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London : H.K. Lewis, 1882.

Paris : Steinheil, 1896.

Paris : Germer-Bailliere, 1877.

London : Philadelphia, 1889.

Detroit, Mich. : G.S. Davis, 1891.

Paris : Béchet jeune, 1838.

Gwenaëlle Castellan, Anthony Cousien, Viet Chi Tran.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 50--81.

Abstract:
Global sensitivity analysis is a set of methods aiming at quantifying the contribution of an uncertain input parameter of the model (or combination of parameters) on the variability of the response. We consider here the estimation of the Sobol indices of order 1 which are commonly-used indicators based on a decomposition of the output’s variance. In a deterministic framework, when the same inputs always give the same outputs, these indices are usually estimated by replicated simulations of the model. In a stochastic framework, when the response given a set of input parameters is not unique due to randomness in the model, metamodels are often used to approximate the mean and dispersion of the response by deterministic functions. We propose a new non-parametric estimator without the need of defining a metamodel to estimate the Sobol indices of order 1. The estimator is based on warped wavelets and is adaptive in the regularity of the model. The convergence of the mean square error to zero, when the number of simulations of the model tend to infinity, is computed and an elbow effect is shown, depending on the regularity of the model. Applications in Epidemiology are carried to illustrate the use of non-parametric estimators.

Zhixin Zhou, Ping Li.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 1302--1347.

Abstract:
The Chernoff coefficient is known to be an upper bound of Bayes error probability in classification problem. In this paper, we will develop a rate optimal Chernoff bound on the Bayes error probability. The new bound is not only an upper bound but also a lower bound of Bayes error probability up to a constant factor. Moreover, we will apply this result to community detection in the stochastic block models. As a clustering problem, the optimal misclassification rate of community detection problem can be characterized by our rate optimal Chernoff bound. This can be formalized by deriving a minimax error rate over certain parameter space of stochastic block models, then achieving such an error rate by a feasible algorithm employing multiple steps of EM type updates.

Alessandro Casa, José E. Chacón, Giovanna Menardi.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 835--856.

Abstract:
Density-based clustering relies on the idea of linking groups to some specific features of the probability distribution underlying the data. The reference to a true, yet unknown, population structure allows framing the clustering problem in a standard inferential setting, where the concept of ideal population clustering is defined as the partition induced by the true density function. The nonparametric formulation of this approach, known as modal clustering, draws a correspondence between the groups and the domains of attraction of the density modes. Operationally, a nonparametric density estimate is required and a proper selection of the amount of smoothing, governing the shape of the density and hence possibly the modal structure, is crucial to identify the final partition. In this work, we address the issue of density estimation for modal clustering from an asymptotic perspective. A natural and easy to interpret metric to measure the distance between density-based partitions is discussed, its asymptotic approximation explored, and employed to study the problem of bandwidth selection for nonparametric modal clustering.

One of the common tasks in unsupervised learning is dimensionality reduction, where the goal is to find meaningful low-dimensional structures hidden in high-dimensional data. Sometimes referred to as manifold learning, this problem is closely related to the problem of localization, which aims at embedding a weighted graph into a low-dimensional Euclidean space. Several methods have been proposed for localization, and also manifold learning. Nonetheless, the robustness property of most of them is little understood. In this paper, we obtain perturbation bounds for classical scaling and trilateration, which are then applied to derive performance bounds for Isomap, Landmark Isomap, and Maximum Variance Unfolding. A new perturbation bound for procrustes analysis plays a key role.

In many applications, observed data are influenced by some combination of latent causes. For example, suppose sensors are placed inside a building to record responses such as temperature, humidity, power consumption and noise levels. These random, observed responses are typically affected by many unobserved, latent factors (or features) within the building such as the number of individuals, the turning on and off of electrical devices, power surges, etc. These latent factors are usually present for a contiguous period of time before disappearing; further, multiple factors could be present at a time. This paper develops new probabilistic methodology and inference methods for random object generation influenced by latent features exhibiting temporal persistence. Every datum is associated with subsets of a potentially infinite number of hidden, persistent features that account for temporal dynamics in an observation. The ensuing class of dynamic models constructed by adapting the Indian Buffet Process — a probability measure on the space of random, unbounded binary matrices — finds use in a variety of applications arising in operations, signal processing, biomedicine, marketing, image analysis, etc. Illustrations using synthetic and real data are provided.

