Federico Castelletti, Guido Consonni.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2289--2311.

Abstract:
A signalling pathway is a sequence of chemical reactions initiated by a stimulus which in turn affects a receptor, and then through some intermediate steps cascades down to the final cell response. Based on the technique of flow cytometry, samples of cell-by-cell measurements are collected under each experimental condition, resulting in a collection of interventional data (assuming no latent variables are involved). Usually several external interventions are applied at different points of the pathway, the ultimate aim being the structural recovery of the underlying signalling network which we model as a causal Directed Acyclic Graph (DAG) using intervention calculus. The advantage of using interventional data, rather than purely observational one, is that identifiability of the true data generating DAG is enhanced. More technically a Markov equivalence class of DAGs, whose members are statistically indistinguishable based on observational data alone, can be further decomposed, using additional interventional data, into smaller distinct Interventional Markov equivalence classes. We present a Bayesian methodology for structural learning of Interventional Markov equivalence classes based on observational and interventional samples of multivariate Gaussian observations. Our approach is objective, meaning that it is based on default parameter priors requiring no personal elicitation; some flexibility is however allowed through a tuning parameter which regulates sparsity in the prior on model space. Based on an analytical expression for the marginal likelihood of a given Interventional Essential Graph, and a suitable MCMC scheme, our analysis produces an approximate posterior distribution on the space of Interventional Markov equivalence classes, which can be used to provide uncertainty quantification for features of substantive scientific interest, such as the posterior probability of inclusion of selected edges, or paths.

Ningshan Zhang, Kyle Schmaus, Patrick O. Perry.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2260--2288.

Abstract:
We consider a particular instance of a common problem in recommender systems, using a database of book reviews to inform user-targeted recommendations. In our dataset, books are categorized into genres and subgenres. To exploit this nested taxonomy, we use a hierarchical model that enables information pooling across across similar items at many levels within the genre hierarchy. The main challenge in deploying this model is computational. The data sizes are large and fitting the model at scale using off-the-shelf maximum likelihood procedures is prohibitive. To get around this computational bottleneck, we extend a moment-based fitting procedure proposed for fitting single-level hierarchical models to the general case of arbitrarily deep hierarchies. This extension is an order of magnitude faster than standard maximum likelihood procedures. The fitting method can be deployed beyond recommender systems to general contexts with deeply nested hierarchical generalized linear mixed models.

Emily Berg, Danhyang Lee.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2158--2188.

Abstract:
Quantiles of the distributions of several measures of erosion are important parameters in the Conservation Effects Assessment Project, a survey intended to quantify soil and nutrient loss on crop fields. Because sample sizes for domains of interest are too small to support reliable direct estimators, model based methods are needed. Quantile regression is appealing for CEAP because finding a single family of parametric models that adequately describes the distributions of all variables is difficult and small area quantiles are parameters of interest. We construct empirical Bayes predictors and bootstrap mean squared error estimators based on the linearly interpolated generalized Pareto distribution (LIGPD). We apply the procedures to predict county-level quantiles for four types of erosion in Wisconsin and validate the procedures through simulation.

Jose Ameijeiras-Alonso, Akli Benali, Rosa M. Crujeiras, Alberto Rodríguez-Casal, José M. C. Pereira.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2120--2139.

Abstract:
Understanding the role of vegetation fires in the Earth system is an important environmental problem. Although fire occurrence is influenced by natural factors, human activity related to land use and management has altered the temporal patterns of fire in several regions of the world. Hence, for a better insight into fires regimes it is of special interest to analyze where human activity has altered fire seasonality. For doing so, multimodality tests are a useful tool for determining the number of annual fire peaks. The periodicity of fires and their complex distributional features motivate the use of nonparametric circular statistics. The unsatisfactory performance of previous circular nonparametric proposals for testing multimodality justifies the introduction of a new approach, considering an adapted version of the excess mass statistic, jointly with a bootstrap calibration algorithm. A systematic application of the test on the Russia–Kazakhstan area is presented in order to determine how many fire peaks can be identified in this region. A False Discovery Rate correction, accounting for the spatial dependence of the data, is also required.

Jason Xu, Samson Koelle, Peter Guttorp, Chuanfeng Wu, Cynthia Dunbar, Janis L. Abkowitz, Vladimir N. Minin.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2091--2119.

Abstract:
Single-cell lineage tracking strategies enabled by recent experimental technologies have produced significant insights into cell fate decisions, but lack the quantitative framework necessary for rigorous statistical analysis of mechanistic models describing cell division and differentiation. In this paper, we develop such a framework with corresponding moment-based parameter estimation techniques for continuous-time, multi-type branching processes. Such processes provide a probabilistic model of how cells divide and differentiate, and we apply our method to study hematopoiesis , the mechanism of blood cell production. We derive closed-form expressions for higher moments in a general class of such models. These analytical results allow us to efficiently estimate parameters of much richer statistical models of hematopoiesis than those used in previous statistical studies. To our knowledge, the method provides the first rate inference procedure for fitting such models to time series data generated from cellular barcoding experiments. After validating the methodology in simulation studies, we apply our estimator to hematopoietic lineage tracking data from rhesus macaques. Our analysis provides a more complete understanding of cell fate decisions during hematopoiesis in nonhuman primates, which may be more relevant to human biology and clinical strategies than previous findings from murine studies. For example, in addition to previously estimated hematopoietic stem cell self-renewal rate, we are able to estimate fate decision probabilities and to compare structurally distinct models of hematopoiesis using cross validation. These estimates of fate decision probabilities and our model selection results should help biologists compare competing hypotheses about how progenitor cells differentiate. The methodology is transferrable to a large class of stochastic compartmental and multi-type branching models, commonly used in studies of cancer progression, epidemiology and many other fields.

