Many methods exist for one dimensional curve registration, and how methods compare has not been made clear in the literature. This special section is a summary of a detailed comparison of a number of major methods, done during a recent workshop. The basis of the comparison was simultaneous analysis of a set of four real data sets, which engendered a high level of informative discussion. Most research groups in this area were represented, and many insights were gained, which are discussed here. The format of this special section is four papers introducing the data, each accompanied by a number of analyses by different groups, plus a discussion summary of the lessons learned.

Proteomics is a rapidly growing research area within bioinformatics which focuses on quantification of peptide concentrations and on the identification of proteins and peptides. In quantitative proteomics the identification of biomarkers from peptide concentrations is important for diagnostic purposes and treatment of diseases.

The goal of this paper is to facilitate research in this area, by providing a test bed for comparison of 1D curve registration methods. This is done in a novel way, by providing not only curves, but also an answer key as to how the peaks should align. In the following papers a number of approaches to this problem are given, and the answer key provides unusually useful insights into how the methods compare.

For this reason, we consider proteomics mass spectrometry profiles which are part of a larger study into the identification of biomarkers in Acute Myeloid Leukaemia (AML). For these profiles large ion counts result in large peaks, but these peaks may occur at different retention times for different profiles. The first step in the quantification of peptides in proteomics profiles is the alignment of the 1D curves of total ion count (TIC).

The paper includes a description of proteomics mass spectrometry profiling, and considers profiles from five patients with AML. It outlines the preprocessing steps we applied to the multiple TIC samples from each patient, and introduces the reference peptides. The retention times of the reference peptides are known for each profile, and using these times as an answer key makes the 1D TIC curves a particularly informative test bed for curve registration.

We analyze the proteomics data introducing a block $k$-mean alignment procedure. This technique is able to jointly align and cluster the data, accounting appropriately for the block structure of these data, that includes measurement repetitions for each patient. An analysis of area-under-peaks, following the alignment, separates patients who respond and those who do not respond to treatment.

We consider the problem of alignment and classification of proteomics data, that is described in Koch et al. [4], using the Extended Fisher-Rao (EFR) framework introduced in [6]. We demonstrate this framework by separating amplitude and phase components of functional data from patients having therapeutic treatments for Acute Myeloid Leukemia (AML). Then, using individual functional principal component analysis, for both the phase and amplitude components [8], we obtain bases for principal subspaces and model the data by imposing probability models on principal coefficients. Lastly, using the distances calculated from individual components, we demonstrate a successful discrimination between responders and non-responders to treatment for AML.

A Bayesian approach to function alignment is introduced. A model is proposed in the ambient space, with a Dirichlet prior for the derivative of the warping function and a Gaussian process for the square root velocity function. Posterior inference is carried out via Markov chain Monte Carlo simulation. The methodology is applied to a dataset of mass spectrometry scans. Good alignment is obtained for most of the known proteins, with more uncertainty at either end of each scan.

The Fisher Rao curve registration is used for curve alignment. Quality of registration is carefully studied using zoomed views. A related linear warp is considered, and seen to give somewhat inferior performance. Alignment is also seen to give large improvements in the ultimate classification problem.

Mass spectrometry (MS) data are becoming common in recent years. Prior to other statistical inferential procedures, alignment of spectra may be needed to ensure that intensities of the same protein/peptide are accurately located/identified. However, the enormous number of peaks poses challenge in handling such data. Direct applications of available curve alignment methods often do not produce satisfactory results. In this work, we propose an Automated Pairwise Piecewise Landmark Registration (APPLR) method for aligning MS data. For a pair of spectra, the most prominent peaks are given the priority to be aligned first. A weighted Gaussian kernel based similarity score is used to test warp these top peaks and spectra are then aligned according to the best match. The algorithm is implemented in an iterative way until all spectra are aligned. We illustrated the new method and two other curve alignment methods to the unlabeled total ion count data.

Statistical analysis of spike trains is one of the central problems in neural coding, and can be pursued in several ways. One option is model-based, i.e. assume a parametric or semi-parametric model, such as the Poisson model, for spike train data and use it in decoding spike trains. The other option is metric-based, i.e. choose a metric for comparing the numbers and the placements of spikes in different trains, and does not need a model. A prominent idea in the latter approach is to derive metrics that are based on measurements of time-warpings of spike trains needed in the alignments of corresponding spikes. We propose the use of ideas developed in functional data analysis, namely the definition and separation of phase-amplitude components, as a novel tool for analyzing spike trains and decoding underlying neural signals. For concreteness, we introduce a real spike train dataset taken from experimental recordings of the primary motor cortex of a monkey while performing certain arm movements. To facilitate functional data analysis, one needs to smooth the observed discrete spike trains with Gaussian kernels.

We analyze the spike train data by means of the $k$-mean alignment algorithm in a double perspective: data as non periodic and data as periodic. In the first analysis, we show that alignment is not needed to identify paths. Indeed, without allowing for warping, we detect four clusters strongly associated to the four possible paths. In the second analysis, by exploiting the circular nature of data and allowing for shifts, we detect two clusters distinguishing between spike trains presenting higher or lower neuronal activity during the bottom-left/bottom-right movement respectively. In this latter case, the alignment procedure is able to match the four movements across paths.