Consider an unknown smooth function $f: [0,1]^d ightarrow mathbb{R}$, and assume we are given $n$ noisy mod 1 samples of $f$, i.e., $y_i = (f(x_i) + eta_i) mod 1$, for $x_i in [0,1]^d$, where $eta_i$ denotes the noise. Given the samples $(x_i,y_i)_{i=1}^{n}$, our goal is to recover smooth, robust estimates of the clean samples $f(x_i) mod 1$. We formulate a natural approach for solving this problem, which works with angular embeddings of the noisy mod 1 samples over the unit circle, inspired by the angular synchronization framework. This amounts to solving a smoothness regularized least-squares problem -- a quadratically constrained quadratic program (QCQP) -- where the variables are constrained to lie on the unit circle. Our proposed approach is based on solving its relaxation, which is a trust-region sub-problem and hence solvable efficiently. We provide theoretical guarantees demonstrating its robustness to noise for adversarial, as well as random Gaussian and Bernoulli noise models. To the best of our knowledge, these are the first such theoretical results for this problem. We demonstrate the robustness and efficiency of our proposed approach via extensive numerical simulations on synthetic data, along with a simple least-squares based solution for the unwrapping stage, that recovers the original samples of $f$ (up to a global shift). It is shown to perform well at high levels of noise, when taking as input the denoised modulo $1$ samples. Finally, we also consider two other approaches for denoising the modulo 1 samples that leverage tools from Riemannian optimization on manifolds, including a Burer-Monteiro approach for a semidefinite programming relaxation of our formulation. For the two-dimensional version of the problem, which has applications in synthetic aperture radar interferometry (InSAR), we are able to solve instances of real-world data with a million sample points in under 10 seconds, on a personal laptop.

Danilo Covaes Nogarotto, Caio Lucidius Naberezny Azevedo, Jorge Luis Bazán.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 304--322.

Abstract:
The interest on the analysis of the zero–one augmented beta regression (ZOABR) model has been increasing over the last few years. In this work, we developed a Bayesian inference for the ZOABR model, providing some contributions, namely: we explored the use of Jeffreys-rule and independence Jeffreys prior for some of the parameters, performing a sensitivity study of prior choice, comparing the Bayesian estimates with the maximum likelihood ones and measuring the accuracy of the estimates under several scenarios of interest. The results indicate, in a general way, that: the Bayesian approach, under the Jeffreys-rule prior, was as accurate as the ML one. Also, different from other approaches, we use the predictive distribution of the response to implement Bayesian residuals. To further illustrate the advantages of our approach, we conduct an analysis of a real psychometric data set including a Bayesian residual analysis, where it is shown that misleading inference can be obtained when the data is transformed. That is, when the zeros and ones are transformed to suitable values and the usual beta regression model is considered, instead of the ZOABR model. Finally, future developments are discussed.

Saralees Nadarajah, Yuancheng Si.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 183--187.

Abstract:
Da Paz, Balakrishnan and Bazan [Braz. J. Probab. Stat. 33 (2019), 455–479] introduced the L-logistic distribution, studied its properties including estimation issues and illustrated a data application. This note derives a closed form expression for moment properties of the distribution. Some computational issues are discussed.

Abedin Haidari, Amir T. Payandeh Najafabadi, Narayanaswamy Balakrishnan.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 150--166.

Abstract:
Consider two parallel systems, say $A$ and $B$, with respective lifetimes $T_{1}$ and $T_{2}$ wherein independent component lifetimes of each system follow exponentiated generalized gamma distribution with possibly different exponential shape and scale parameters. We show here that $T_{2}$ is smaller than $T_{1}$ with respect to the usual stochastic order (reversed hazard rate order) if the vector of logarithm (the main vector) of scale parameters of System $B$ is weakly weighted majorized by that of System $A$, and if the vector of exponential shape parameters of System $A$ is unordered mojorized by that of System $B$. By means of some examples, we show that the above results can not be extended to the hazard rate and likelihood ratio orders. However, when the scale parameters of each system divide into two homogeneous groups, we verify that the usual stochastic and reversed hazard rate orders can be extended, respectively, to the hazard rate and likelihood ratio orders. The established results complete and strengthen some of the known results in the literature.

