Electronic Journal of Statistics Articles (Project Euclid)
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The bias and skewness of M -estimators in regression
http://projecteuclid.org/euclid.ejs/1262876992
<strong>Christopher Withers</strong>, <strong>Saralees Nadarajah</strong><p><strong>Source: </strong>Electron. J. Statist., Volume 4, 1--14.</p><p><strong>Abstract:</strong><br/>
We consider M estimation of a regression model with a nuisance parameter and a vector of other parameters. The unknown distribution of the residuals is not assumed to be normal or symmetric. Simple and easily estimated formulas are given for the dominant terms of the bias and skewness of the parameter estimates. For the linear model these are proportional to the skewness of the ‘independent’ variables. For a nonlinear model, its linear component plays the role of these independent variables, and a second term must be added proportional to the covariance of its linear and quadratic components. For the least squares estimate with normal errors this term was derived by Box [1]. We also consider the effect of a large number of parameters, and the case of random independent variables.
</p>projecteuclid.org/euclid.ejs/1262876992_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTBeyond sigmoids: How to obtain well-calibrated probabilities from binary classifiers with beta calibrationhttps://projecteuclid.org/euclid.ejs/1513306867<strong>Meelis Kull</strong>, <strong>Telmo M. Silva Filho</strong>, <strong>Peter Flach</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 5052--5080.</p><p><strong>Abstract:</strong><br/> For optimal decision making under variable class distributions and misclassification costs a classifier needs to produce well-calibrated estimates of the posterior probability. Isotonic calibration is a powerful non-parametric method that is however prone to overfitting on smaller datasets; hence a parametric method based on the logistic sigmoidal curve is commonly used. While logistic calibration is designed for normally distributed per-class scores, we demonstrate experimentally that many classifiers including Naive Bayes and Adaboost suffer from a particular distortion where these score distributions are heavily skewed. In such cases logistic calibration can easily yield probability estimates that are worse than the original scores. Moreover, the logistic curve family does not include the identity function, and hence logistic calibration can easily uncalibrate a perfectly calibrated classifier. In this paper we solve all these problems with a richer class of parametric calibration maps based on the beta distribution. We derive the method from first principles and show that fitting it is as easy as fitting a logistic curve. Extensive experiments show that beta calibration is superior to logistic calibration for a wide range of classifiers: Naive Bayes, Adaboost, random forest, logistic regression, support vector machine and multi-layer perceptron. If the original classifier is already calibrated, then beta calibration learns a function close to the identity. On this we build a statistical test to recognise if the model deviates from being well-calibrated. </p>projecteuclid.org/euclid.ejs/1513306867_20171214220223Thu, 14 Dec 2017 22:02 ESTPoisson intensity estimation with reproducing kernelshttps://projecteuclid.org/euclid.ejs/1513306868<strong>Seth Flaxman</strong>, <strong>Yee Whye Teh</strong>, <strong>Dino Sejdinovic</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 5081--5104.</p><p><strong>Abstract:</strong><br/>
Despite the fundamental nature of the inhomogeneous Poisson process in the theory and application of stochastic processes, and its attractive generalizations (e.g. Cox process), few tractable nonparametric modeling approaches of intensity functions exist, especially when observed points lie in a high-dimensional space. In this paper we develop a new, computationally tractable Reproducing Kernel Hilbert Space (RKHS) formulation for the inhomogeneous Poisson process. We model the square root of the intensity as an RKHS function. Whereas RKHS models used in supervised learning rely on the so-called representer theorem, the form of the inhomogeneous Poisson process likelihood means that the representer theorem does not apply. However, we prove that the representer theorem does hold in an appropriately transformed RKHS, guaranteeing that the optimization of the penalized likelihood can be cast as a tractable finite-dimensional problem. The resulting approach is simple to implement, and readily scales to high dimensions and large-scale datasets.
</p>projecteuclid.org/euclid.ejs/1513306868_20171214220223Thu, 14 Dec 2017 22:02 ESTAsymptotically exact inference in differentiable generative modelshttps://projecteuclid.org/euclid.ejs/1513306869<strong>Matthew M. Graham</strong>, <strong>Amos J. Storkey</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 5105--5164.</p><p><strong>Abstract:</strong><br/>
Many generative models can be expressed as a differentiable function applied to input variables sampled from a known probability distribution. This framework includes both the generative component of learned parametric models such as variational autoencoders and generative adversarial networks, and also procedurally defined simulator models which involve only differentiable operations. Though the distribution on the input variables to such models is known, often the distribution on the output variables is only implicitly defined. We present a method for performing efficient Markov chain Monte Carlo inference in such models when conditioning on observations of the model output. For some models this offers an asymptotically exact inference method where approximate Bayesian computation might otherwise be employed. We use the intuition that computing conditional expectations is equivalent to integrating over a density defined on the manifold corresponding to the set of inputs consistent with the observed outputs. This motivates the use of a constrained variant of Hamiltonian Monte Carlo which leverages the smooth geometry of the manifold to move between inputs exactly consistent with observations. We validate the method by performing inference experiments in a diverse set of models.