We present a metric-based framework for analyzing statistical variability of the neural spike train data that was introduced in an earlier paper on this section [14]. Treating the smoothed spike trains as functional data, we apply the extended Fisher-Rao Riemannian metric, first introduced in Srivastava et al. [9], to perform: (1) pairwise alignment of spike functions, (2) averaging of multiple functions, and (3) alignment of spike functions to the mean. The last item results in separation phase and amplitude components from the functional data. Further, we utilize proper metrics on these components for classification of activities represented by spike trains. This approach is based on the square-root slope function (SRSF) representation of functions that transforms the Fisher-Rao metric into the standard $\mathbb{L}^{2}$ metric and, thus, simplifies computations. We compare our registration results with some current methods and demonstrate an application of our approach in neural decoding to infer motor behaviors.

We analyze a data set of spike trains obtained under four different experimental conditions. We model the data curves via mixtures of normal densities. The peak locations in the fitted curves are modeled via a non-homogeneous Poisson process and classification of the spike trains into groups may be done based on the estimated spacings between peaks. We employ a Bayesian, MCMC-based registration method to align the fitted curves and summarize the data using meaningful functional statistics and posterior intervals.

This paper compares the real experimental data with the data simulated by Wu and Srivastava, 2011. It turns out that the real data exhibit a different composition of the phase and the amplitude variation from the simulated data, where these two types of variation are separated using the Fisher Rao curve registration. As a result, for the real data the original functions are a better choice of data objects for path discrimination, while for the simulated data the domain warping functions are better.

This discussion of the spike train data applies the approach of Hadjipantelis et al. developed for Linguistic data to jointly model both the phase and amplitude functions via a mixed effects model. The approach shows that care needs to be taken when assessing amplitude and phase functions, particularly if separate analysis is undertaken, as there can be high levels of correlation between the warping and amplitude component functions.

In this paper we summarize and comment on different techniques presented for analyzing spike train data.

In this data introduction paper we discuss different aspects of a juggling dataset collected in 1998 at the motor control laboratory in the McGill University Department of Psychology. The juggler was asked to juggle three balls and the x, y, z position of his index finger was recorded. This data consists of ten juggling trials, each lasting 10 seconds, with 11–13 cycles per trial. We also describe a set of processing steps applied to this dataset. In particular, the raw data was smoothed and a new coordinate system was defined. Finally, we provide a few initial observations from examining the raw data.

We analyze the juggling data by means of the $k$-mean alignment algorithm using cycles as the experimental units of the analysis. Allowing for affine warping, we detect two clusters distinguishing between mainly-planar trajectories and trajectories tilted toward the body of the juggler in the lower part of the cycle. In particular we detect an anomalous presence of tilted trajectories among the trial third cycles. We also find warping functions to be clustered according to trials suggesting that each trial is performed at a different pace and thus associated to a different typical cycle-duration.

The paper considers an analysis of the juggling dataset based on registration. An elementary landmark registration is used to extract the juggling cycles from the data. The resulting cycles are then registered to functional principal components. After the registration step the paper then lays its focus on a functional principal component analysis to explain the amplitude variation of the cycles. More results about the behavior of the juggler’s movements of the hand during the juggling trials are obtained by a further investigation of the principal scores.

This paper focuses on the two one-dimensional summaries of the three-dimensional juggling data: tangential velocity and tangential acceleration. These are used jointly to define the beginnings and endings of the 123 cycles in the data divided unevenly over ten trials. Two levels of registration were used. The first was a landmark registration of each trial to a periodic image of itself with the period fixed at 712 milliseconds. The 123 tangential velocity cycles were then subjected to a continuous registration over this fixed cycle length. The amounts of across-trial and within-cycle phase variation, respectively, were surprisingly small, indicating tight neural control over the behavior. Within-cycle phase variation was primarily due to variation in the trajectories of the ball between throw and catch.

This paper describes an analysis of acceleration variability among the juggling cycles. The Fisher Rao curve registration is used for curve alignment. Five different choices of data objects are considered in this paper. We show that one of these choices of data objects leads to a much better clustering into two distinct types of juggling cycles than the other choices.

One of the purposes of the curve alignment has been to recover a structural mean of the curves by taking into account the common structural information or shape. Borrowing ideas from shape analysis, we introduce the Frenet-Serret framework to remove phase variation and to define a mean shape for three dimensional curves. Our method effectively regularizes the estimation of the geometry through curvature and torsion, and does not require curve alignment to define a mean. The method is demonstrated with the juggling data set.

We illustrate how physical constraints of a biomechanical system can be taken into account when registering functional data from juggling trials. We define an idealized model of juggling, based on a periodic joint movement in a low-dimensional space and a periodic position vector (from an undefined joint to the finger tip) of approximately constant length along the observed trajectory. Our registration procedure first warps the cycles in the trial to each other and computes a periodic average, and then estimates the joint movement and the position vector of the abovementioned model.

In this paper we present results from alignment, extraction, and statistical analysis of juggling trajectories using an elastic functional data analysis framework. This framework, specifically adapted for analyzing cyclostationary signals using an elastic Riemannian metric, was introduced recently by Kurtek et al. [2]. It relies on a special representation of curves called the square-root velocity function to pose the alignment problem as an optimization over the re-parametrization space. The cost function for alignment is a proper metric and is used to separate phase and amplitude components of juggling cycles. We present results of segmenting juggling trials into cycles, separating phase and amplitude components of cycles, and developing principal component analysis (PCA) based statistical models for these individual components.

We describe the AneuRisk65 data, obtained from image reconstruction of three-dimensional cerebral angiographies. This dataset was collected for the study of the aneurysmal pathology, within the AneuRisk Project. It includes the geometrical reconstructions of one of the main cerebral vessels, the Inner Carotid Artery, described in terms of the vessel centreline and of the vessel radius profile. We briefly illustrate the data derivation and processing, explaining various aspects that are of interest for this applied problem, while also discussing the peculiarities and critical issues concerning the definition of phase and amplitude variabilities for these three-dimensional functional data.