Gabriela V. Cohen Freue, David Kepplinger, Matías Salibián-Barrera, Ezequiel Smucler.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2065--2090.

Abstract:
In large-scale quantitative proteomic studies, scientists measure the abundance of thousands of proteins from the human proteome in search of novel biomarkers for a given disease. Penalized regression estimators can be used to identify potential biomarkers among a large set of molecular features measured. Yet, the performance and statistical properties of these estimators depend on the loss and penalty functions used to define them. Motivated by a real plasma proteomic biomarkers study, we propose a new class of penalized robust estimators based on the elastic net penalty, which can be tuned to keep groups of correlated variables together in the selected model and maintain robustness against possible outliers. We also propose an efficient algorithm to compute our robust penalized estimators and derive a data-driven method to select the penalty term. Our robust penalized estimators have very good robustness properties and are also consistent under certain regularity conditions. Numerical results show that our robust estimators compare favorably to other robust penalized estimators. Using our proposed methodology for the analysis of the proteomics data, we identify new potentially relevant biomarkers of cardiac allograft vasculopathy that are not found with nonrobust alternatives. The selected model is validated in a new set of 52 test samples and achieves an area under the receiver operating characteristic (AUC) of 0.85.

Ye Zhang, Zhigang Yao, Patrik Forssén, Torgny Fornstedt.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2011--2042.

Abstract:
The means to obtain the rate constants of a chemical reaction is a fundamental open problem in both science and the industry. Traditional techniques for finding rate constants require either chemical modifications of the reactants or indirect measurements. The rate constant map method is a modern technique to study binding equilibrium and kinetics in chemical reactions. Finding a rate constant map from biosensor data is an ill-posed inverse problem that is usually solved by regularization. In this work, rather than finding a deterministic regularized rate constant map that does not provide uncertainty quantification of the solution, we develop an adaptive variational Bayesian approach to estimate the distribution of the rate constant map, from which some intrinsic properties of a chemical reaction can be explored, including information about rate constants. Our new approach is more realistic than the existing approaches used for biosensors and allows us to estimate the dynamics of the interactions, which are usually hidden in a deterministic approximate solution. We verify the performance of the new proposed method by numerical simulations, and compare it with the Markov chain Monte Carlo algorithm. The results illustrate that the variational method can reliably capture the posterior distribution in a computationally efficient way. Finally, the developed method is also tested on the real biosensor data (parathyroid hormone), where we provide two novel analysis tools—the thresholding contour map and the high order moment map—to estimate the number of interactions as well as their rate constants.

Bret Zeldow, Vincent Lo Re III, Jason Roy.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1989--2010.

Abstract:
Bayesian Additive Regression Trees (BART) is a flexible machine learning algorithm capable of capturing nonlinearities between an outcome and covariates and interactions among covariates. We extend BART to a semiparametric regression framework in which the conditional expectation of an outcome is a function of treatment, its effect modifiers, and confounders. The confounders are allowed to have unspecified functional form, while treatment and effect modifiers that are directly related to the research question are given a linear form. The result is a Bayesian semiparametric linear regression model where the posterior distribution of the parameters of the linear part can be interpreted as in parametric Bayesian regression. This is useful in situations where a subset of the variables are of substantive interest and the others are nuisance variables that we would like to control for. An example of this occurs in causal modeling with the structural mean model (SMM). Under certain causal assumptions, our method can be used as a Bayesian SMM. Our methods are demonstrated with simulation studies and an application to dataset involving adults with HIV/Hepatitis C coinfection who newly initiate antiretroviral therapy. The methods are available in an R package called semibart.

Youyi Zhang, Jeffrey S. Morris, Shivali Narang Aerry, Arvind U. K. Rao, Veerabhadran Baladandayuthapani.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1957--1988.

Abstract:
Technological innovations have produced large multi-modal datasets that include imaging and multi-platform genomics data. Integrative analyses of such data have the potential to reveal important biological and clinical insights into complex diseases like cancer. In this paper, we present Bayesian approaches for integrative analysis of radiological imaging and multi-platform genomic data, where-in our goals are to simultaneously identify genomic and radiomic, that is, radiology-based imaging markers, along with the latent associations between these two modalities, and to detect the overall prognostic relevance of the combined markers. For this task, we propose Radio-iBAG: Radiomics-based Integrative Bayesian Analysis of Multiplatform Genomic Data , a multi-scale Bayesian hierarchical model that involves several innovative strategies: it incorporates integrative analysis of multi-platform genomic data sets to capture fundamental biological relationships; explores the associations between radiomic markers accompanying genomic information with clinical outcomes; and detects genomic and radiomic markers associated with clinical prognosis. We also introduce the use of sparse Principal Component Analysis (sPCA) to extract a sparse set of approximately orthogonal meta-features each containing information from a set of related individual radiomic features, reducing dimensionality and combining like features. Our methods are motivated by and applied to The Cancer Genome Atlas glioblastoma multiforme data set, where-in we integrate magnetic resonance imaging-based biomarkers along with genomic, epigenomic and transcriptomic data. Our model identifies important magnetic resonance imaging features and the associated genomic platforms that are related with patient survival times.

Arkaprava Roy, Subhashis Ghosal, Jeffrey Prescott, Kingshuk Roy Choudhury.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1791--1816.