</p>projecteuclid.org/euclid.ejs/1513306869_20171214220223Thu, 14 Dec 2017 22:02 ESTLinear Thompson sampling revisitedhttps://projecteuclid.org/euclid.ejs/1513306870<strong>Marc Abeille</strong>, <strong>Alessandro Lazaric</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 5165--5197.</p><p><strong>Abstract:</strong><br/>
We derive an alternative proof for the regret of Thompson sampling (TS) in the stochastic linear bandit setting. While we obtain a regret bound of order $\widetilde{O}(d^{3/2}\sqrt{T})$ as in previous results, the proof sheds new light on the functioning of the TS. We leverage the structure of the problem to show how the regret is related to the sensitivity (i.e., the gradient) of the objective function and how selecting optimal arms associated to optimistic parameters does control it. Thus we show that TS can be seen as a generic randomized algorithm where the sampling distribution is designed to have a fixed probability of being optimistic, at the cost of an additional $\sqrt{d}$ regret factor compared to a UCB-like approach. Furthermore, we show that our proof can be readily applied to regularized linear optimization and generalized linear model problems.
</p>projecteuclid.org/euclid.ejs/1513306870_20171214220223Thu, 14 Dec 2017 22:02 ESTOn the interpretability of conditional probability estimates in the agnostic settinghttps://projecteuclid.org/euclid.ejs/1513306871<strong>Yihan Gao</strong>, <strong>Aditya Parameswaran</strong>, <strong>Jian Peng</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 5198--5231.</p><p><strong>Abstract:</strong><br/>
We study the interpretability of conditional probability estimates for binary classification under the agnostic setting or scenario. Under the agnostic setting, conditional probability estimates do not necessarily reflect the true conditional probabilities. Instead, they have a certain calibration property: among all data points that the classifier has predicted $\mathcal{P}(Y=1|X)=p$, $p$ portion of them actually have label $Y=1$. For cost-sensitive decision problems, this calibration property provides adequate support for us to use Bayes Decision Rule. In this paper, we define a novel measure for the calibration property together with its empirical counterpart, and prove a uniform convergence result between them. This new measure enables us to formally justify the calibration property of conditional probability estimations. It also provides new insights on the problem of estimating and calibrating conditional probabilities, and allows us to reliably estimate the expected cost of decision rules when applied to an unlabeled dataset.
</p>projecteuclid.org/euclid.ejs/1513306871_20171214220223Thu, 14 Dec 2017 22:02 ESTRandom consensus robust PCAhttps://projecteuclid.org/euclid.ejs/1513306872<strong>Daniel Pimentel-Alarcón</strong>, <strong>Robert Nowak</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 5232--5253.</p><p><strong>Abstract:</strong><br/>
This paper presents R2PCA, a random consensus method for robust principal component analysis. R2PCA takes RANSAC’s principle of using as little data as possible one step further. It iteratively selects small subsets of the data to identify pieces of the principal components, to then stitch them together. We show that if the principal components are in general position and the errors are sufficiently sparse, R2PCA will exactly recover the principal components with probability $1$, in lieu of assumptions on coherence or the distribution of the sparse errors, and even under adversarial settings. R2PCA enjoys many advantages: it works well under noise, its computational complexity scales linearly in the ambient dimension, it is easily parallelizable, and due to its low sample complexity, it can be used in settings where data is so large it cannot even be stored in memory. We complement our theoretical findings with synthetic and real data experiments showing that R2PCA outperforms state-of-the-art methods in a broad range of settings.