We describe the $k$-mean alignment procedure, for the joint alignment and clustering of functional data and we apply it to the analysis of the AneuRisk65 data. Thanks to the efficient separation of the variability in phase variability and within/between clusters amplitude variability, we are able to discriminate subjects having aneurysms in different cerebral districts and identifying different morphological shapes of Inner Carotid Arteries, unveiling a strong association between arteries morphologies and the aneurysmal pathology.

The AneuRisk65 data are analysed using methodology from statistical shape analysis. The internal carotid arteries are aligned using translation and rotation in three dimensions, together with shifts of the abscissa coordinate. Spline interpolation and weighted Procrustes methods are used to estimate the mean size-and-shapes in each of the six groups. Differences in torsion and curvature of the group means are highlighted, and permutation and bootstrap tests confirm there is weak evidence for differences in shape between the upper aneurysm group compared to the others. Finally shape variability, analysis of mean radii and classification are explored.

This paper concerns the relationship between the geometry of the Inner Carotid Artery, as described by its centerline curvature and its radius, and the location of the aneurysm for the AneuRisk65 data. Fisher Rao curve registration is used to align the curvature of the artery, and this alignment is then used to register both the curvature and the radius profiles. Based on this alignment, interesting results are found regarding the discrepancy between the arteries of patients with aneurysms at or after the terminal bifurcation (upper group) and the arteries of subjects with aneurysms before bifurcation, or without aneurysms (lower-no group).

Shapes of carotid arteries are analyzed here as elastic curves under the framework introduced in [2]. Using a mathematical representation of parameterized curves, termed square-root velocity function (SRVF), in conjunction with an elastic Riemannian metric, the framework provides (1) parameterization-invariant shape metrics for comparing curves, (2) simultaneous registration of coordinate functions across curves, and (3) computation of statistical summaries and models of shapes of given curves. The method is applicable to curves in $\mathbb{R} ^{n}$ for $n\geq1$. Thus, we study the shapes and alignments of carotid arteries using their 3D coordinates and other geometric properties along the curves, such as radii and curvatures. The results show a significant improvement in curve alignments, leading to a compact phase-amplitude PCA representation and modeling of artery data.

We analyze the AneuRisk65 curvature functions using a likelihood-based warping method for sparsely sampled curves, and combine it with logistic regression in order to discriminate subjects with aneurysms at or after the terminal bifurcation of the internal carotid artery (the most life-threatening) from subjects with no aneurysms or aneurysms along the carotid artery (the less serious). Significantly lower misclassification rates are obtained when the warping functions are included in the logistic discrimination model, rather than being treated as mere nuisance parameters.

The identification of causal effects between two groups of time series has been an important topic in a wide range of applications such as economics, engineering, medicine, neuroscience, and biology. In this paper, a simplified causal relationship (called trimmed Granger causality) based on the context of Granger causality and vector autoregressive (VAR) model is introduced. The idea is to characterize a subset of “important variables” for both groups of time series so that the underlying causal structure can be presented based on minimum variable information. When the VAR model is specified, explicit solutions are provided for the identification of important variables. When the parameters of the VAR model are unknown, an efficient statistical hypothesis testing procedure is introduced to estimate the solution. An example representing the stock indices of different countries is used to illustrate the proposed methods. In addition, a simulation study shows that the proposed methods significantly outperform the Lasso-type methods in terms of the accuracy of characterizing the simplified causal relationship.

The normal-error-based simulation extrapolation (N-SIMEX) procedure provides a simulation-based method to remove or reduce the bias in estimators of parameters in measurement error models. This paper shows that the N-SIMEX procedure only works for the normal measurement error and does not work for Laplace measurement error. A new L-SIMEX procedure is particularly designed to remove or reduce the Laplace measurement errors in parametric models. Unlike in the N-SIMEX procedure where the measurement error is removed or reduced by adding independent normal errors controlled by the scale parameter, the proposed procedure removes or reduces the Laplace measurement error by adding a noise variable which is a difference between two independent gamma random variables, and where the noise level is governed by the shape parameter. Heuristic and rigorous arguments are provided to justify the proposed method and a Jackknife-type estimation procedure is provided to estimate the variance of the L-SIMEX estimate. Simulation studies and a real data example are presented to demonstrate the proposed estimation procedure. A finite sample comparison with some revised moment estimators of Hong and Tamer is also included.

We introduce the notion of relative volatility/intermittency and demonstrate how relative volatility statistics can be used to estimate consistently the temporal variation of volatility/intermittency when the data of interest are generated by a non-semimartingale, or a Brownian semistationary process in particular. This estimation method is motivated by the assessment of relative energy dissipation in empirical data of turbulence, but it is also applicable in other areas. We develop a probabilistic asymptotic theory for realised relative power variations of Brownian semistationary processes, and introduce inference methods based on the theory. We also discuss how to extend the asymptotic theory to other classes of processes exhibiting stochastic volatility/intermittency. As an empirical application, we study relative energy dissipation in data of atmospheric turbulence.

We propose a nonparametric approach to link prediction in large-scale dynamic networks. Our model uses graph-based features of pairs of nodes as well as those of their local neighborhoods to predict whether those nodes will be linked at each time step. The model allows for different types of evolution in different parts of the graph (e.g, growing or shrinking communities). We focus on large-scale graphs and present an implementation of our model that makes use of locality-sensitive hashing to allow it to be scaled to large problems. Experiments with simulated data as well as five real-world dynamic graphs show that we outperform the state of the art, especially when sharp fluctuations or nonlinearities are present. We also establish theoretical properties of our estimator, in particular consistency and weak convergence, the latter making use of an elaboration of Stein’s method for dependency graphs.

