Supplementary Material Online Research Articles

Improving on Estimation for the Generalized Pareto Distribution

The generalized Pareto distribution (GPD) was widely used to model exceedances over thresholds, such as flood levels of rivers. Zhang and Stephens (2009) proposed a new estimation method for parameters of the GPD, which, based on the likelihood method and empirical Bayesian method, is free from the theoretical and computational problems suffered by traditional estimation approaches. In terms of estimation efficiency and bias, the new method outperforms other existing methods in common situations, but it may perform poorly for very heavy-tailed distributions. The new method is modified in this article to significantly improve its adaptivity. This article has supplementary material online.

Covered stents in giant coronary artery aneurysm

A 71-year-old woman presented with stable angina pectoris. On coronary angiography, a giant aneurysm of the left anterior descending artery (LAD) measuring 70 × 40 mm was seen. The aneurysm was lined with thrombus ( Panel A : aneurysm indicated by large arrows and lumen indicated by small arrows; see Supplementary material online, Video S1 ). Proximal and distal to the aneurysm, the LAD was …

Global Partial Likelihood for Nonparametric Proportional Hazards Models

As an alternative to the local partial likelihood method of Tibshirani and Hastie and Fan, Gijbels, and King, a global partial likelihood method is proposed to estimate the covariate effect in a nonparametric proportional hazards model, λ(t|x)=exp{ψ(x)}λ0(t). The estimator, ψˆ(x), reduces to the Cox partial likelihood estimator if the covariate is discrete. The estimator is shown to be consistent and semiparametrically efficient for linear functionals of ψ(x). Moreover, Breslow-type estimation of the cumulative baseline hazard function, using the proposed estimator ψˆ(x), is proved to be efficient. The asymptotic bias and variance are derived under regularity conditions. Computation of the estimator involves an iterative but simple algorithm. Extensive simulation studies provide evidence supporting the theory. The method is illustrated with the Stanford heart transplant data set. The proposed global approach is also extended to a partially linear proportional hazards model and found to provide efficient estimation of the slope parameter. This article has the supplementary materials online.

Using Evidence of Mixed Populations to Select Variables for Clustering Very High-Dimensional Data

In this paper we develop a nonparametric approach to clustering very high-dimensional data, designed particularly for problems where the mixture nature of a population is expressed through multimodality of its density. Therefore, a technique based implicitly on mode testing can be particularly effective. In principle, several alternative approaches could be used to assess the extent of multimodality, but in the present problem the excess mass method has important advantages. We show that the resulting methodology for determining clusters is particularly effective in cases where the data are relatively heavy tailed or show a moderate to high degree of correlation, or when the number of important components is relatively small. Conversely, in the case of light-tailed, almost-independent components when there are many clusters, clustering in terms of modality can be less reliable than more conventional approaches. This article has supplementary material online.

Bayesian Inference With Incomplete Multinomial Data: A Problem in Pathogen Diversity

With recent advances in genetic analysis, it has become feasible to classify a pathogen into genetically distinct variants even though they apparently cause an infected subject similar symptoms. The availability of such data opens up the interesting problem of studying the spatiotemporal variation in the diversity of variants of a pathogen. Data on pathogen variants often suffer the problems of (i) low cell counts, (ii) incomplete classification due to laboratory problems (e.g., contamination), and (iii) unseen variants. Shannon’s entropy may be used as a measure of variant diversity. A Bayesian approach can be used to deal with the problems of low cell counts and unseen variants. Bayesian analysis of incomplete multinomial data may be carried out by Markov chain Monte Carlo techniques. However, for pathogen-variant data, there often is only one source of missingness—namely, some subjects are known to be infected by some unidentified pathogen variant. We point out that for incomplete data with disjoint sources of missingness, Bayesian analysis can be done more efficiently using an iid sampling scheme from the posterior distribution. We illustrate the method by analyzing a data set on the prevalence of bartonella infection among individual colonies of prairie dogs at the study site in Colorado between 2003 and 2006. We compare the result from the proposed Monte Carlo method with the results from other methods, including a model that entertains within-variant spatial correlation but no between-variant spatial correlation. This article has supplementary material online.