</p>projecteuclid.org/euclid.ejs/1513306872_20171214220223Thu, 14 Dec 2017 22:02 ESTA flexible convex relaxation for phase retrievalhttps://projecteuclid.org/euclid.ejs/1513306873<strong>Sohail Bahmani</strong>, <strong>Justin Romberg</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 5254--5281.</p><p><strong>Abstract:</strong><br/> We propose a flexible convex relaxation for the phase retrieval problem that operates in the natural domain of the signal. Therefore, we avoid the prohibitive computational cost associated with “lifting” and semidefinite programming (SDP) in methods such as PhaseLift and compete with recently developed non-convex techniques for phase retrieval. We relax the quadratic equations for phaseless measurements to inequality constraints each of which representing a symmetric “slab”. Through a simple convex program, our proposed estimator finds an extreme point of the intersection of these slabs that is best aligned with a given anchor vector . We characterize geometric conditions that certify success of the proposed estimator. Furthermore, using classic results in statistical learning theory, we show that for random measurements the geometric certificates hold with high probability at an optimal sample complexity. We demonstrate the effectiveness of the proposed method through numerical simulations using both independent random measurements and coded diffraction patterns. We also extend this formulation to include sparsity constraints on the target vector. With this additional constraint, we show that, considering “nested” measurements, the number of phaseless measurements needed to recover the sparse vector is essentially the same (to within a logarithmic factor) as the number of linear measurements needed by standard compressive sensing techniques. </p>projecteuclid.org/euclid.ejs/1513306873_20171214220223Thu, 14 Dec 2017 22:02 ESTOnline learning for changing environments using coin bettinghttps://projecteuclid.org/euclid.ejs/1513306874<strong>Kwang-Sung Jun</strong>, <strong>Francesco Orabona</strong>, <strong>Stephen Wright</strong>, <strong>Rebecca Willett</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 5282--5310.</p><p><strong>Abstract:</strong><br/>
A key challenge in online learning is that classical algorithms can be slow to adapt to changing environments. Recent studies have proposed “meta” algorithms that convert any online learning algorithm to one that is adaptive to changing environments, where the adaptivity is analyzed in a quantity called the strongly-adaptive regret. This paper describes a new meta algorithm that has a strongly-adaptive regret bound that is a factor of $\sqrt{\log (T)}$ better than other algorithms with the same time complexity, where $T$ is the time horizon. We also extend our algorithm to achieve a first-order (i.e., dependent on the observed losses) strongly-adaptive regret bound for the first time, to our knowledge. At its heart is a new parameter-free algorithm for the learning with expert advice (LEA) problem in which experts sometimes do not output advice for consecutive time steps (i.e., sleeping experts). This algorithm is derived by a reduction from optimal algorithms for the so-called coin betting problem. Empirical results show that our algorithm outperforms state-of-the-art methods in both learning with expert advice and metric learning scenarios.
</p>projecteuclid.org/euclid.ejs/1513306874_20171214220223Thu, 14 Dec 2017 22:02 ESTAttributing hacks with survival trend filteringhttps://projecteuclid.org/euclid.ejs/1513306875<strong>Ziqi Liu</strong>, <strong>Alexander Smola</strong>, <strong>Kyle Soska</strong>, <strong>Yu-Xiang Wang</strong>, <strong>Qinghua Zheng</strong>, <strong>Jun Zhou</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 5311--5341.</p><p><strong>Abstract:</strong><br/>
In this paper we describe an algorithm for estimating the provenance of hacks on websites. That is, given properties of sites and the temporal occurrence of attacks, we are able to attribute individual attacks to joint causes and vulnerabilities, as well as estimate the evolution of these vulnerabilities over time. Specifically, we use hazard regression with a time-varying additive hazard function parameterized in a generalized linear form. The activation coefficients on each feature are continuous-time functions over time. We formulate the problem of learning these functions as a constrained variational maximum likelihood estimation problem with total variation penalty and show that the optimal solution is a $0$th order spline (a piecewise constant function) with a finite number of adaptively chosen knots. This allows the inference problem to be solved efficiently and at scale by solving a finite dimensional optimization problem. Extensive experiments on real data sets show that our method significantly outperforms Cox’s proportional hazard model. We also conduct case studies and verify that the fitted functions of the features respond to real-life campaigns.
</p>projecteuclid.org/euclid.ejs/1513306875_20171214220223Thu, 14 Dec 2017 22:02 ESTAsymptotic properties of quasi-maximum likelihood estimators in observation-driven time series modelshttps://projecteuclid.org/euclid.ejs/1499133752<strong>Randal Douc</strong>, <strong>Konstantinos Fokianos</strong>, <strong>Eric Moulines</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 2707--2740.</p><p><strong>Abstract:</strong><br/>
We study a general class of quasi-maximum likelihood estimators for observation-driven time series models. Our main focus is on models related to the exponential family of distributions like Poisson based models for count time series or duration models. However, the proposed approach is more general and covers a variety of time series models including the ordinary GARCH model which has been studied extensively in the literature. We provide general conditions under which quasi-maximum likelihood estimators can be analyzed for this class of time series models and we prove that these estimators are consistent and asymptotically normally distributed regardless of the true data generating process. We illustrate our results using classical examples of quasi-maximum likelihood estimation including standard GARCH models, duration models, Poisson type autoregressions and ARMA models with GARCH errors. Our contribution unifies the existing theory and gives conditions for proving consistency and asymptotic normality in a variety of situations.