Abstract:
We study possible relations between Alzheimer’s disease progression and the structure of the connectome which is white matter connecting different regions of the brain. Regression models in covariates including age, gender and disease status for the extent of white matter connecting each pair of regions of the brain are proposed. Subject inhomogeneity is also incorporated in the model through random effects with an unknown distribution. As there is a large number of pairs of regions, we also adopt a dimension reduction technique through graphon ( J. Combin. Theory Ser. B 96 (2006) 933–957) functions which reduces the functions of pairs of regions to functions of regions. The connecting graphon functions are considered unknown but the assumed smoothness allows putting priors of low complexity on these functions. We pursue a nonparametric Bayesian approach by assigning a Dirichlet process scale mixture of zero to mean normal prior on the distributions of the random effects and finite random series of tensor products of B-splines priors on the underlying graphon functions. We develop efficient Markov chain Monte Carlo techniques for drawing samples for the posterior distributions using Hamiltonian Monte Carlo (HMC). The proposed Bayesian method overwhelmingly outperforms a competing method based on ANCOVA models in the simulation setup. The proposed Bayesian approach is applied on a dataset of 100 subjects and 83 brain regions and key regions implicated in the changing connectome are identified.

Yiyi Liu, Joshua L. Warren, Hongyu Zhao.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1733--1752.

Abstract:
Understanding the heterogeneity of cells is an important biological question. The development of single-cell RNA-sequencing (scRNA-seq) technology provides high resolution data for such inquiry. A key challenge in scRNA-seq analysis is the high variability of measured RNA expression levels and frequent dropouts (missing values) due to limited input RNA compared to bulk RNA-seq measurement. Existing clustering methods do not perform well for these noisy and zero-inflated scRNA-seq data. In this manuscript we propose a Bayesian hierarchical model, called BasClu, to appropriately characterize important features of scRNA-seq data in order to more accurately cluster cells. We demonstrate the effectiveness of our method with extensive simulation studies and applications to three real scRNA-seq datasets.

Seong-Hwan Jun, Samuel W. K. Wong, James V. Zidek, Alexandre Bouchard-Côté.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1678--1707.

Abstract:
In this paper, we consider the knot-matching problem arising in computational forestry. The knot-matching problem is an important problem that needs to be solved to advance the state of the art in automatic strength prediction of lumber. We show that this problem can be formulated as a quadripartite matching problem and develop a sequential decision model that admits efficient parameter estimation along with a sequential Monte Carlo sampler on graph matching that can be utilized for rapid sampling of graph matching. We demonstrate the effectiveness of our methods on 30 manually annotated boards and present findings from various simulation studies to provide further evidence supporting the efficacy of our methods.

Ying Chen, J. S. Marron, Jiejie Zhang.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1590--1616.

Abstract:
Electricity prices are high dimensional, serially dependent and have seasonal variations. We propose a Warping Functional AutoRegressive (WFAR) model that simultaneously accounts for the cross time-dependence and seasonal variations of the large dimensional data. In particular, electricity price curves are obtained by smoothing over the $24$ discrete hourly prices on each day. In the functional domain, seasonal phase variations are separated from level amplitude changes in a warping process with the Fisher–Rao distance metric, and the aligned (season-adjusted) electricity price curves are modeled in the functional autoregression framework. In a real application, the WFAR model provides superior out-of-sample forecast accuracy in both a normal functioning market, Nord Pool, and an extreme situation, the California market. The forecast performance as well as the relative accuracy improvement are stable for different markets and different time periods.

Johann Gagnon-Bartsch, Yotam Shem-Tov.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1464--1483.

Abstract:
The gold standard for identifying causal relationships is a randomized controlled experiment. In many applications in the social sciences and medicine, the researcher does not control the assignment mechanism and instead may rely upon natural experiments or matching methods as a substitute to experimental randomization. The standard testable implication of random assignment is covariate balance between the treated and control units. Covariate balance is commonly used to validate the claim of as good as random assignment. We propose a new nonparametric test of covariate balance. Our Classification Permutation Test (CPT) is based on a combination of classification methods (e.g., random forests) with Fisherian permutation inference. We revisit four real data examples and present Monte Carlo power simulations to demonstrate the applicability of the CPT relative to other nonparametric tests of equality of multivariate distributions.

Jeng-Min Chiou, Yu-Ting Chen, Tailen Hsing.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1430--1463.

Abstract:
Motivated by the study of road segmentation partitioned by shifts in traffic conditions along a freeway, we introduce a two-stage procedure, Dynamic Segmentation and Backward Elimination (DSBE), for identifying multiple changes in the mean functions for a sequence of functional data. The Dynamic Segmentation procedure searches for all possible changepoints using the derived global optimality criterion coupled with the local strategy of at-most-one-changepoint by dividing the entire sequence into individual subsequences that are recursively adjusted until convergence. Then, the Backward Elimination procedure verifies these changepoints by iteratively testing the unlikely changes to ensure their significance until no more changepoints can be removed. By combining the local strategy with the global optimal changepoint criterion, the DSBE algorithm is conceptually simple and easy to implement and performs better than the binary segmentation-based approach at detecting small multiple changes. The consistency property of the changepoint estimators and the convergence of the algorithm are proved. We apply DSBE to detect changes in traffic streams through real freeway traffic data. The practical performance of DSBE is also investigated through intensive simulation studies for various scenarios.

Lin Liu, Yuqi Qiu, Loki Natarajan, Karen Messer.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1370--1396.