Inference in Semiparametric Regression Models Under Partial Questionnaire Design and Nonmonotone Missing Data

In epidemiologic studies, partial questionnaire design (PQD) can reduce cost, time, and other practical burdens associated with lengthy questionnaires by assigning different subsets of the questionnaire to different, but overlapping, subsets of the study participants. In this article, we describe methods for semiparametric inference for regression model under PQD and other study settings that can generate nonmonotone missing data in covariates. In particular, motivated from methods for multiphase designs, we develop three estimators, namely mean score, pseudo-likelihood, and semiparametric maximum likelihood, each of which has some unique advantages. We develop the asymptotic theory and a sandwich variance estimator for each of the estimators under the underlying semiparametric model that allows the distribution of the covariates to remain nonparametric. We study the finite sample performances and relative efficiencies of the methods using simulation studies. We illustrate the methods using data from a case-control study of non-Hodgkin’s lymphoma where the data on the main chemical exposures of interest are collected using two different instruments on two different, but overlapping, subsets of the participants. This article has supplementary material online.

Latent Stick-Breaking Processes

We develop a model for stochastic processes with random marginal distributions. Our model relies on a stick-breaking construction for the marginal distribution of the process, and introduces dependence across locations by using a latent Gaussian copula model as the mechanism for selecting the atoms. The resulting latent stick-breaking process (LaSBP) induces a random partition of the index space, with points closer in space having a higher probability of being in the same cluster. We develop an efficient and straightforward Markov chain Monte Carlo (MCMC) algorithm for computation and discuss applications in financial econometrics and ecology. This article has supplementary material online.

Field-Failure and Warranty Prediction Based on Auxiliary Use-Rate Information

Assessment of risk due to product failure is important both for purposes of finance (e.g., warranty costs) and safety (e.g., potential loss of human life). In many applications a prediction of the number of future failures is an important input to such an assessment.Usually the field-data response used to make predictions of future failures is the number of weeks (or another unit of real time) in service. Use-rate information usually is not available (automobile warranty data are an exception, where both weeks in service and number of miles driven are available for units returned for warranty repair). With new technology, however, sensors and smart chips are being installed in many modern products ranging from computers and printers to automobiles and aircraft engines. Thus the coming generations of field data for many products will provide information on how the product was used and the environment in which it was used. This article was motivated by the need to predict warranty returns for a product with multiple failure modes. For this product, cycles-to-failure/use-rate information was available for those units that were connected to the network. We show how to use a cycles-to-failure model to compute predictions and prediction intervals for the number of warranty returns. We also present prediction methods for units not connected to the network. To provide insight into the reasons that use-rate models provide better predictions, we also present a comparison of asymptotic variances comparing the cycles-to-failure and time-to-failure models. This article has supplementary material online.

Is the t Confidence Interval X̅ ± tα (n − 1)s / √n Optimal?

The classical t confidence interval (CI) for unknown mean θ, , is so fundamental in statistics and so prevailing in practice, it is regarded as an optimal procedure for many practitioners. In this article we describe a new procedure that can outperform not only this classical CI but also bootstrap CIs. This article has supplementary material online.

Main-Effect Plans Orthogonal Through the Block Factor

Many industrial experiments require different number of levels for different factors. Orthogonal main-effect plans (OMEPs) for such asymmetrical experiments often require large run size. If blocking is needed, then run size usually becomes even larger. In this article we show that there are situations when use of blocks may in fact be helpful in finding an OMEP with small run size. Explicitly, we obtained saturated OMEPs for a 33n⋅23n experiment in 3n blocks of size 4 each for each Hadamard number n. In each of these plans, the three-level factors are nonorthogonal to the block factor but are pairwise orthogonal through the block factor. The two-level factors are orthogonal to the block factor. The data analysis for such a plan is illustrated with the help of examples. This article has supplementary material online.

The First (Known) Statistical Graph: Michael Florent van Langren and the “Secret” of Longitude

A 1644 diagram by Michael Florent van Langren, showing estimates of the difference in longitude between Toledo and Rome, is sometimes considered to be the first known instance of a graph of statistical data. Some recently discovered documents help to date the genesis of this graphic to before March 1628, and shed some light on why van Langren chose to display this information in this form. In the process, we discovered three earlier versions of the 1644 graph and one slightly later reproduction. This article describes these early attempts on the solution of “the problem of longitude” from the perspective of a history of data visualization. This article has supplementary material online.