</p>projecteuclid.org/euclid.ejs/1499133752_20171227220736Wed, 27 Dec 2017 22:07 ESTA Wald-type test statistic for testing linear hypothesis in logistic regression models based on minimum density power divergence estimatorhttps://projecteuclid.org/euclid.ejs/1499133753<strong>Ayanendranath Basu</strong>, <strong>Abhik Ghosh</strong>, <strong>Abhijit Mandal</strong>, <strong>Nirian Martín</strong>, <strong>Leandro Pardo</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 2741--2772.</p><p><strong>Abstract:</strong><br/>
In this paper a robust version of the classical Wald test statistics for linear hypothesis in the logistic regression model is introduced and its properties are explored. We study the problem under the assumption of random covariates although some ideas with non random covariates are also considered. A family of robust Wald type tests are considered here, where the minimum density power divergence estimator is used instead of the maximum likelihood estimator. We obtain the asymptotic distribution and also study the robustness properties of these Wald type test statistics. The robustness of the tests is investigated theoretically through the influence function analysis as well as suitable practical examples. It is theoretically established that the level as well as the power of the Wald-type tests are stable against contamination, while the classical Wald type test breaks down in this scenario. Some classical examples are presented which numerically substantiate the theory developed. Finally a simulation study is included to provide further confirmation of the validity of the theoretical results established in the paper.
</p>projecteuclid.org/euclid.ejs/1499133753_20171227220736Wed, 27 Dec 2017 22:07 ESTParametrically guided local quasi-likelihood with censored datahttps://projecteuclid.org/euclid.ejs/1499133754<strong>Majda Talamakrouni</strong>, <strong>Anouar El Ghouch</strong>, <strong>Ingrid Van Keilegom</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 2773--2799.</p><p><strong>Abstract:</strong><br/>
It is widely pointed out in the literature that misspecification of a parametric model can lead to inconsistent estimators and wrong inference. However, even a misspecified model can provide some valuable information about the phenomena under study. This is the main idea behind the development of an approach known, in the literature, as parametrically guided nonparametric estimation. Due to its promising bias reduction property, this approach has been investigated in different frameworks such as density estimation, least squares regression and local quasi-likelihood. Our contribution is concerned with parametrically guided local quasi-likelihood estimation adapted to randomly right censored data. The generalization to censored data involves synthetic data and local linear fitting. The asymptotic properties of the guided estimator as well as its finite sample performance are studied and compared with the unguided local quasi-likelihood estimator. The results confirm the bias reduction property and show that, using an appropriate guide and an appropriate bandwidth, the proposed estimator outperforms the classical local quasi-likelihood estimator.
</p>projecteuclid.org/euclid.ejs/1499133754_20171227220736Wed, 27 Dec 2017 22:07 ESTConverting high-dimensional regression to high-dimensional conditional density estimationhttps://projecteuclid.org/euclid.ejs/1499133755<strong>Rafael Izbicki</strong>, <strong>Ann B. Lee</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 2800--2831.</p><p><strong>Abstract:</strong><br/>
There is a growing demand for nonparametric conditional density estimators (CDEs) in fields such as astronomy and economics. In astronomy, for example, one can dramatically improve estimates of the parameters that dictate the evolution of the Universe by working with full conditional densities instead of regression (i.e., conditional mean) estimates. More generally, standard regression falls short in any prediction problem where the distribution of the response is more complex with multi-modality, asymmetry or heteroscedastic noise. Nevertheless, much of the work on high-dimensional inference concerns regression and classification only, whereas research on density estimation has lagged behind. Here we propose FlexCode , a fully nonparametric approach to conditional density estimation that reformulates CDE as a non-parametric orthogonal series problem where the expansion coefficients are estimated by regression. By taking such an approach, one can efficiently estimate conditional densities and not just expectations in high dimensions by drawing upon the success in high-dimensional regression. Depending on the choice of regression procedure, our method can adapt to a variety of challenging high-dimensional settings with different structures in the data (e.g., a large number of irrelevant components and nonlinear manifold structure) as well as different data types (e.g., functional data, mixed data types and sample sets). We study the theoretical and empirical performance of our proposed method, and we compare our approach with traditional conditional density estimators on simulated as well as real-world data, such as photometric galaxy data, Twitter data, and line-of-sight velocities in a galaxy cluster.