Rosineide F. da Paz, Narayanaswamy Balakrishnan, Jorge Luis Bazán.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 3, 455--479.

Abstract:
Tadikamalla and Johnson [ Biometrika 69 (1982) 461–465] developed the $L_{B}$ distribution to variables with bounded support by considering a transformation of the standard Logistic distribution. In this manuscript, a convenient parametrization of this distribution is proposed in order to develop regression models. This distribution, referred to here as L-Logistic distribution, provides great flexibility and includes the uniform distribution as a particular case. Several properties of this distribution are studied, and a Bayesian approach is adopted for the parameter estimation. Simulation studies, considering prior sensitivity analysis, recovery of parameters and comparison of algorithms, and robustness to outliers are all discussed showing that the results are insensitive to the choice of priors, efficiency of the algorithm MCMC adopted, and robustness of the model when compared with the beta distribution. Applications to estimate the vulnerability to poverty and to explain the anxiety are performed. The results to applications show that the L-Logistic regression models provide a better fit than the corresponding beta regression models.

Francisco Cribari-Neto, Rodney V. Fonseca.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 329--355.

Abstract:
The log-linear Birnbaum–Saunders model has been widely used in empirical applications. We introduce an extension of this model based on a recently proposed version of the Birnbaum–Saunders distribution which is more flexible than the standard Birnbaum–Saunders law since its density may assume both unimodal and bimodal shapes. We show how to perform point estimation, interval estimation and hypothesis testing inferences on the parameters that index the regression model we propose. We also present a number of diagnostic tools, such as residual analysis, local influence, generalized leverage, generalized Cook’s distance and model misspecification tests. We investigate the usefulness of model selection criteria and the accuracy of prediction intervals for the proposed model. Results of Monte Carlo simulations are presented. Finally, we also present and discuss an empirical application.

Nancy Heckman

Source: Statist. Surv., Volume 6, 113--141.

Abstract:
The popular cubic smoothing spline estimate of a regression function arises as the minimizer of the penalized sum of squares $sum_{j}(Y_{j}-mu(t_{j}))^{2}+lambda int_{a}^{b}[mu''(t)]^{2},dt$, where the data are $t_{j},Y_{j}$, $j=1,ldots,n$. The minimization is taken over an infinite-dimensional function space, the space of all functions with square integrable second derivatives. But the calculations can be carried out in a finite-dimensional space. The reduction from minimizing over an infinite dimensional space to minimizing over a finite dimensional space occurs for more general objective functions: the data may be related to the function $mu$ in another way, the sum of squares may be replaced by a more suitable expression, or the penalty, $int_{a}^{b}[mu''(t)]^{2},dt$, might take a different form. This paper reviews the Reproducing Kernel Hilbert Space structure that provides a finite-dimensional solution for a general minimization problem. Particular attention is paid to the construction and study of the Reproducing Kernel Hilbert Space corresponding to a penalty based on a linear differential operator. In this case, one can often calculate the minimizer explicitly, using Green’s functions.

In this paper we propose a bimodal gamma distribution using a quadratic transformation based on the alpha-skew-normal model. We discuss several properties of this distribution such as mean, variance, moments, hazard rate and entropy measures. Further, we propose a new regression model with censored data based on the bimodal gamma distribution. This regression model can be very useful to the analysis of real data and could give more realistic fits than other special regression models. Monte Carlo simulations were performed to check the bias in the maximum likelihood estimation. The proposed models are applied to two real data sets found in literature.

A graph homomorphism is a map between two graphs that preserves adjacency relations. We consider the problem of sampling a random graph homomorphism from a graph $F$ into a large network $mathcal{G}$. We propose two complementary MCMC algorithms for sampling a random graph homomorphisms and establish bounds on their mixing times and concentration of their time averages. Based on our sampling algorithms, we propose a novel framework for network data analysis that circumvents some of the drawbacks in methods based on independent and neigborhood sampling. Various time averages of the MCMC trajectory give us various computable observables, including well-known ones such as homomorphism density and average clustering coefficient and their generalizations. Furthermore, we show that these network observables are stable with respect to a suitably renormalized cut distance between networks. We provide various examples and simulations demonstrating our framework through synthetic networks. We also apply our framework for network clustering and classification problems using the Facebook100 dataset and Word Adjacency Networks of a set of classic novels.