We study the problem of constructing uniform and adaptive confidence intervals for the tail coefficient in a second order Pareto model, when the second order coefficient is unknown. This problem is translated into a testing problem on the second order parameter. By constructing an appropriate model and an associated test statistic, we provide a uniform and adaptive confidence interval for the first order parameter. We also provide an almost matching lower bound, which proves that the result is minimax optimal up to a logarithmic factor.

We consider Bayesian estimation of a $p\times p$ precision matrix, when $p$ can be much larger than the available sample size $n$. It is well known that consistent estimation in such ultra-high dimensional situations requires regularization such as banding, tapering or thresholding. We consider a banding structure in the model and induce a prior distribution on a banded precision matrix through a Gaussian graphical model, where an edge is present only when two vertices are within a given distance. For a proper choice of the order of graph, we obtain the convergence rate of the posterior distribution and Bayes estimators based on the graphical model in the $L_{\infty}$-operator norm uniformly over a class of precision matrices, even if the true precision matrix may not have a banded structure. Along the way to the proof, we also compute the convergence rate of the maximum likelihood estimator (MLE) under the same set of condition, which is of independent interest. The graphical model based MLE and Bayes estimators are automatically positive definite, which is a desirable property not possessed by some other estimators in the literature. We also conduct a simulation study to compare finite sample performance of the Bayes estimators and the MLE based on the graphical model with that obtained by using a Cholesky decomposition of the precision matrix. Finally, we discuss a practical method of choosing the order of the graphical model using the marginal likelihood function.

We consider the likelihood ratio test (LRT) process related to the test of the absence of QTL (i.e. a gene with quantitative effect on a trait) on a chromosome. We consider two different recombination models. We prove that even if the LRT is constructed from the false recombination model (i.e. the model which does not correspond to the one of the data), the maximum of the LRT process converges asymptotically to the maximum of the LRT process constructed from the true recombination model. We also prove that under some conditions, the arg max of the LRT processes will be different.

The aim of this paper is to build estimators of the infection parameter in the different phases of an epidemic (growth and decay phases). The epidemic is modeled by a Markovian process of order $d\geqslant1$, and can be written as a multitype branching process. We propose three estimators suitable for the different classes of criticality of the process, and consequently for different phases of the epidemic. We prove their consistency and asymptotic normality when the number of ancestors (resp. number of generations) tends to infinity.

For the important classical problem of inference on a sparse high-dimensional normal mean vector, we propose a novel empirical Bayes model that admits a posterior distribution with desirable properties under mild conditions. In particular, our empirical Bayes posterior distribution concentrates on balls, centered at the true mean vector, with squared radius proportional to the minimax rate, and its posterior mean is an asymptotically minimax estimator. We also show that, asymptotically, the support of our empirical Bayes posterior has roughly the same effective dimension as the true sparse mean vector. Simulation from our empirical Bayes posterior is straightforward, and our numerical results demonstrate the quality of our method compared to others having similar large-sample properties.

We are concerned with the false discovery rate (FDR) of the linear step-up test $\varphi^{LSU}$ considered by Benjamini and Hochberg (1995). It is well known that $\varphi^{LSU}$ controls the FDR at level $m_{0}q/m$ if the joint distribution of $p$-values is multivariate totally positive of order $2$. In this, $m$ denotes the total number of hypotheses, $m_{0}$ the number of true null hypotheses, and $q$ the nominal FDR level. Under the assumption of an Archimedean $p$-value copula with completely monotone generator, we derive a sharper upper bound for the FDR of $\varphi^{LSU}$ as well as a non-trivial lower bound. Application of the sharper upper bound to parametric subclasses of Archimedean $p$-value copulae allows us to increase the power of $\varphi^{LSU}$ by pre-estimating the copula parameter and adjusting $q$. Based on the lower bound, a sufficient condition is obtained under which the FDR of $\varphi^{LSU}$ is exactly equal to $m_{0}q/m$, as in the case of stochastically independent $p$-values. Finally, we deal with high-dimensional multiple test problems with exchangeable test statistics by drawing a connection between infinite sequences of exchangeable $p$-values and Archimedean copulae with completely monotone generators. Our theoretical results are applied to important copula families, including Clayton copulae and Gumbel-Hougaard copulae.

We consider the problem of estimating the parameters of discrete Markov random fields from partially observed data in a high-dimensional setting. Using a $\ell^{1}$-penalized pseudo-likelihood approach, we fit a misspecified model obtained by ignoring the missing data problem. We derive an estimation error bound that highlights the effect of the misspecification. We report some simulation results that illustrate the theoretical findings.

This paper presents a complete methodology for Bayesian inference on a semi-Markov process, from the elicitation of the prior distribution, to the computation of posterior summaries, including a guidance for its implementation. The inter-occurrence times (conditional on the transition between two given states) are assumed to be Weibull-distributed. We examine the elicitation of the joint prior density of the shape and scale parameters of the Weibull distributions, deriving a specific class of priors in a natural way, along with a method for the determination of hyperparameters based on “learning data” and moment existence conditions. This framework is applied to data of earthquakes of three types of severity (low, medium and high size) that occurred in the central Northern Apennines in Italy and collected by the CPTI04(2004) catalogue. Assumptions on two types of energy accumulation and release mechanisms are evaluated.

We establish a Glivenko-Cantelli and a Donsker theorem for a class of random discrete measures which generalize the empirical measure, under conditions on uniform entropy numbers that are common in empirical processes theory. Some illustrative applications in nonparametric Bayesian theory and randomly sized sampling are provided.