</p>projecteuclid.org/euclid.ejs/1499133755_20171227220736Wed, 27 Dec 2017 22:07 ESTError bounds for the convex loss Lasso in linear modelshttps://projecteuclid.org/euclid.ejs/1502157624<strong>Mark Hannay</strong>, <strong>Pierre-Yves Deléamont</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 2832--2875.</p><p><strong>Abstract:</strong><br/>
In this paper we investigate error bounds for convex loss functions for the Lasso in linear models, by first establishing a gap in the theory with respect to the existing error bounds. Then, under the compatibility condition, we recover bounds for the absolute value estimation error and the squared prediction error under mild conditions, which appear to be far more appropriate than the existing bounds for the convex loss Lasso. Interestingly, asymptotically the only difference between the new bounds of the convex loss Lasso and the classical Lasso is a term solely depending on a well-known expression in the robust statistics literature appearing multiplicatively in the bounds. We show that this result holds whether or not the scale parameter needs to be estimated jointly with the regression coefficients. Finally, we use the ratio to optimize our bounds in terms of minimaxity.
</p>projecteuclid.org/euclid.ejs/1502157624_20171227220736Wed, 27 Dec 2017 22:07 ESTKernel estimates of nonparametric functional autoregression models and their bootstrap approximationhttps://projecteuclid.org/euclid.ejs/1502157625<strong>Tingyi Zhu</strong>, <strong>Dimitris N. Politis</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 2876--2906.</p><p><strong>Abstract:</strong><br/>
This paper considers a nonparametric functional autoregression model of order one. Existing contributions addressing the problem of functional time series prediction have focused on the linear model and literatures are rather lacking in the context of nonlinear functional time series. In our nonparametric setting, we define the functional version of kernel estimator for the autoregressive operator and develop its asymptotic theory under the assumption of a strong mixing condition on the sample. The results are general in the sense that high-order autoregression can be naturally written as a first-order AR model. In addition, a component-wise bootstrap procedure is proposed that can be used for estimating the distribution of the kernel estimation and its asymptotic validity is theoretically justified. The bootstrap procedure is implemented to construct prediction regions that achieve good coverage rate. A supporting simulation study is presented in the end to illustrate the theoretical advances in the paper.
</p>projecteuclid.org/euclid.ejs/1502157625_20171227220736Wed, 27 Dec 2017 22:07 ESTVariable selection for partially linear models via learning gradientshttps://projecteuclid.org/euclid.ejs/1502157627<strong>Lei Yang</strong>, <strong>Yixin Fang</strong>, <strong>Junhui Wang</strong>, <strong>Yongzhao Shao</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 2907--2930.</p><p><strong>Abstract:</strong><br/>
Partially linear models (PLMs) are important generalizations of linear models and are very useful for analyzing high-dimensional data. Compared to linear models, the PLMs possess desirable flexibility of non-parametric regression models because they have both linear and non-linear components. Variable selection for PLMs plays an important role in practical applications and has been extensively studied with respect to the linear component. However, for the non-linear component, variable selection has been well developed only for PLMs with extra structural assumptions such as additive PLMs and generalized additive PLMs. There is currently an unmet need for variable selection methods applicable to general PLMs without structural assumptions on the non-linear component. In this paper, we propose a new variable selection method based on learning gradients for general PLMs without any assumption on the structure of the non-linear component. The proposed method utilizes the reproducing-kernel-Hilbert-space tool to learn the gradients and the group-lasso penalty to select variables. In addition, a block-coordinate descent algorithm is suggested and some theoretical properties are established including selection consistency and estimation consistency. The performance of the proposed method is further evaluated via simulation studies and illustrated using real data.
</p>projecteuclid.org/euclid.ejs/1502157627_20171227220736Wed, 27 Dec 2017 22:07 ESTCox Markov models for estimating single cell growthhttps://projecteuclid.org/euclid.ejs/1502416820<strong>Federico Bassetti</strong>, <strong>Ilenia Epifani</strong>, <strong>Lucia Ladelli</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 2931--2977.</p><p><strong>Abstract:</strong><br/>
Recent experimental techniques produce thousands of data of single cell growth, consequently stochastic models of growth can be validated on true data and used to understand the main mechanisms that control the cell cycle. A sequence of growing cells is usually modeled by a suitable Markov chain. In this framework, the most interesting goal is to infer the distribution of the doubling time (or of the added size ) of a cell given its initial size and its elongation rate. In the literature, these distributions are described in terms of the corresponding conditional hazard function, referred as division hazard rate . In this work we propose a simple but effective way to estimate the division hazard by using extended Cox modeling . We investigate the convergence to the stationary distribution of the Markov chain describing the sequence of growing cells and we prove that, under reasonable conditions, the proposed estimators of the division hazard rates are asymptotically consistent. Finally, we apply our model to study some published datasets of E-Coli cells.