This paper considers the problem of estimation of the Fisher information for location from a random sample of size $n$. First, an estimator proposed by Bhattacharya is revisited and improved convergence rates are derived. Second, a new estimator, termed a clipped estimator, is proposed. Superior upper bounds on the rates of convergence can be shown for the new estimator compared to the Bhattacharya estimator, albeit with different regularity conditions. Third, both of the estimators are evaluated for the practically relevant case of a random variable contaminated by Gaussian noise. Moreover, using Brown's identity, which relates the Fisher information and the minimum mean squared error (MMSE) in Gaussian noise, two corresponding consistent estimators for the MMSE are proposed. Simulation examples for the Bhattacharya estimator and the clipped estimator as well as the MMSE estimators are presented. The examples demonstrate that the clipped estimator can significantly reduce the required sample size to guarantee a specific confidence interval compared to the Bhattacharya estimator.

Jere Koskela, Paul A. Jenkins, Adam M. Johansen, Dario Spanò.

Source: The Annals of Statistics, Volume 48, Number 1, 560--583.

Abstract:
We study weighted particle systems in which new generations are resampled from current particles with probabilities proportional to their weights. This covers a broad class of sequential Monte Carlo (SMC) methods, widely-used in applied statistics and cognate disciplines. We consider the genealogical tree embedded into such particle systems, and identify conditions, as well as an appropriate time-scaling, under which they converge to the Kingman $n$-coalescent in the infinite system size limit in the sense of finite-dimensional distributions. Thus, the tractable $n$-coalescent can be used to predict the shape and size of SMC genealogies, as we illustrate by characterising the limiting mean and variance of the tree height. SMC genealogies are known to be connected to algorithm performance, so that our results are likely to have applications in the design of new methods as well. Our conditions for convergence are strong, but we show by simulation that they do not appear to be necessary.

Monika Bhattacharjee, Arup Bose.

Source: The Annals of Statistics, Volume 47, Number 6, 3470--3503.

Abstract:
Consider a high-dimensional linear time series model where the dimension $p$ and the sample size $n$ grow in such a way that $p/n o 0$. Let $hat{Gamma }_{u}$ be the $u$th order sample autocovariance matrix. We first show that the LSD of any symmetric polynomial in ${hat{Gamma }_{u},hat{Gamma }_{u}^{*},ugeq 0}$ exists under independence and moment assumptions on the driving sequence together with weak assumptions on the coefficient matrices. This LSD result, with some additional effort, implies the asymptotic normality of the trace of any polynomial in ${hat{Gamma }_{u},hat{Gamma }_{u}^{*},ugeq 0}$. We also study similar results for several independent MA processes. We show applications of the above results to statistical inference problems such as in estimation of the unknown order of a high-dimensional MA process and in graphical and significance tests for hypotheses on coefficient matrices of one or several such independent processes.

Benjamin Eltzner, Stephan F. Huckemann.

Source: The Annals of Statistics, Volume 47, Number 6, 3360--3381.

Abstract:
The (CLT) central limit theorems for generalized Fréchet means (data descriptors assuming values in manifolds, such as intrinsic means, geodesics, etc.) on manifolds from the literature are only valid if a certain empirical process of Hessians of the Fréchet function converges suitably, as in the proof of the prototypical BP-CLT [ Ann. Statist. 33 (2005) 1225–1259]. This is not valid in many realistic scenarios and we provide for a new very general CLT. In particular, this includes scenarios where, in a suitable chart, the sample mean fluctuates asymptotically at a scale $n^{alpha }$ with exponents $alpha <1/2$ with a nonnormal distribution. As the BP-CLT yields only fluctuations that are, rescaled with $n^{1/2}$, asymptotically normal, just as the classical CLT for random vectors, these lower rates, somewhat loosely called smeariness, had to date been observed only on the circle. We make the concept of smeariness on manifolds precise, give an example for two-smeariness on spheres of arbitrary dimension, and show that smeariness, although “almost never” occurring, may have serious statistical implications on a continuum of sample scenarios nearby. In fact, this effect increases with dimension, striking in particular in high dimension low sample size scenarios.

Joshua Cape, Minh Tang, Carey E. Priebe.

Source: The Annals of Statistics, Volume 47, Number 5, 2405--2439.