Abstract:
It is common to encounter missing data among the potential predictor variables in the setting of model selection. For example, in a recent study we attempted to improve the US guidelines for risk stratification after screening colonoscopy ( Cancer Causes Control 27 (2016) 1175–1185), with the aim to help reduce both overuse and underuse of follow-on surveillance colonoscopy. The goal was to incorporate selected additional informative variables into a neoplasia risk-prediction model, going beyond the three currently established risk factors, using a large dataset pooled from seven different prospective studies in North America. Unfortunately, not all candidate variables were collected in all studies, so that one or more important potential predictors were missing on over half of the subjects. Thus, while variable selection was a main focus of the study, it was necessary to address the substantial amount of missing data. Multiple imputation can effectively address missing data, and there are also good approaches to incorporate the variable selection process into model-based confidence intervals. However, there is not consensus on appropriate methods of inference which address both issues simultaneously. Our goal here is to study the properties of model-based confidence intervals in the setting of imputation for missing data followed by variable selection. We use both simulation and theory to compare three approaches to such post-imputation-selection inference: a multiple-imputation approach based on Rubin’s Rules for variance estimation ( Comput. Statist. Data Anal. 71 (2014) 758–770); a single imputation-selection followed by bootstrap percentile confidence intervals; and a new bootstrap model-averaging approach presented here, following Efron ( J. Amer. Statist. Assoc. 109 (2014) 991–1007). We investigate relative strengths and weaknesses of each method. The “Rubin’s Rules” multiple imputation estimator can have severe undercoverage, and is not recommended. The imputation-selection estimator with bootstrap percentile confidence intervals works well. The bootstrap-model-averaged estimator, with the “Efron’s Rules” estimated variance, may be preferred if the true effect sizes are moderate. We apply these results to the colorectal neoplasia risk-prediction problem which motivated the present work.

Jong Yun Hwang, Ji Oon Lee, Wooseok Yang.

Source: Bernoulli, Volume 26, Number 3, 2400--2435.

Abstract:
We consider the spectral properties of sparse stochastic block models, where $N$ vertices are partitioned into $K$ balanced communities. Under an assumption that the intra-community probability and inter-community probability are of similar order, we prove a local semicircle law up to the spectral edges, with an explicit formula on the deterministic shift of the spectral edge. We also prove that the fluctuation of the extremal eigenvalues is given by the GOE Tracy–Widom law after rescaling and centering the entries of sparse stochastic block models. Applying the result to sparse stochastic block models, we rigorously prove that there is a large gap between the outliers and the spectral edge without centering.

Piotr Kokoszka, Neda Mohammadi Jouzdani.

Source: Bernoulli, Volume 26, Number 3, 2383--2399.

Abstract:
This paper is concerned with frequency domain theory for functional time series, which are temporally dependent sequences of functions in a Hilbert space. We consider a variance decomposition, which is more suitable for such a data structure than the variance decomposition based on the Karhunen–Loéve expansion. The decomposition we study uses eigenvalues of spectral density operators, which are functional analogs of the spectral density of a stationary scalar time series. We propose estimators of the variance components and derive convergence rates for their mean square error as well as their asymptotic normality. The latter is derived from a frequency domain invariance principle for the estimators of the spectral density operators. This principle is established for a broad class of linear time series models. It is a main contribution of the paper.

Bo Ning, Seonghyun Jeong, Subhashis Ghosal.

Source: Bernoulli, Volume 26, Number 3, 2353--2382.

Abstract:
We study frequentist properties of a Bayesian high-dimensional multivariate linear regression model with correlated responses. The predictors are separated into many groups and the group structure is pre-determined. Two features of the model are unique: (i) group sparsity is imposed on the predictors; (ii) the covariance matrix is unknown and its dimensions can also be high. We choose a product of independent spike-and-slab priors on the regression coefficients and a new prior on the covariance matrix based on its eigendecomposition. Each spike-and-slab prior is a mixture of a point mass at zero and a multivariate density involving the $ell_{2,1}$-norm. We first obtain the posterior contraction rate, the bounds on the effective dimension of the model with high posterior probabilities. We then show that the multivariate regression coefficients can be recovered under certain compatibility conditions. Finally, we quantify the uncertainty for the regression coefficients with frequentist validity through a Bernstein–von Mises type theorem. The result leads to selection consistency for the Bayesian method. We derive the posterior contraction rate using the general theory by constructing a suitable test from the first principle using moment bounds for certain likelihood ratios. This leads to posterior concentration around the truth with respect to the average Rényi divergence of order $1/2$. This technique of obtaining the required tests for posterior contraction rate could be useful in many other problems.

Sarat B. Moka, Dirk P. Kroese.

Source: Bernoulli, Volume 26, Number 3, 2082--2104.

Abstract:
We present a perfect sampling algorithm for Gibbs point processes, based on the partial rejection sampling of Guo, Jerrum and Liu (In STOC’17 – Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (2017) 342–355 ACM). Our particular focus is on pairwise interaction processes, penetrable spheres mixture models and area-interaction processes, with a finite interaction range. For an interaction range $2r$ of the target process, the proposed algorithm can generate a perfect sample with $O(log(1/r))$ expected running time complexity, provided that the intensity of the points is not too high and $Theta(1/r^{d})$ parallel processor units are available.

Adriana Coutinho, Rodrigo Lambert, Jérôme Rousseau.

Source: Bernoulli, Volume 26, Number 3, 2021--2050.

Abstract:
We investigate the length of the longest common substring for encoded sequences and its asymptotic behaviour. The main result is a strong law of large numbers for a re-scaled version of this quantity, which presents an explicit relation with the Rényi entropy of the source. We apply this result to the zero-inflated contamination model and the stochastic scrabble. In the case of dynamical systems, this problem is equivalent to the shortest distance between two observed orbits and its limiting relationship with the correlation dimension of the pushforward measure. An extension to the shortest distance between orbits for random dynamical systems is also provided.

O. Collier, L. Comminges, A.B. Tsybakov.

Source: Bernoulli, Volume 26, Number 3, 1989--2020.

Abstract:
We study the problem of estimation of $N_{gamma }( heta )=sum_{i=1}^{d}| heta _{i}|^{gamma }$ for $gamma >0$ and of the $ell _{gamma }$-norm of $ heta $ for $gamma ge 1$ based on the observations $y_{i}= heta _{i}+varepsilon xi _{i}$, $i=1,ldots,d$, where $ heta =( heta _{1},dots , heta _{d})$ are unknown parameters, $varepsilon >0$ is known, and $xi _{i}$ are i.i.d. standard normal random variables. We find the non-asymptotic minimax rate for estimation of these functionals on the class of $s$-sparse vectors $ heta $ and we propose estimators achieving this rate.

