Peter J. Bickel

Simultaneous analysis of Lasso and Dantzig selector

We show that, under a sparsity scenario, the Lasso estimator and the Dantzig selector exhibit similar behavior. For both methods, we derive, in parallel, oracle inequalities for the prediction risk in the general nonparametric regression model, as well as bounds on the l p estimation loss for 1 ≤ p ≤ 2 in the linear model when the number of variables can be much larger than the sample size.

Efficient and adaptive estimation for semiparametric models

Introduction.- Asymptotic Inference for (Finite-Dimensional) Parametric Models.- Information Bounds for Euclidean Parameters in Infinite-Dimensional Models.- Euclidean Parameters: Further Examples.- Information Bounds for Infinite-Dimensional Parameters.- Infinite-Dimensional Parameters: Further Examples: Construction of Examples.

The developmental transcriptome of Drosophila melanogaster

Drosophila melanogaster is one of the most well studied genetic model organisms, nonetheless its genome still contains unannotated coding and non-coding genes, transcripts, exons, and RNA editing sites. Full discovery and annotation are prerequisites for understanding how the regulation of transcription, splicing, and RNA editing directs development of this complex organism. We used RNA-Seq, tiling microarrays, and cDNA sequencing to explore the transcriptome in 30 distinct developmental stages. We identified 111,195 new elements, including thousands of genes, coding and non-coding transcripts, exons, splicing and editing events and inferred protein isoforms that previously eluded discovery using established experimental, prediction and conservation-based approaches. Together, these data substantially expand the number of known transcribed elements in the Drosophila genome and provide a high-resolution view of transcriptome dynamics throughout development.

ChIP-seq guidelines and practices of the ENCODE and modENCODE consortia

Chromatin immunoprecipitation (ChIP) followed by high-throughput DNA sequencing (ChIP-seq) has become a valuable and widely used approach for mapping the genomic location of transcription-factor binding and histone modifications in living cells. Despite its widespread use, there are considerable differences in how these experiments are conducted, how the results are scored and evaluated for quality, and how the data and metadata are archived for public use. These practices affect the quality and utility of any global ChIP experiment. Through our experience in performing ChIP-seq experiments, the ENCODE and modENCODE consortia have developed a set of working standards and guidelines for ChIP experiments that are updated routinely. The current guidelines address antibody validation, experimental replication, sequencing depth, data and metadata reporting, and data quality assessment. We discuss how ChIP quality, assessed in these ways, affects different uses of ChIP-seq data. All data sets used in the analysis have been deposited for public viewing and downloading at the ENCODE (http://encodeproject.org/ENCODE/) and modENCODE (http://www.modencode.org/) portals.

Regularized estimation of large covariance matrices

This paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance. We show that these estimates are consistent in the operator norm as long as (logp) 2 =n ! 0, and obtain explicit rates. The results are uniform over some fairly natural well-conditioned families of covariance matrices. We also introduce an analogue of the Gaussian white noise model and show that if the population covariance is embeddable in that model and well-conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix. The results can be extended to smooth versions of banding and to non-Gaussian distributions with su‐ciently short tails. A resampling approach is proposed for choosing the banding parameter in practice. This approach is illustrated numerically on both simulated and real data.

Covariance regularization by thresholding

This paper considers regularizing a covariance matrix of p variables estimated from n observations, by hard thresholding. We show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense, the variables are Gaussian or sub-Gaussian, and (log p)/n → 0, and obtain explicit rates. The results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity. We discuss an intuitive resampling scheme for threshold selection and prove a general cross-validation result that justifies this approach. We also compare thresholding to other covariance estimators in simulations and on an example from climate data. 1. Introduction. Estimation of covariance matrices is important in a number of areas of statistical analysis, including dimension reduction by principal component analysis (PCA), classification by linear or quadratic discriminant analysis (LDA and QDA), establishing independence and conditional independence relations in the context of graphical models, and setting confidence intervals on linear functions of the means of the components. In recent years, many application areas where these tools are used have been dealing with very high-dimensional datasets, and sample sizes can be very small relative to dimension. Examples include genetic data, brain imaging, spectroscopic imaging, climate data and many others. It is well known by now that the empirical covariance matrix for samples of size n from a p-variate Gaussian distribution, Np(μ, � p), is not a good estimator of the population covariance if p is large. Many results in random matrix theory illustrate this, from the classical Mary law [29] to the more recent work of Johnstone and his students on the theory of the largest eigenvalues [12, 23, 30] and associated eigenvectors [24]. However, with the exception of a method for estimating the covariance spectrum [11], these probabilistic results do not offer alternatives to the sample covariance matrix. Alternative estimators for large covariance matrices have therefore attracted a lot of attention recently. Two broad classes of covariance estimators have emerged: those that rely on a natural ordering among variables, and assume that variables

Sparse permutation invariant covariance estimation

The paper proposes a method for constructing a sparse estima- tor for the inverse covariance (concentration) matrix in high-dimensional settings. The estimator uses a penalized normal likelihood approach and forces sparsity by using a lasso-type penalty. We establish a rate of con- vergence in the Frobenius norm as both data dimension p and sample size n are allowed to grow, and show that the rate depends explicitly on how sparse the true concentration matrix is. We also show that a correlation- based version of the method exhibits better rates in the operator norm. We also derive a fast iterative algorithm for computing the estimator, which relies on the popular Cholesky decomposition of the inverse but produces a permutation-invariant estimator. The method is compared to other es- timators on simulated data and on a real data example of tumor tissue classification using gene expression data.

An Analysis of Transformations Revisited

Abstract Following Box and Cox (1964), we assume that a transform Z i = h(Yi , λ) of our original data {Yi } satisfies a linear model. Consistency properties of the Box-Cox estimates (MLEs) of λ and the parameters in the linear model, as well as the asymptotic variances of these estimates, are considered. We find that in some structured models such as transformed linear regression with small to moderate error variances, the asymptotic variances of the estimates of the parameters in the linear model are much larger when the transformation parameter λ is unknown than when it is known. In some unstructured models such as transformed one-way analysis of variance with moderate to large error variances, the cost of not knowing λ is moderate to small. The case where the error distribution in the linear model is not normal but actually unknown is considered, and robust methods in the presence of transformations are introduced for this case. Asymptotics and simulation results for the transformed additive two-way ...

Obstacles to High-Dimensional Particle Filtering

Abstract Particle filters are ensemble-based assimilation schemes that, unlike the ensemble Kalman filter, employ a fully nonlinear and non-Gaussian analysis step to compute the probability distribution function (pdf) of a system’s state conditioned on a set of observations. Evidence is provided that the ensemble size required for a successful particle filter scales exponentially with the problem size. For the simple example in which each component of the state vector is independent, Gaussian, and of unit variance and the observations are of each state component separately with independent, Gaussian errors, simulations indicate that the required ensemble size scales exponentially with the state dimension. In this example, the particle filter requires at least 1011 members when applied to a 200-dimensional state. Asymptotic results, following the work of Bengtsson, Bickel, and collaborators, are provided for two cases: one in which each prior state component is independent and identically distributed, and ...

A nonparametric view of network models and Newman–Girvan and other modularities

Prompted by the increasing interest in networks in many fields, we present an attempt at unifying points of view and analyses of these objects coming from the social sciences, statistics, probability and physics communities. We apply our approach to the Newman–Girvan modularity, widely used for “community” detection, among others. Our analysis is asymptotic but we show by simulation and application to real examples that the theory is a reasonable guide to practice.