The item count technique (ICT) is a helpful tool to conduct surveys on sensitive characteristics such as tax evasion, corruption, insurance fraud, social fraud or drug consumption. The ICT yields cooperation of the respondents by protecting their privacy. There have been several interesting developments on the ICT in recent years. However, some approaches are incomplete while some research questions can not be tackled by the ICT so far. For these reasons, we broaden the existing literature in two main directions. First, we generalize the single sample count (SSC) technique, which is a simplified version of the original ICT, and derive an admissible estimate for the proportion of persons bearing a stigmatizing attribute, bootstrap variance estimates and bootstrap confidence intervals. Moreover, we present both a Bayesian and a covariate extension of the generalized SSC technique. The Bayesian set up allows the incorporation of prior information (e.g., available from a previous study) into the estimation and thus can lead to more efficient estimates. Our covariate extension is useful to conduct regression analysis, i.e., to estimate the effects of explanatory variables on the sensitive characteristic. Second, we establish a new ICT that is applicable to multicategorical sensitive variables such as the number of times a respondent has evaded taxes or the amount of money earned by undeclared work (recorded in classes). The estimation of the distribution of such attributes was not at all treated in the literature on the ICT so far. Therefore, we derive estimates for the marginal distribution of the sensitive characteristic, Bayesian estimates and regression estimates corresponding to our multicategorical ICT.

We consider the nonparametric kernel estimation of the conditional cumulative distribution function given a functional covariate. Given the bias-variance trade-off of the risk, we first propose a totally data-driven bandwidth selection mechanism in the spirit of the recent Goldenshluger-Lepski method and of model selection tools. The resulting estimator is shown to be adaptive and minimax optimal: we establish nonasymptotic risk bounds and compute rates of convergence under various assumptions on the decay of the small ball probability of the functional variable. We also prove lower bounds. Both pointwise and integrated criteria are considered. Finally, the choice of the norm or semi-norm involved in the definition of the estimator is also discussed, as well as the projection of the data on finite dimensional subspaces. Numerical results illustrate the method.

We consider quantile regression processes from censored data under dependent data structures and derive a uniform Bahadur representation for those processes. We also consider cases where the dimension of the parameter in the quantile regression model is large. It is demonstrated that traditional penalization methods such as the adaptive lasso yield sub-optimal rates if the coefficients of the quantile regression cross zero. New penalization techniques are introduced which are able to deal with specific problems of censored data and yield estimates with an optimal rate. In contrast to most of the literature, the asymptotic analysis does not require the assumption of independent observations, but is based on rather weak assumptions, which are satisfied for many kinds of dependent data.

We consider quantile estimation using Markov chain Monte Carlo and establish conditions under which the sampling distribution of the Monte Carlo error is approximately Normal. Further, we investigate techniques to estimate the associated asymptotic variance, which enables construction of an asymptotically valid interval estimator. Finally, we explore the finite sample properties of these methods through examples and provide some recommendations to practitioners.

We consider Grenander type estimators for monotone functions $f$ in a very general setting, which includes estimation of monotone regression curves, monotone densities, and monotone failure rates. These estimators are defined as the left-hand slope of the least concave majorant $\widehat{F}_{n}$ of a naive estimator $F_{n}$ of the integrated curve $F$ corresponding to $f$. We prove that the supremum distance between $\widehat{F}_{n}$ and $F_{n}$ is of the order $O_{p}(n^{-1}\logn)^{2/(4-\tau)}$, for some $\tau\in[0,4)$ that characterizes the tail probabilities of an approximating process for $F_{n}$. In typical examples, the approximating process is Gaussian and $\tau=1$, in which case the convergence rate $n^{-2/3}(\log n)^{2/3}$ is in the same spirit as the one obtained by Kiefer and Wolfowitz [9] for the special case of estimating a decreasing density. We also obtain a similar result for the primitive of $F_{n}$, in which case $\tau=2$, leading to a faster rate $n^{-1}\log n$, also found by Wang and Woodfroofe [22]. As an application in our general setup, we show that a smoothed Grenander type estimator and its derivative are asymptotically equivalent to the ordinary kernel estimator and its derivative in first order.

Recent years have seen a growing body of literature on the analysis of length-biased data. Much of this literature adopts the accelerated failure time or proportional hazards models as the basis of study. The overwhelming majority of the existing work also assumes independence between the censoring variable and covariates. In this paper, we develop a varying-coefficient quantile regression approach to model length-biased data. Our approach does not only allow the direct estimation of the conditional quantiles of survival times based on a flexible model structure, but also has the important appeal of permitting dependence between the censoring variable and the covariates. We develop local linear estimators of the coefficients using a local inverse probability weighted estimating equation approach, and examine these estimators’ asymptotic properties. Moreover, we develop a resampling method for computing the estimators’ covariances. The small sample properties of the proposed methods are investigated in a simulation study. A real data example illustrates the application of the methods in practice.

Consider a regression model in which the response is subject to random right censoring. The main goal of this paper concerns the kernel estimation of the conditional density function in the case of censored interest variable. We employ a recursive version of the Nadaraya-Watson estimator in this context. The uniform strong consistency of the recursive kernel conditional density estimator is derived. Also, we prove the asymptotic normality of this estimator.