Tom Alberts, Firas Rassoul-Agha, Mackenzie Simper.

Source: Bernoulli, Volume 26, Number 3, 1927--1955.

Abstract:
We prove that given any fixed asymptotic velocity, the finite length O’Connell–Yor polymer has an infinite length limit satisfying the law of large numbers with this velocity. By a Markovian property of the quenched polymer this reduces to showing the existence of Busemann functions : almost sure limits of ratios of random point-to-point partition functions. The key ingredients are the Burke property of the O’Connell–Yor polymer and a comparison lemma for the ratios of partition functions. We also show the existence of infinite length limits in the Brownian last passage percolation model.

Ulrike Mayer, Henryk Zähle, Zhou Zhou.

Source: Bernoulli, Volume 26, Number 3, 1891--1911.

Abstract:
We derive a functional weak limit theorem for a local empirical process of a wide class of piece-wise locally stationary (PLS) time series. The latter result is applied to derive the asymptotics of weighted empirical quantiles and weighted V-statistics of non-stationary time series. The class of admissible underlying time series is illustrated by means of PLS linear processes and PLS ARCH processes.

Sébastien Gadat, Sébastien Gerchinovitz, Clément Marteau.

Source: Bernoulli, Volume 26, Number 3, 1797--1831.

Abstract:
We consider the binary supervised classification problem with the Gaussian functional model introduced in ( Math. Methods Statist. 22 (2013) 213–225). Taking advantage of the Gaussian structure, we design a natural plug-in classifier and derive a family of upper bounds on its worst-case excess risk over Sobolev spaces. These bounds are parametrized by a separation distance quantifying the difficulty of the problem, and are proved to be optimal (up to logarithmic factors) through matching minimax lower bounds. Using the recent works of (In Advances in Neural Information Processing Systems (2014) 3437–3445 Curran Associates) and ( Ann. Statist. 44 (2016) 982–1009), we also derive a logarithmic lower bound showing that the popular $k$-nearest neighbors classifier is far from optimality in this specific functional setting.

David Corlin Marchand, Ioan Manolescu.

Source: Bernoulli, Volume 26, Number 3, 1665--1705.

Abstract:
We study randomly growing trees governed by the affine preferential attachment rule. Starting with a seed tree $S$, vertices are attached one by one, each linked by an edge to a random vertex of the current tree, chosen with a probability proportional to an affine function of its degree. This yields a one-parameter family of preferential attachment trees $(T_{n}^{S})_{ngeq |S|}$, of which the linear model is a particular case. Depending on the choice of the parameter, the power-laws governing the degrees in $T_{n}^{S}$ have different exponents. We study the problem of the asymptotic influence of the seed $S$ on the law of $T_{n}^{S}$. We show that, for any two distinct seeds $S$ and $S'$, the laws of $T_{n}^{S}$ and $T_{n}^{S'}$ remain at uniformly positive total-variation distance as $n$ increases. This is a continuation of Curien et al. ( J. Éc. Polytech. Math. 2 (2015) 1–34), which in turn was inspired by a conjecture of Bubeck et al. ( IEEE Trans. Netw. Sci. Eng. 2 (2015) 30–39). The technique developed here is more robust than previous ones and is likely to help in the study of more general attachment mechanisms.

Kamran Kalbasi, Thomas Mountford.

Source: Bernoulli, Volume 26, Number 2, 1504--1534.

Abstract:
In this paper, we study the local times of vector-valued Gaussian fields that are ‘diagonally operator-self-similar’ and whose increments are stationary. Denoting the local time of such a Gaussian field around the spatial origin and over the temporal unit hypercube by $Z$, we show that there exists $lambdain(0,1)$ such that under some quite weak conditions, $lim_{n ightarrow+infty}frac{sqrt[n]{mathbb{E}(Z^{n})}}{n^{lambda}}$ and $lim_{x ightarrow+infty}frac{-logmathbb{P}(Z>x)}{x^{frac{1}{lambda}}}$ both exist and are strictly positive (possibly $+infty$). Moreover, we show that if the underlying Gaussian field is ‘strongly locally nondeterministic’, the above limits will be finite as well. These results are then applied to establish similar statements for the intersection local times of diagonally operator-self-similar Gaussian fields with stationary increments.

Martin Kolb, Mladen Savov.

Source: Bernoulli, Volume 26, Number 2, 1453--1472.

Abstract:
We derive a criterium for the almost sure finiteness of perpetual integrals of Lévy processes for a class of real functions including all continuous functions and for general one-dimensional Lévy processes that drifts to plus infinity. This generalizes previous work of Döring and Kyprianou, who considered Lévy processes having a local time, leaving the general case as an open problem. It turns out, that the criterium in the general situation simplifies significantly in the situation, where the process has a local time, but we also demonstrate that in general our criterium can not be reduced. This answers an open problem posed in ( J. Theoret. Probab. 29 (2016) 1192–1198).

Pasha Tkachov.

Source: Bernoulli, Volume 26, Number 2, 1354--1380.

Abstract:
The aim of this paper is to prove stability of traveling waves for integro-differential equations connected with branching Markov processes. In other words, the limiting law of the left-most particle of a (time-continuous) branching Markov process with a Lévy non-branching part is demonstrated. The key idea is to approximate the branching Markov process by a branching random walk and apply the result of Aïdékon [ Ann. Probab. 41 (2013) 1362–1426] on the limiting law of the latter one.

Denis Talay, Milica Tomašević.

Source: Bernoulli, Volume 26, Number 2, 1323--1353.