</p>projecteuclid.org/euclid.ejs/1502416820_20171227220736Wed, 27 Dec 2017 22:07 ESTMaximum likelihood estimation for a bivariate Gaussian process under fixed domain asymptoticshttps://projecteuclid.org/euclid.ejs/1502416821<strong>Daira Velandia</strong>, <strong>François Bachoc</strong>, <strong>Moreno Bevilacqua</strong>, <strong>Xavier Gendre</strong>, <strong>Jean-Michel Loubes</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 2978--3007.</p><p><strong>Abstract:</strong><br/>
We consider maximum likelihood estimation with data from a bivariate Gaussian process with a separable exponential covariance model under fixed domain asymptotics. We first characterize the equivalence of Gaussian measures under this model. Then consistency and asymptotic normality for the maximum likelihood estimator of the microergodic parameters are established. A simulation study is presented in order to compare the finite sample behavior of the maximum likelihood estimator with the given asymptotic distribution.
</p>projecteuclid.org/euclid.ejs/1502416821_20171227220736Wed, 27 Dec 2017 22:07 ESTWeak convergence of the least concave majorant of estimators for a concave distribution functionhttps://projecteuclid.org/euclid.ejs/1508292528<strong>Brendan K. Beare</strong>, <strong>Zheng Fang</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 3841--3870.</p><p><strong>Abstract:</strong><br/>
We study the asymptotic behavior of the least concave majorant of an estimator of a concave distribution function under general conditions. The true concave distribution function is permitted to violate strict concavity, so that the empirical distribution function and its least concave majorant are not asymptotically equivalent. Our results are proved by demonstrating the Hadamard directional differentiability of the least concave majorant operator. Standard approaches to bootstrapping fail to deliver valid inference when the true distribution function is not strictly concave. While the rescaled bootstrap of Dümbgen delivers asymptotically valid inference, its performance in small samples can be poor, and depends upon the selection of a tuning parameter. We show that two alternative bootstrap procedures—one obtained by approximating a conservative upper bound, the other by resampling from the Grenander estimator—can be used to construct reliable confidence bands for the true distribution. Some related results on isotonic regression are provided.
</p>projecteuclid.org/euclid.ejs/1508292528_20171227220736Wed, 27 Dec 2017 22:07 ESTSparse transition matrix estimation for high-dimensional and locally stationary vector autoregressive modelshttps://projecteuclid.org/euclid.ejs/1508292529<strong>Xin Ding</strong>, <strong>Ziyi Qiu</strong>, <strong>Xiaohui Chen</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 3871--3902.</p><p><strong>Abstract:</strong><br/>
We consider the estimation of the transition matrix in the high-dimensional time-varying vector autoregression (TV-VAR) models. Our model builds on a general class of locally stationary VAR processes that evolve smoothly in time. We propose a hybridized kernel smoothing and $\ell^{1}$-regularized method to directly estimate the sequence of time-varying transition matrices. Under the sparsity assumption on the transition matrix, we establish the rate of convergence of the proposed estimator and show that the convergence rate depends on the smoothness of the locally stationary VAR processes only through the smoothness of the transition matrix function. In addition, for our estimator followed by thresholding, we prove that the false positive rate (type I error) and false negative rate (type II error) in the pattern recovery can asymptotically vanish in the presence of weak signals without assuming the minimum nonzero signal strength condition. Favorable finite sample performances over the $\ell^{2}$-penalized least-squares estimator and the unstructured maximum likelihood estimator are shown on simulated data. We also provide two real examples on estimating the dependence structures on financial stock prices and economic exchange rates datasets.