Marked point processes are commonly used stochastic models for representing a finite number of natural hazard events located in space and time, because these kinds of data often associate measurements (i.e. marks) with locations (i.e. points) of events. Methods of marked point processes when marks and points are interacting have been proposed, but it is still necessary to know whether the interaction must be considered. This article presents a Kolmogorov-Smirnov type method to test the independence between points and marks of marked point processes. The asymptotic distribution of the test statistic under a few weak regularity conditions is derived. According to the asymptotic result, a specific way to construct the test statistic is recommended as its null distribution can be approximated by the absolute maximum of the two-dimensional standard Brownian pillow. The simulation results and real data analyses demonstrated that the proposed method is powerful in detecting weak dependence between marks and points and performs well with a moderate sample size.

We consider the horseshoe estimator due to Carvalho, Polson and Scott (2010) for the multivariate normal mean model in the situation that the mean vector is sparse in the nearly black sense. We assume the frequentist framework where the data is generated according to a fixed mean vector. We show that if the number of nonzero parameters of the mean vector is known, the horseshoe estimator attains the minimax $\ell_{2}$ risk, possibly up to a multiplicative constant. We provide conditions under which the horseshoe estimator combined with an empirical Bayes estimate of the number of nonzero means still yields the minimax risk. We furthermore prove an upper bound on the rate of contraction of the posterior distribution around the horseshoe estimator, and a lower bound on the posterior variance. These bounds indicate that the posterior distribution of the horseshoe prior may be more informative than that of other one-component priors, including the Lasso.

We propose notions of calibration for probabilistic forecasts of general multivariate quantities. Probabilistic copula calibration is a natural analogue of probabilistic calibration in the univariate setting. It can be assessed empirically by checking for the uniformity of the copula probability integral transform (CopPIT), which is invariant under coordinate permutations and coordinatewise strictly monotone transformations of the predictive distribution and the outcome. The CopPIT histogram can be interpreted as a generalization and variant of the multivariate rank histogram, which has been used to check the calibration of ensemble forecasts. Kendall calibration is an analogue of marginal calibration in the univariate case. Methods and tools are illustrated in simulation studies and applied to compare raw numerical model and statistically postprocessed ensemble forecasts of bivariate wind vectors.

In this work, consistent estimators and simultaneous confidence bands for the derivatives of mean functions are proposed when curves are repeatedly recorded for each subject. The within-curve correlation of trajectories has been considered while the proposed novel confidence bands still enjoys semiparametric efficiency. The proposed methods lead to a straightforward extension of the two-sample case in which we compare the derivatives of mean functions from two populations. We demonstrate in simulations that the proposed confidence bands are superior to existing approaches which ignore the within-curve dependence. The proposed methods are applied to investigate the derivatives of mortality rates from period lifetables that are repeatedly collected over many years for various countries.

Monte Carlo methods (based on iid sampling or Markov chains) for estimating integrals with respect to a proper target distribution (that is, a probability distribution) are well known in the statistics literature. If the target distribution $\pi$ happens to be improper then it is shown here that the standard time average estimator based on Markov chains with $\pi$ as its stationary distribution will converge to zero with probability 1, and hence is not appropriate. In this paper, we present some limit theorems for regenerative sequences and use these to develop some algorithms to produce strongly consistent estimators (called regeneration and ratio estimators) that work whether $\pi$ is proper or improper. These methods may be referred to as regenerative sequence Monte Carlo (RSMC) methods. The regenerative sequences include Markov chains as a special case. We also present an algorithm that uses the domination of the given target $\pi$ by a probability distribution $\pi_{0}$. Examples are given to illustrate the use and limitations of our algorithms.

We establish a necessary and sufficient condition for the existence of the precision matrix estimator obtained by minimizing the negative Gaussian log-likelihood plus a weighted bridge penalty. This condition enables us to connect the literature on Gaussian graphical models to the literature on penalized Gaussian likelihood.

This paper considers the problems of computing and estimating the asymptotic variance matrix of the least squares (LS) and/or the quasi-maximum likelihood (QML) estimators of vector autoregressive moving-average (VARMA) models under the assumption that the errors are uncorrelated but not necessarily independent. We firstly give expressions for the derivatives of the VARMA residuals in terms of the parameters of the models. Secondly we give an explicit expression of the asymptotic variance matrix of the QML/LS estimator, in terms of the VAR and MA polynomials, and of the second and fourth-order structure of the noise. We then deduce a consistent estimator of this asymptotic variance matrix. Modified versions of the Wald, Lagrange Multiplier and Likelihood Ratio tests are proposed for testing linear restrictions on the parameters. The theoretical results are illustrated by means Monte Carlo experiments.

This study is devoted to comparing the most popular circle fits (the geometric fit, Pratt’s, Taubin’s, Kåsa’s) and the most recently developed algebraic circle fits: hyperaccurate fit and HyperLS fit. Even though hyperaccurate fit has zero essential bias and HyperLS fit is unbiased up to order $\sigma^{4}$, the geometric fit still outperforms them in some circumstances. Since the first-order leading term of the MSE for all fits are equal, we go one step further and derive all terms of order $\sigma^{4}$, which come from essential bias, as well as all terms of order $\sigma^{4}/n$, which come from two sources: the variance and the outer product of the essential bias and the nonessential bias.

Our analysis shows that when data are distributed along a short circular arc, the covariance part is the dominant part of the second-order term in the MSE. Accordingly, the geometric fit outperforms all existing methods. However, for a long circular arc, the bias becomes the most dominant part of the second-order term, and as such, hyperaccurate fit and HyperLS fit outperform the geometric fit. We finally propose a ‘bias correction’ version of the geometric fit, which in turn, outperforms all existing methods. The new method has two features. Its variance is the smallest and has zero bias up to order $\sigma^{4}$. Our numerical tests confirm the superiority of the proposed fit over the existing fits.