Abstract:
In this paper, we analyze a stochastic interpretation of the one-dimensional parabolic–parabolic Keller–Segel system without cut-off. It involves an original type of McKean–Vlasov interaction kernel. At the particle level, each particle interacts with all the past of each other particle by means of a time integrated functional involving a singular kernel. At the mean-field level studied here, the McKean–Vlasov limit process interacts with all the past time marginals of its probability distribution in a similarly singular way. We prove that the parabolic–parabolic Keller–Segel system in the whole Euclidean space and the corresponding McKean–Vlasov stochastic differential equation are well-posed for any values of the parameters of the model.

Emanuele Dolera, Stefano Favaro.

Source: Bernoulli, Volume 26, Number 2, 1294--1322.

Abstract:
Given a sequence ${X_{n}}_{ngeq 1}$ of exchangeable Bernoulli random variables, the celebrated de Finetti representation theorem states that $frac{1}{n}sum_{i=1}^{n}X_{i}stackrel{a.s.}{longrightarrow }Y$ for a suitable random variable $Y:Omega ightarrow [0,1]$ satisfying $mathsf{P}[X_{1}=x_{1},dots ,X_{n}=x_{n}|Y]=Y^{sum_{i=1}^{n}x_{i}}(1-Y)^{n-sum_{i=1}^{n}x_{i}}$. In this paper, we study the rate of convergence in law of $frac{1}{n}sum_{i=1}^{n}X_{i}$ to $Y$ under the Kolmogorov distance. After showing that a rate of the type of $1/n^{alpha }$ can be obtained for any index $alpha in (0,1]$, we find a sufficient condition on the distribution of $Y$ for the achievement of the optimal rate of convergence, that is $1/n$. Besides extending and strengthening recent results under the weaker Wasserstein distance, our main result weakens the regularity hypotheses on $Y$ in the context of the Hausdorff moment problem.

Nina Gantert, Markus Heydenreich, Timo Hirscher.

Source: Bernoulli, Volume 26, Number 2, 1269--1293.

Abstract:
We investigate a model for opinion dynamics, where individuals (modeled by vertices of a graph) hold certain abstract opinions. As time progresses, neighboring individuals interact with each other, and this interaction results in a realignment of opinions closer towards each other. This mechanism triggers formation of consensus among the individuals. Our main focus is on strong consensus (i.e., global agreement of all individuals) versus weak consensus (i.e., local agreement among neighbors). By extending a known model to a more general opinion space, which lacks a “central” opinion acting as a contraction point, we provide an example of an opinion formation process on the one-dimensional lattice $mathbb{Z}$ with weak consensus but no strong consensus.

Yating Liu, Gilles Pagès.

Source: Bernoulli, Volume 26, Number 2, 1171--1204.

Abstract:
We establish conditions to characterize probability measures by their $L^{p}$-quantization error functions in both $mathbb{R}^{d}$ and Hilbert settings. This characterization is two-fold: static (identity of two distributions) and dynamic (convergence for the $L^{p}$-Wasserstein distance). We first propose a criterion on the quantization level $N$, valid for any norm on $mathbb{R}^{d}$ and any order $p$ based on a geometrical approach involving the Voronoï diagram. Then, we prove that in the $L^{2}$-case on a (separable) Hilbert space, the condition on the level $N$ can be reduced to $N=2$, which is optimal. More quantization based characterization cases in dimension 1 and a discussion of the completeness of a distance defined by the quantization error function can be found at the end of this paper.

Giacomo Aletti, Irene Crimaldi, Andrea Ghiglietti.

Source: Bernoulli, Volume 26, Number 2, 1098--1138.

Abstract:
This work deals with a system of interacting reinforced stochastic processes , where each process $X^{j}=(X_{n,j})_{n}$ is located at a vertex $j$ of a finite weighted directed graph, and it can be interpreted as the sequence of “actions” adopted by an agent $j$ of the network. The interaction among the dynamics of these processes depends on the weighted adjacency matrix $W$ associated to the underlying graph: indeed, the probability that an agent $j$ chooses a certain action depends on its personal “inclination” $Z_{n,j}$ and on the inclinations $Z_{n,h}$, with $h eq j$, of the other agents according to the entries of $W$. The best known example of reinforced stochastic process is the Pólya urn. The present paper focuses on the weighted empirical means $N_{n,j}=sum_{k=1}^{n}q_{n,k}X_{k,j}$, since, for example, the current experience is more important than the past one in reinforced learning. Their almost sure synchronization and some central limit theorems in the sense of stable convergence are proven. The new approach with weighted means highlights the key points in proving some recent results for the personal inclinations $Z^{j}=(Z_{n,j})_{n}$ and for the empirical means $overline{X}^{j}=(sum_{k=1}^{n}X_{k,j}/n)_{n}$ given in recent papers (e.g. Aletti, Crimaldi and Ghiglietti (2019), Ann. Appl. Probab. 27 (2017) 3787–3844, Crimaldi et al. Stochastic Process. Appl. 129 (2019) 70–101). In fact, with a more sophisticated decomposition of the considered processes, we can understand how the different convergence rates of the involved stochastic processes combine. From an application point of view, we provide confidence intervals for the common limit inclination of the agents and a test statistics to make inference on the matrix $W$, based on the weighted empirical means. In particular, we answer a research question posed in Aletti, Crimaldi and Ghiglietti (2019).

Pier Luigi Conti, Daniela Marella, Fulvia Mecatti, Federico Andreis.

Source: Bernoulli, Volume 26, Number 2, 1044--1069.