Approximate Methods for State-Space Models

State-space models provide an important body of techniques for analyzing time series, but their use requires estimating unobserved states. The optimal estimate of the state is its conditional expectation given the observation histories, and computing this expectation is hard when there are nonlinearities. Existing filtering methods, including sequential Monte Carlo, tend to be either inaccurate or slow. In this paper, we study a nonlinear filter for nonlinear/non-Gaussian state-space models, which uses Laplace’s method, an asymptotic series expansion, to approximate the state’s conditional mean and variance, together with a Gaussian conditional distribution. This Laplace Gaussian filter (LGF) gives fast, recursive, deterministic state estimates, with an error which is set by the stochastic characteristics of the model and is, we show, stable over time. We illustrate the estimation ability of the LGF by applying it to the problem of neural decoding and compare it to sequential Monte Carlo both in simulations and with real data. We find that the LGF can deliver superior results in a small fraction of the computing time. This article has supplementary material online.

Demonstrating Single and Multiple Currents Through the E. coli-SecYEG-Pore: Testing for the Number of Modes of Noisy Observations

We analyze a new dataset from an electrophysiological recording of transmembrane currents through a bacterial membrane channel to demonstrate the existence of single and multiple channel currents. Protein channels mediate transport through biological membranes; knowledge of the channel properties gained from electrophysiological recordings is important for a targeted drug design. We investigate the bacterial membrane protein SecYEG which is of essential importance for the secretory pathway for sorting of newly synthesized proteins to their place of function in the cell. Our results strongly indicate that in the SecYEG pore the different modes of the density of channel currents are approximately equidistant and correspond to different numbers of open channels in the membrane. A current of ≈12 pA under the present experimental conditions turns out to be characteristic of the presence of a single open SecYEG pore, a fact that had not been electrophysiologically characterized so far. Electrophysiological recordings of single protein channels show a substantial amount of background noise. The data at our disposal can be modeled as the independent sum of an error variable and the realization of the ionic current. Thus, we are led to deconvoluting the density of the observations in order to recover the density f of the ionic currents, and then investigating the number of modes of f. To this end we propose an extension of Silverman’s (1981) test for the number of modes to deconvolution kernel density estimation, and develop the relevant theory. The finite sample performance of the test is investigated in a simulation study.Technical details for the proofs in this article are available as supplementary material online.

Cardiovascular magnetic resonance prior to surgical treatment of chronic thrombo-embolic pulmonary hypertension

A 72-year-old lady was referred to our Pulmonary Hypertension Unit with WHO Class IV dyspnoea. Previous computed tomography pulmonary angiography had revealed chronic thrombo-embolism. The degree and distribution of disease was confirmed with gadolinium-enhanced cardiac magnetic resonance (CMR) pulmonary angiography ( Panel A , see also Supplementary material online, Movie 1 ). The right ventricle was severely dilated with an ejection fraction of 32% by CMR. Functional testing demonstrated reduced 6 min walk …

Fast Nonparametric Quantile Regression With Arbitrary Smoothing Methods

The calculation of nonparametric quantile regression curve estimates is often computationally intensive, as typically an expensive nonlinear optimization problem is involved. This article proposes a fast and easy-to-implement method for computing such estimates. The main idea is to approximate the costly nonlinear optimization by a sequence of well-studied penalized least squares-type nonparametric mean regression estimation problems. The new method can be paired with different nonparametric smoothing methods and can also be applied to higher dimensional settings. Therefore, it provides a unified framework for computing different types of nonparametric quantile regression estimates, and it also greatly broadens the scope of the applicability of quantile regression methodology. This wide applicability and the practical performance of the proposed method are illustrated with smoothing spline and wavelet curve estimators, for both uni- and bivariate settings. Results from numerical experiments suggest that estimates obtained from the proposed method are superior to many competitors. This article has supplementary material online.

