensional populations where the principal differences are in location. The scale adjustment removes the tendency of scale differences to confound differences in means. Certain other distance-based methods, for example, those founded on nearest-neighbor distance, do not have optimal performance in the sense that we propose. Our results permit varying degrees of sparsity and signal strength to be treated, and require only mild conditions on dependence of vector components. Additionally, we permit the marginal distributions of vector components to vary extensively. In addition to providing theory we explore numerical properties of a centroid-based classifier, and show that these features reflect theoretical accounts of performance.

We consider dimension reduction for regression or classification in which the predictors are matrix- or array-valued. This type of predictor arises when measurements are obtained for each combination of two or more underlying variables—for example, the voltage measured at different channels and times in electroencephalography data. For these applications, it is desirable to preserve the array structure of the reduced predictor (e.g., time versus channel), but this cannot be achieved within the conventional dimension reduction formulation. In this paper, we introduce a dimension reduction method, to be called dimension folding, for matrix- and array-valued predictors that preserves the array structure. In an application of dimension folding to an electroencephalography data set, we correctly classify 97 out of 122 subjects as alcoholic or nonalcoholic based on their electroencephalography in a cross-validation sample.

Given a sample from some unknown continuous density f : ℝ→ℝ, we construct adaptive confidence bands that are honest for all densities in a “generic” subset of the union of t-Hölder balls, 0<t≤r, where r is a fixed but arbitrary integer. The exceptional (“nongeneric”) set of densities for which our results do not hold is shown to be nowhere dense in the relevant Hölder-norm topologies. In the course of the proofs we also obtain limit theorems for maxima of linear wavelet and kernel density estimators, which are of independent interest.

The notion of probability density for a random function is not as straightforward as in finite-dimensional cases. While a probability density function generally does not exist for functional data, we show that it is possible to develop the notion of density when functional data are considered in the space determined by the eigenfunctions of principal component analysis. This leads to a transparent and meaningful surrogate for density defined in terms of the average value of the logarithms of the densities of the distributions of principal components for a given dimension. This density approximation is estimable readily from data. It accurately represents, in a monotone way, key features of small-ball approximations to density. Our results on estimators of the densities of principal component scores are also of independent interest; they reveal interesting shape differences that have not previously been considered. The statistical implications of these results and properties are identified and discussed, and practical ramifications are illustrated in numerical work.

Classical multivariate principal component analysis has been extended to functional data and termed functional principal component analysis (FPCA). Most existing FPCA approaches do not accommodate covariate information, and it is the goal of this paper to develop two methods that do. In the first approach, both the mean and covariance functions depend on the covariate Z and time scale t while in the second approach only the mean function depends on the covariate Z. Both new approaches accommodate additional measurement errors and functional data sampled at regular time grids as well as sparse longitudinal data sampled at irregular time grids. The first approach to fully adjust both the mean and covariance functions adapts more to the data but is computationally more intensive than the approach to adjust the covariate effects on the mean function only. We develop general asymptotic theory for both approaches and compare their performance numerically through simulation studies and a data set.

For any class of one-sided 1−α confidence intervals with a certain monotonicity ordering on the random confidence limit, the smallest interval, in the sense of the set inclusion for the difference of two proportions of two independent binomial random variables, is constructed based on a direct analysis of coverage probability function. A special ordering on the confidence limit is developed and the corresponding smallest confidence interval is derived. This interval is then applied to identify the minimum effective dose (MED) for binary data in dose-response studies, and a multiple test procedure that controls the familywise error rate at level α is obtained. A generalization of constructing the smallest one-sided confidence interval to other discrete sample spaces is discussed in the presence of nuisance parameters.

We introduce new quantile estimators with adaptive importance sampling. The adaptive estimators are based on weighted samples that are neither independent nor identically distributed. Using a new law of iterated logarithm for martingales, we prove the convergence of the adaptive quantile estimators for general distributions with nonunique quantiles thereby extending the work of Feldman and Tucker [Ann. Math. Statist. 37 (1996) 451–457]. We illustrate the algorithm with an example from credit portfolio risk analysis.