Statistical inference in competing risks models is often based on the famous Aalen-Johansen estimator. Since the corresponding limit process lacks independent increments, it is typically applied together with Lin’s (1997) resampling technique involving standard normal multipliers. Recently, it has been seen that this approach can be interpreted as a wild bootstrap technique and that other multipliers, e.g. centered Poissons, may lead to better finite sample performances, see Beyersmann et al. (2013). Since the latter is closely related to Efron’s classical bootstrap, the question arises whether this or more general weighted bootstrap versions of Aalen-Johansen processes lead to valid results. Here we analyze their asymptotic behaviour and it turns out that such weighted bootstrap versions in general possess the wrong covariance structure in the limit. However, we demonstrate that the weighted bootstrap can nevertheless be applied for specific null hypotheses of interest and also discuss its limitations for statistical inference. To this end, we introduce different consistent weighted bootstrap tests for the null hypothesis of stochastically ordered cumulative incidence functions and compare their finite sample performance in a simulation study.

A fully Bayesian approach is proposed for ultrahigh-dimensional nonparametric additive models in which the number of additive components may be larger than the sample size, though ideally the true model is believed to include only a small number of components. Bayesian approaches can conduct stochastic model search and fulfill flexible parameter estimation by stochastic draws. The theory shows that the proposed model selection method has satisfactory properties. For instance, when the hyperparameter associated with the model prior is correctly specified, the true model has posterior probability approaching one as the sample size goes to infinity; when this hyperparameter is incorrectly specified, the selected model is still acceptable since asymptotically it is shown to be nested in the true model. To enhance model flexibility, two new $g$-priors are proposed and their theoretical performance is investigated. We also propose an efficient reversible jump MCMC algorithm to handle the computational issues. Several simulation examples are provided to demonstrate the advantages of our method.

We propose an adaptive smoothing method for nonparametric regression. The central idea of the proposed method is to “calibrate” the estimated function through an adaptive bandwidth function, which is a kind of intermediate solution between the global bandwidth (constant on the support) and the local bandwidth (variable with $x$). This also allows to correct the bias of the local polynomial estimator, with some benefits for the inference based on such estimators. Our method, which uses the Neural Network technique in a preliminary (pilot) stage, is based on a rolling, plug-in, bandwidth selection procedure. It automatically reaches a trade-off between the efficiency of global smoothing and the adaptability of local smoothing. The consistency and the optimal convergence rate of the resulting bandwidth estimators are shown theoretically. A simulation study shows the performance of our method for finite sample size.

We investigate the data driven choice of the cutoff parameter in density deconvolution problems with unknown error distribution. To make the target density identifiable, one has to assume that some additional information on the noise is available. We consider two different models: the framework where some additional sample of the pure noise is available, as well as the model of repeated measurements, where the contaminated random variables of interest can be observed repeatedly, with independent errors. We introduce spectral cutoff estimators and present upper risk bounds. The focus of this work lies on the optimal choice of the bandwidth by penalization strategies, leading to non-asymptotic oracle bounds.

Vertex clustering in a stochastic blockmodel graph has wide applicability and has been the subject of extensive research. In this paper, we provide a short proof that the adjacency spectral embedding can be used to obtain perfect clustering for the stochastic blockmodel and the degree-corrected stochastic blockmodel. We also show an analogous result for the more general random dot product graph model.

Minimal balanced cross over designs having lesser, equal and more periods than the number of treatments are constructed using directed m-terraces and their modified forms. A complementary pair and trio of the terraces constructs a cross over design with lesser periods while a uniform terrace yields a uniform cross over design. Two new series of cross over designs in even number of treatments have been obtained. All the designs possess good efficiency of separability and therefore they are suitable for the estimation of direct and first order carry over effects of treatments. A list of terraces for the construction of minimal balanced cross over designs having three to nine treatments is given.

The shoulder tip pain study of Lumley [13] is re-investigated. It is shown that the new logarithmic quantile estimation (LQE) technique in [9] applies and behaves well under singular covariance structure and small sample sizes as in the shoulder tip pain study. The findings in [6] can be assured under weaker assumptions using a combination of LQE and an ANOVA type statistic.

A smoothed rank-based procedure is developed for the accelerated failure time model to overcome computational issues. The proposed estimator is based on an EM-type procedure coupled with the induced smoothing. The proposed iterative approach converges provided the initial value is based on a consistent estimator, and the limiting covariance matrix can be obtained from a sandwich-type formula. The consistency and asymptotic normality of the proposed estimator are also established. Extensive simulations show that the new estimator is not only computationally less demanding but also more reliable than the other existing estimators.

Time series data obtained from neurophysiological signals is often high-dimensional and the length of the time series is often short relative to the number of dimensions. Thus, it is difficult or sometimes impossible to compute statistics that are based on the spectral density matrix because estimates of these matrices are often numerically unstable. In this work, we discuss the importance of regularization for spectral analysis of high-dimensional time series and propose shrinkage estimation for estimating high-dimensional spectral density matrices. We use and develop the multivariate Time-frequency Toggle (TFT) bootstrap procedure for multivariate time series to estimate the shrinkage parameters, and show that the multivariate TFT bootstrap is theoretically valid. We show via simulations and an fMRI data set that failure to regularize the estimates of the spectral density matrix can yield unstable statistics, and that this can be alleviated by shrinkage estimation.