</p>projecteuclid.org/euclid.ejs/1508292529_20171227220736Wed, 27 Dec 2017 22:07 ESTRobust PCA and pairs of projections in a Hilbert spacehttps://projecteuclid.org/euclid.ejs/1508292530<strong>Ilaria Giulini</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 3903--3926.</p><p><strong>Abstract:</strong><br/> This is a study of principal component analysis performed on a statistical sample. We assume that this data sample is made of independent copies of some random variable ranging in a separable real Hilbert space. This covers data in function spaces as well as data represented in reproducing kernel Hilbert spaces. Based on some new inequalities about the perturbation of nonnegative self-adjoint operators, we provide new bounds for the statistical fluctuations of the principal component representation with the draw of the statistical sample. We suggest two kinds of improvements to decrease these fluctuations: the first is to use a robust estimate of the covariance operator, for which non-asymptotic bounds of the estimation error are available under weak polynomial moment assumptions. The second improvement is to use some modification of the projection on the principal components based on functional calculus applied to the covariance operator. Using this modified projection, we can obtain bounds that do not depend on the spectral gap but on some more favorable factor. In appendix, we provide a new approach to the analysis of the relative positions of two orthogonal projections that is useful for our proofs and that has an interest of its own. </p>projecteuclid.org/euclid.ejs/1508292530_20171227220736Wed, 27 Dec 2017 22:07 ESTTree based weighted learning for estimating individualized treatment rules with censored datahttps://projecteuclid.org/euclid.ejs/1508292531<strong>Yifan Cui</strong>, <strong>Ruoqing Zhu</strong>, <strong>Michael Kosorok</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 3927--3953.</p><p><strong>Abstract:</strong><br/>
Estimating individualized treatment rules is a central task for personalized medicine. [25] and [23] proposed outcome weighted learning to estimate individualized treatment rules directly through maximizing the expected outcome without modeling the response directly. In this paper, we extend the outcome weighted learning to right censored survival data without requiring either inverse probability of censoring weighting or semiparametric modeling of the censoring and failure times as done in [26]. To accomplish this, we take advantage of the tree based approach proposed in [29] to nonparametrically impute the survival time in two different ways. The first approach replaces the reward of each individual by the expected survival time, while in the second approach only the censored observations are imputed by their conditional expected failure times. We establish consistency and convergence rates for both estimators. In simulation studies, our estimators demonstrate improved performance compared to existing methods. We also illustrate the proposed method on a phase III clinical trial of non-small cell lung cancer.
</p>projecteuclid.org/euclid.ejs/1508292531_20171227220736Wed, 27 Dec 2017 22:07 ESTThe Kato–Temple inequality and eigenvalue concentration with applications to graph inferencehttps://projecteuclid.org/euclid.ejs/1508292532<strong>Joshua Cape</strong>, <strong>Minh Tang</strong>, <strong>Carey E. Priebe</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 3954--3978.</p><p><strong>Abstract:</strong><br/>
We present an adaptation of the Kato–Temple inequality for bounding perturbations of eigenvalues with applications to statistical inference for random graphs, specifically hypothesis testing and change-point detection. We obtain explicit high-probability bounds for the individual distances between certain signal eigenvalues of a graph’s adjacency matrix and the corresponding eigenvalues of the model’s edge probability matrix, even when the latter eigenvalues have multiplicity. Our results extend more broadly to the perturbation of singular values in the presence of quite general random matrix noise.
</p>projecteuclid.org/euclid.ejs/1508292532_20171227220736Wed, 27 Dec 2017 22:07 ESTConsistent estimation in general sublinear preferential attachment treeshttps://projecteuclid.org/euclid.ejs/1508292533<strong>Fengnan Gao</strong>, <strong>Aad van der Vaart</strong>, <strong>Rui Castro</strong>, <strong>Remco van der Hofstad</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 3979--3999.</p><p><strong>Abstract:</strong><br/>
We propose an empirical estimator of the preferential attachment function $f$ in the setting of general sublinear preferential attachment trees. Using a supercritical continuous-time branching process framework, we prove the almost sure consistency of the proposed estimator. We perform simulations to study the empirical properties of our estimators.
</p>projecteuclid.org/euclid.ejs/1508292533_20171227220736Wed, 27 Dec 2017 22:07 ESTConstrained parameter estimation with uncertain priors for Bayesian networkshttps://projecteuclid.org/euclid.ejs/1508378636<strong>Ali Karimnezhad</strong>, <strong>Peter J. F. Lucas</strong>, <strong>Ahmad Parsian</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4000--4032.</p><p><strong>Abstract:</strong><br/>
In this paper we investigate the task of parameter learning of Bayesian networks and, in particular, we deal with the prior uncertainty of learning using a Bayesian framework. Parameter learning is explored in the context of Bayesian inference and we subsequently introduce Bayes, constrained Bayes and robust Bayes parameter learning methods. Bayes and constrained Bayes estimates of parameters are obtained to meet the twin objective of simultaneous estimation and closeness between the histogram of the estimates and the posterior estimates of the parameter histogram. Treating the prior uncertainty, we consider some classes of prior distributions and derive simultaneous Posterior Regret Gamma Minimax estimates of parameters. Evaluation of the merits of the various procedures was done using synthetic data and a real clinical dataset.