Abstract:
In this paper, a class of resampling techniques for finite populations under $pi $ps sampling design is introduced. The basic idea on which they rest is a two-step procedure consisting in: (i) constructing a “pseudo-population” on the basis of sample data; (ii) drawing a sample from the predicted population according to an appropriate resampling design. From a logical point of view, this approach is essentially based on the plug-in principle by Efron, at the “sampling design level”. Theoretical justifications based on large sample theory are provided. New approaches to construct pseudo populations based on various forms of calibrations are proposed. Finally, a simulation study is performed.

Sumit Mukherjee.

Source: Bernoulli, Volume 26, Number 2, 1016--1043.

Abstract:
A sufficient criterion for “non-degeneracy” is given for Exponential Random Graph Models on sparse graphs with sufficient statistics which are functions of the degree sequence. This criterion explains why statistics such as alternating $k$-star are non-degenerate, whereas subgraph counts are degenerate. It is further shown that this criterion is “almost” tight. Existence of consistent estimates is then proved for non-degenerate Exponential Random Graph Models.

Leif Döring, Philip Weissmann.

Source: Bernoulli, Volume 26, Number 2, 980--1015.

Abstract:
Conditioning stable Lévy processes on zero probability events recently became a tractable subject since several explicit formulas emerged from a deep analysis using the Lamperti transformations for self-similar Markov processes. In this article, we derive new harmonic functions and use them to explain how to condition stable processes to hit continuously a compact interval from the outside.

Jie Yen Fan, Kais Hamza, Peter Jagers, Fima Klebaner.

Source: Bernoulli, Volume 26, Number 2, 893--926.

Abstract:
We consider a family of general branching processes with reproduction parameters depending on the age of the individual as well as the population age structure and a parameter $K$, which may represent the carrying capacity. These processes are Markovian in the age structure. In a previous paper ( Proc. Steklov Inst. Math. 282 (2013) 90–105), the Law of Large Numbers as $K o infty $ was derived. Here we prove the central limit theorem, namely the weak convergence of the fluctuation processes in an appropriate Skorokhod space. We also show that the limit is driven by a stochastic partial differential equation.

Peggy Cénac, Basile de Loynes, Yoann Offret, Arnaud Rousselle.

Source: Bernoulli, Volume 26, Number 2, 858--892.

Abstract:
The recurrence and transience of persistent random walks built from variable length Markov chains are investigated. It turns out that these stochastic processes can be seen as Lévy walks for which the persistence times depend on some internal Markov chain: they admit Markov random walk skeletons. A recurrence versus transience dichotomy is highlighted. Assuming the positive recurrence of the driving chain, a sufficient Fourier criterion for the recurrence, close to the usual Chung–Fuchs one, is given and a series criterion is derived. The key tool is the Nagaev–Guivarc’h method. Finally, we focus on particular two-dimensional persistent random walks, including directionally reinforced random walks, for which necessary and sufficient Fourier and series criteria are obtained. Inspired by ( Adv. Math. 208 (2007) 680–698), we produce a genuine counterexample to the conjecture of ( Adv. Math. 117 (1996) 239–252). As for the one-dimensional case studied in ( J. Theoret. Probab. 31 (2018) 232–243), it is easier for a persistent random walk than its skeleton to be recurrent. However, such examples are much more difficult to exhibit in the higher dimensional context. These results are based on a surprisingly novel – to our knowledge – upper bound for the Lévy concentration function associated with symmetric distributions.

Richard A. Davis, Mikkel Slot Nielsen, Victor Rohde.

Source: Bernoulli, Volume 26, Number 2, 799--827.

Abstract:
In this paper, we introduce a model, the stochastic fractional delay differential equation (SFDDE), which is based on the linear stochastic delay differential equation and produces stationary processes with hyperbolically decaying autocovariance functions. The model departs from the usual way of incorporating this type of long-range dependence into a short-memory model as it is obtained by applying a fractional filter to the drift term rather than to the noise term. The advantages of this approach are that the corresponding long-range dependent solutions are semimartingales and the local behavior of the sample paths is unaffected by the degree of long memory. We prove existence and uniqueness of solutions to the SFDDEs and study their spectral densities and autocovariance functions. Moreover, we define a subclass of SFDDEs which we study in detail and relate to the well-known fractionally integrated CARMA processes. Finally, we consider the task of simulating from the defining SFDDEs.

Jing Lei.

Source: Bernoulli, Volume 26, Number 1, 767--798.

Abstract:
We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization can cover Euclidean spaces with large dimensionality, with the optimal dependence on the dimensionality. Our method also covers the important case of Gaussian processes in separable Hilbert spaces, with rate-optimal upper bounds for functional data distributions whose coordinates decay geometrically or polynomially. Moreover, our bounds of the expected value can be combined with mean-concentration results to yield improved exponential tail probability bounds for the Wasserstein error of empirical measures under Bernstein-type or log Sobolev-type conditions.

Elena Issoglio, Francesco Russo.

Source: Bernoulli, Volume 26, Number 1, 728--766.

Abstract:
In this paper, we investigate BSDEs where the driver contains a distributional term (in the sense of generalised functions) and derive general Feynman–Kac formulae related to these BSDEs. We introduce an integral operator to give sense to the equation and then we show the existence of a strong solution employing results on a related PDE. Due to the irregularity of the driver, the $Y$-component of a couple $(Y,Z)$ solving the BSDE is not necessarily a semimartingale but a weak Dirichlet process.

Mingjie Liang, René L. Schilling, Jian Wang.

Source: Bernoulli, Volume 26, Number 1, 664--693.

Abstract:
We present a general method to construct couplings of stochastic differential equations driven by Lévy noise in terms of coupling operators. This approach covers both coupling by reflection and refined basic coupling which are often discussed in the literature. As applications, we prove regularity results for the transition semigroups and obtain successful couplings for the solutions to stochastic differential equations driven by additive Lévy noise.

Alexander Schnurr.