Dasgupta and Shulman [1] showed that a two-round variant of the EM algorithm can learn mixture of Gaussian distributions with near optimal precision with high probability if the Gaussian distributions are well separated and if the dimension is sufficiently high. In this paper, we generalize their theory to learning mixture of high-dimensional Bernoulli templates. Each template is a binary vector, and a template generates examples by randomly switching its binary components independently with a certain probability. In computer vision applications, a binary vector is a feature map of an image, where each binary component indicates whether a local feature or structure is present or absent within a certain cell of the image domain. A Bernoulli template can be considered as a statistical model for images of objects (or parts of objects) from the same category. We show that the two-round EM algorithm can learn mixture of Bernoulli templates with near optimal precision with high probability, if the Bernoulli templates are sufficiently different and if the number of features is sufficiently high. We illustrate the theoretical results by synthetic and real examples.

This study aims at contributing to lower bounds for empirical compatibility constants or empirical restricted eigenvalues. This is of importance in compressed sensing and theory for $\ell_{1}$-regularized estimators. Let $X$ be an $n\times p$ data matrix with rows being independent copies of a $p$-dimensional random variable. Let $\hat{\Sigma}:=X^{T}X/n$ be the inner product matrix. We show that the quadratic forms $u^{T}\hat{\Sigma}u$ are lower bounded by a value converging to one, uniformly over the set of vectors $u$ with $u^{T}\Sigma_{0}u$ equal to one and $\ell_{1}$-norm at most $M$. Here $\Sigma_{0}:=\mathbb{E} \hat{\Sigma}$ is the theoretical inner product matrix, which we assume to exist. The constant $M$ is required to be of small order $\sqrt{n/\log p}$. We assume moreover $m$-th order isotropy for some $m>2$ and sub-exponential tails or moments up to order $\log p$ for the entries in $X$. As a consequence, we obtain convergence of the empirical compatibility constant to its theoretical counterpart, and similarly for the empirical restricted eigenvalue. If the data matrix $X$ is first normalized so that its columns all have equal length we obtain lower bounds assuming only isotropy and no further moment conditions on its entries. The isotropy condition is shown to hold for certain martingale situations.

Linear mixed-effect models with two variance components are often used when variability comes from two sources. In genetics applications, variation in observed traits can be attributed to biological and environmental effects, and the heritability coefficient is a fundamental quantity that measures the proportion of total variability due to the biological effect. We propose a new inferential model approach which yields exact prior-free probabilistic inference on the heritability coefficient. In particular we construct exact confidence intervals and demonstrate numerically our method’s efficiency compared to that of existing methods.

This paper revisits the classical inference results for profile quasi maximum likelihood estimators (profile MLE) in semiparametric models. We mainly focus on two prominent theorems: the Wilks phenomenon and Fisher expansion for the profile MLE are stated in a new fashion allowing finite samples and model misspecification. The method of study is also essentially different from the usual analysis of the semiparametric problem based on the notion of the hardest parametric submodel. Instead we derive finite sample deviation bounds for the linear approximation error for the gradient of the loglikelihood. This novel approach particularly allows to address the impact of the effective target and nuisance dimension on the accuracy of the results. The obtained nonasymptotic results are surprisingly sharp and yield the classical asymptotic statements including the asymptotic normality and efficiency of the profile MLE. The general results are specified for the important special case of an i.i.d. sample and the analysis is exemplified with a single index model.

Let $M_{n}$ be the maximum of $n$ unit Gaussian variables $X_{1},\ldots,X_{n}$ with correlation matrix having minimum eigenvalue $\lambda_{n}$. Then \[M_{n}\ge\lambda_{n}\sqrt{2\log n}+o_{p}(1).\] As an application, the log likelihood ratio statistic testing for the presence of two components in a 1-dimensional exponential family mixture, with one component known, is shown to be at least $\frac{1} {2}\log\log n(1+o_{p}(n))$ under the null hypothesis that there is only one component.

In this paper, we consider the problem of selecting the most appropriate model, amongst a given collection of mixture models, to describe datasets likely drawn from mixture of distributions. The proposed method consists of finding the quasi-maximum likelihood estimators (QMLEs) of the various models in competition, using the Expectation-Maximization (EM) type algorithms, and subsequently estimating, for every model, a statistical distance to the true model based on the empirical cumulative distribution function (cdf) of the original dataset and the QMLE-fitted cdf. To evaluate the goodness of fit, a new metric, the Integrated Cumulative Error ($ICE$) is proposed and compared with other existing metrics for accuracy of detecting the appropriate model. We state, under mild conditions, that our estimator of the $ICE$ distance converges at the rate $\sqrt{n}$ in probability along with the consistency of our model selection procedure (ability to detect asymptotically the right model). The $ICE$ criterion shows, over a set of benchmark examples, numerically improved performance from the existing distance-based criteria in identifying the correct model. The method is applied in a material fatigue life context to model the distribution of indicators of the fatigue crack formation potency, obtained from numerical experiments.

A simulation model of a complex system is considered, for which the outcome is described by $m(p,X)$, where $p$ is a parameter of the system, $X$ is a random input of the system and $m$ is a real-valued function. The maximum (with respect to $p$) of the quantiles of $m(p,X)$ is estimated. The quantiles of $m(p,X)$ of a given level are estimated for various values of $p$ from an order statistic of values $m(p_{i},X_{i})$ where $X,X_{1},X_{2},\dots$ are independent and identically distributed and where $p_{i}$ is close to $p$, and the maximal quantile is estimated by the maximum of these quantile estimates. Under assumptions on the smoothness of the function describing the dependency of the values of the quantiles on the parameter $p$ the rate of convergence of this estimate is analyzed. The finite sample size behavior of the estimate is illustrated by simulated data and by applying it in a simulation model of a real mechanical system.