Multiscale Exploratory Analysis of Regression Quantiles Using Quantile SiZer

The SiZer methodology proposed by Chaudhuri and Marron (1999) is a valuable tool for conducting exploratory data analysis. Since its inception different versions of SiZer have been proposed in the literature. Most of these SiZer variants are targeting the mean structure of the data, and are incapable of providing any information about the quantile composition of the data. To fill this need, this article proposes a quantile version of SiZer for the regression setting. By inspecting the SiZer maps produced by this new SiZer, real quantile structures hidden in a dataset can be more effectively revealed, while at the same time spurious features can be filtered out. The utility of this quantile SiZer is illustrated via applications to both real data and simulated examples. This article has supplementary material online.

On the Utility of Graphics Cards to Perform Massively Parallel Simulation of Advanced Monte Carlo Methods

We present a case study on the utility of graphics cards to perform massively parallel simulation of advanced Monte Carlo methods. Graphics cards, containing multiple Graphics Processing Units (GPUs), are self-contained parallel computational devices that can be housed in conventional desktop and laptop computers and can be thought of as prototypes of the next generation of many-core processors. For certain classes of population-based Monte Carlo algorithms they offer massively parallel simulation, with the added advantage over conventional distributed multicore processors that they are cheap, easily accessible, easy to maintain, easy to code, dedicated local devices with low power consumption. On a canonical set of stochastic simulation examples including population-based Markov chain Monte Carlo methods and Sequential Monte Carlo methods, we find speedups from 35- to 500-fold over conventional single-threaded computer code. Our findings suggest that GPUs have the potential to facilitate the growth of statistical modeling into complex data-rich domains through the availability of cheap and accessible many-core computation. We believe the speedup we observe should motivate wider use of parallelizable simulation methods and greater methodological attention to their design. This article has supplementary material online.

Visualizing Influential Observations in Dependent Data

We introduce the hair-plot to visualize influential observations in dependent data. It consists of all trajectories of the value of an estimator when each observation is modified in turn by an additive perturbation. We define two measures of influence: the local influence which describes the rate of departure from the original estimate due to a small perturbation of each observation; and the asymptotic influence which indicates the influence on the original estimate of the most extreme contamination for each observation. The cases of estimators defined as quadratic forms or ratios of quadratic forms are investigated in detail. Sample autocovariances, covariograms, and variograms belong to the first case. Sample autocorrelations, correlograms, and indices of spatial autocorrelation such as Moran’s I belong to the second case. We illustrate our approach on various datasets from time series analysis and spatial statistics. This article has supplementary material online.

Solving ℓ1 Regularization Problems With Piecewise Linear Losses

This article is concerned with the computational aspect of ℓ1 regularization problems with a certain class of piecewise linear loss functions. The problem of computing the ℓ1 regularization path for a piecewise linear loss can be formalized as a parametric linear programming problem. We propose an efficient implementation method of the parametric simplex algorithm for such a problem. We also conduct a simulation study to investigate the behavior of the number of “breakpoints” of the regularization path when both the number of observations and the number of explanatory variables vary. Our method is also applicable to the computation of the regularization path for a piecewise linear loss and the blockwise ℓ∞ penalty. This article has supplementary material online.

Nonparametric Assessment of Contamination in Multivariate Data Using Generalized Quantile Sets and FDR

Large, multivariate datasets from high-throughput instrumentation have become ubiquitous in the sciences. Frequently, it is of interest to characterize the measurements in these datasets by the extent to which they represent ‘nominal’ versus ‘contaminated’ instances. However, often the nature of even the nominal patterns in the data is unknown and potentially quite complex, making their explicit parametric modeling a daunting task. In this article, we introduce a nonparametric method for the simultaneous annotation of multivariate data (called MN-SCAnn), by which one may produce an annotated ranking of the observations, indicating the relative extent to which each may or may not be considered nominal, while making minimal assumptions on the nature of the nominal distribution. In our framework each observation is linked to a corresponding generalized quantile set and, implicitly adopting a hypothesis testing perspective, each set is associated with a test, which in turn is accompanied by a certain false discovery rate. The combination of generalized quantile set methods with false discovery rate principles, in the context of contaminated data, is new, and estimation of the key underlying quantities requires that a number of issues be addressed. We illustrate MN-SCAnn through examples in two contexts: the preprocessing of flow cytometry data in bioinformatics, and the detection of anomalous traffic patterns in Internet measurement studies. This article has supplementary material online.