Source: Bernoulli, Volume 26, Number 1, 642--663.

Abstract:
We extend the class of semimartingales in a natural way. This allows us to incorporate processes having paths that leave the state space $mathbb{R}^{d}$. In particular, Markov processes related to sub-Markovian kernels, but also non-Markovian processes with path-dependent behavior. By carefully distinguishing between two killing states, we are able to introduce a fourth semimartingale characteristic which generalizes the fourth part of the Lévy quadruple. Using the probabilistic symbol, we analyze the close relationship between the generators of certain Markov processes with killing and their (now four) semimartingale characteristics.

Keisuke Yano, Kengo Kato.

Source: Bernoulli, Volume 26, Number 1, 616--641.

Abstract:
In this paper, we study frequentist coverage errors of Bayesian credible sets for an approximately linear regression model with (moderately) high dimensional regressors, where the dimension of the regressors may increase with but is smaller than the sample size. Specifically, we consider quasi-Bayesian inference on the slope vector under the quasi-likelihood with Gaussian error distribution. Under this setup, we derive finite sample bounds on frequentist coverage errors of Bayesian credible rectangles. Derivation of those bounds builds on a novel Berry–Esseen type bound on quasi-posterior distributions and recent results on high-dimensional CLT on hyperrectangles. We use this general result to quantify coverage errors of Castillo–Nickl and $L^{infty}$-credible bands for Gaussian white noise models, linear inverse problems, and (possibly non-Gaussian) nonparametric regression models. In particular, we show that Bayesian credible bands for those nonparametric models have coverage errors decaying polynomially fast in the sample size, implying advantages of Bayesian credible bands over confidence bands based on extreme value theory.

Eric Beutner, Laurent Bordes, Laurent Doyen.

Source: Bernoulli, Volume 26, Number 1, 557--586.

Abstract:
Virtual age models are very useful to analyse recurrent events. Among the strengths of these models is their ability to account for treatment (or intervention) effects after an event occurrence. Despite their flexibility for modeling recurrent events, the number of applications is limited. This seems to be a result of the fact that in the semiparametric setting all the existing results assume the virtual age function that describes the treatment (or intervention) effects to be known. This shortcoming can be overcome by considering semiparametric virtual age models with parametrically specified virtual age functions. Yet, fitting such a model is a difficult task. Indeed, it has recently been shown that for these models the standard profile likelihood method fails to lead to consistent estimators. Here we show that consistent estimators can be constructed by smoothing the profile log-likelihood function appropriately. We show that our general result can be applied to most of the relevant virtual age models of the literature. Our approach shows that empirical process techniques may be a worthwhile alternative to martingale methods for studying asymptotic properties of these inference methods. A simulation study is provided to illustrate our consistency results together with an application to real data.

Abdelaati Daouia, Stéphane Girard, Gilles Stupfler.

Source: Bernoulli, Volume 26, Number 1, 531--556.

Abstract:
Expectiles define a least squares analogue of quantiles. They are determined by tail expectations rather than tail probabilities. For this reason and many other theoretical and practical merits, expectiles have recently received a lot of attention, especially in actuarial and financial risk management. Their estimation, however, typically requires to consider non-explicit asymmetric least squares estimates rather than the traditional order statistics used for quantile estimation. This makes the study of the tail expectile process a lot harder than that of the standard tail quantile process. Under the challenging model of heavy-tailed distributions, we derive joint weighted Gaussian approximations of the tail empirical expectile and quantile processes. We then use this powerful result to introduce and study new estimators of extreme expectiles and the standard quantile-based expected shortfall, as well as a novel expectile-based form of expected shortfall. Our estimators are built on general weighted combinations of both top order statistics and asymmetric least squares estimates. Some numerical simulations and applications to actuarial and financial data are provided.

Luca M. Giordano, Maria Jolis, Lluís Quer-Sardanyons.

Source: Bernoulli, Volume 26, Number 1, 352--386.

Abstract:
In this article, we consider the quasi-linear stochastic wave and heat equations on the real line and with an additive Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index $Hin (0,1)$. The drift term is assumed to be globally Lipschitz. We prove that the solution of each of the above equations is continuous in terms of the index $H$, with respect to the convergence in law in the space of continuous functions.

Tobias Zwingmann, Hajo Holzmann.

Source: Bernoulli, Volume 26, Number 1, 323--351.

Abstract:
We show weak convergence of quantile and expectile processes to Gaussian limit processes in the space of bounded functions endowed with an appropriate semimetric which is based on the concepts of epi- and hypo- convergence as introduced in A. Bücher, J. Segers and S. Volgushev (2014), ‘ When Uniform Weak Convergence Fails: Empirical Processes for Dependence Functions and Residuals via Epi- and Hypographs ’, Annals of Statistics 42 . We impose assumptions for which it is known that weak convergence with respect to the supremum norm generally fails to hold. For quantiles, we consider stationary observations, where the marginal distribution function is assumed to be strictly increasing and continuous except for finitely many points and to admit strictly positive – possibly infinite – left- and right-sided derivatives. For expectiles, we focus on independent and identically distributed (i.i.d.) observations. Only a finite second moment and continuity at the boundary points but no further smoothness properties of the distribution function are required. We also show consistency of the bootstrap for this mode of convergence in the i.i.d. case for quantiles and expectiles.

Irina Gaynanova.

Source: Bernoulli, Volume 26, Number 1, 286--322.

Abstract:
Sparse linear discriminant analysis via penalized optimal scoring is a successful tool for classification in high-dimensional settings. While the variable selection consistency of sparse optimal scoring has been established, the corresponding prediction and estimation consistency results have been lacking. We bridge this gap by providing probabilistic bounds on out-of-sample prediction error and estimation error of multi-class penalized optimal scoring allowing for diverging number of classes.