Abstract:
The singular value matrix decomposition plays a ubiquitous role throughout statistics and related fields. Myriad applications including clustering, classification, and dimensionality reduction involve studying and exploiting the geometric structure of singular values and singular vectors. This paper provides a novel collection of technical and theoretical tools for studying the geometry of singular subspaces using the two-to-infinity norm. Motivated by preliminary deterministic Procrustes analysis, we consider a general matrix perturbation setting in which we derive a new Procrustean matrix decomposition. Together with flexible machinery developed for the two-to-infinity norm, this allows us to conduct a refined analysis of the induced perturbation geometry with respect to the underlying singular vectors even in the presence of singular value multiplicity. Our analysis yields singular vector entrywise perturbation bounds for a range of popular matrix noise models, each of which has a meaningful associated statistical inference task. In addition, we demonstrate how the two-to-infinity norm is the preferred norm in certain statistical settings. Specific applications discussed in this paper include covariance estimation, singular subspace recovery, and multiple graph inference. Both our Procrustean matrix decomposition and the technical machinery developed for the two-to-infinity norm may be of independent interest.

Peng Shi, Zifeng Zhao.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 357--380.

Abstract:
In actuarial research a task of particular interest and importance is to predict the loss cost for individual risks so that informative decisions are made in various insurance operations such as underwriting, ratemaking and capital management. The loss cost is typically viewed to follow a compound distribution where the summation of the severity variables is stopped by the frequency variable. A challenging issue in modeling such outcomes is to accommodate the potential dependence between the number of claims and the size of each individual claim. In this article we introduce a novel regression framework for compound distributions that uses a copula to accommodate the association between the frequency and the severity variables and, thus, allows for arbitrary dependence between the two components. We further show that the new model is very flexible and is easily modified to account for incomplete data due to censoring or truncation. The flexibility of the proposed model is illustrated using both simulated and real data sets. In the analysis of granular claims data from property insurance, we find substantive negative relationship between the number and the size of insurance claims. In addition, we demonstrate that ignoring the frequency-severity association could lead to biased decision-making in insurance operations.

Wanghuan Chu, Runze Li, Jingyuan Liu, Matthew Reimherr.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 276--298.

Abstract:
Motivated by an empirical analysis of data from a genome-wide association study on obesity, measured by the body mass index (BMI), we propose a two-step gene-detection procedure for generalized varying coefficient mixed-effects models with ultrahigh dimensional covariates. The proposed procedure selects significant single nucleotide polymorphisms (SNPs) impacting the mean BMI trend, some of which have already been biologically proven to be “fat genes.” The method also discovers SNPs that significantly influence the age-dependent variability of BMI. The proposed procedure takes into account individual variations of genetic effects and can also be directly applied to longitudinal data with continuous, binary or count responses. We employ Monte Carlo simulation studies to assess the performance of the proposed method and further carry out causal inference for the selected SNPs.

Wenjie Wang, Robert Aseltine, Kun Chen, Jun Yan.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 51--73.

Abstract:
The concept of integrating data from disparate sources to accelerate scientific discovery has generated tremendous excitement in many fields. The potential benefits from data integration, however, may be compromised by the uncertainty due to incomplete/imperfect record linkage. Motivated by a suicide risk study, we propose an approach for analyzing survival data with uncertain event times arising from data integration. Specifically, in our problem deaths identified from the hospital discharge records together with reported suicidal deaths determined by the Office of Medical Examiner may still not include all the death events of patients, and the missing deaths can be recovered from a complete database of death records. Since the hospital discharge data can only be linked to the death record data by matching basic patient characteristics, a patient with a censored death time from the first dataset could be linked to multiple potential event records in the second dataset. We develop an integrative Cox proportional hazards regression in which the uncertainty in the matched event times is modeled probabilistically. The estimation procedure combines the ideas of profile likelihood and the expectation conditional maximization algorithm (ECM). Simulation studies demonstrate that under realistic settings of imperfect data linkage the proposed method outperforms several competing approaches including multiple imputation. A marginal screening analysis using the proposed integrative Cox model is performed to identify risk factors associated with death following suicide-related hospitalization in Connecticut. The identified diagnostics codes are consistent with existing literature and provide several new insights on suicide risk, prediction and prevention.