</p>projecteuclid.org/euclid.ejs/1508378636_20171227220736Wed, 27 Dec 2017 22:07 ESTGeometric ergodicity of Gibbs samplers in Bayesian penalized regression modelshttps://projecteuclid.org/euclid.ejs/1508378637<strong>Dootika Vats</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4033--4064.</p><p><strong>Abstract:</strong><br/>
We consider three Bayesian penalized regression models and show that the respective deterministic scan Gibbs samplers are geometrically ergodic regardless of the dimension of the regression problem. We prove geometric ergodicity of the Gibbs samplers for the Bayesian fused lasso, the Bayesian group lasso, and the Bayesian sparse group lasso. Geometric ergodicity along with a moment condition results in the existence of a Markov chain central limit theorem for Monte Carlo averages and ensures reliable output analysis. Our results of geometric ergodicity allow us to also provide default starting values for the Gibbs samplers.
</p>projecteuclid.org/euclid.ejs/1508378637_20171227220736Wed, 27 Dec 2017 22:07 ESTPosterior concentration rates for mixtures of normals in random design regressionhttps://projecteuclid.org/euclid.ejs/1508810899<strong>Zacharie Naulet</strong>, <strong>Judith Rousseau</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4065--4102.</p><p><strong>Abstract:</strong><br/>
Previous works on location and location-scale mixtures of normals have shown different upper bounds on the posterior rates of contraction, either in a density estimation context or in nonlinear regression. In both cases, the observations were assumed not too spread by considering either the true density has light tails or the regression function has compact support. It has been conjectured that in a situation where the data are diffuse, location-scale mixtures may benefit from allowing a spatially varying order of approximation. Here we test the argument on the mean regression with normal errors and random design model. Although we cannot invalidate the conjecture due to the lack of lower bound, we find slower upper bounds for location-scale mixtures, even under heavy tails assumptions on the design distribution. However, the proofs suggest to introduce hybrid location-scale mixtures for which faster upper bounds are derived. Finally, we show that all tails assumptions on the design distribution can be released at the price of making the prior distribution covariate dependent.
</p>projecteuclid.org/euclid.ejs/1508810899_20171227220736Wed, 27 Dec 2017 22:07 ESTEstimation of the Hurst and the stability indices of a $H$-self-similar stable processhttps://projecteuclid.org/euclid.ejs/1508810900<strong>Thi To Nhu Dang</strong>, <strong>Jacques Istas</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4103--4150.</p><p><strong>Abstract:</strong><br/>
In this paper we estimate both the Hurst and the stability indices of a $H$-self-similar stable process. More precisely, let $X$ be a $H$-sssi (self-similar stationary increments) symmetric $\alpha$-stable process. The process $X$ is observed at points $\frac{k}{n}$, $k=0,\ldots,n$. Our estimate is based on $\beta$-negative power variations with $-\frac{1}{2}<\beta<0$. We obtain consistent estimators, with rate of convergence, for several classical $H$-sssi $\alpha$-stable processes (fractional Brownian motion, well-balanced linear fractional stable motion, Takenaka’s process, Lévy motion). Moreover, we obtain asymptotic normality of our estimators for fractional Brownian motion and Lévy motion.
</p>projecteuclid.org/euclid.ejs/1508810900_20171227220736Wed, 27 Dec 2017 22:07 ESTStructured regression models for high-dimensional spatial spectroscopy datahttps://projecteuclid.org/euclid.ejs/1508918427<strong>Arash A. Amini</strong>, <strong>Elizaveta Levina</strong>, <strong>Kerby A. Shedden</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4151--4178.</p><p><strong>Abstract:</strong><br/>
Modeling and analysis of spectroscopy data is an active area of research with applications to chemistry and biology. This paper focuses on modelling high-dimensional spectra for the purpose of noise reduction and prediction in problems where the spectra can be used as covariates. We propose a functional representation of the spectra as well as functional regression model that accommodates multiple spatial dimensions. Both steps emphasize sparsity to reduce the number of parameters and mitigate over-fitting. The motivating application for these models, discussed in some detail, is predicting bone-mineral-density (BMD), an important indicator of fracture healing, from Raman spectra, in both the in vivo and ex vivo settings of a bone fracture healing experiment. To illustrate the general applicability of the method, we also use it to predict lipoprotein concentrations from spectra obtained by nuclear magnetic resonance (NMR) spectroscopy.
</p>projecteuclid.org/euclid.ejs/1508918427_20171227220736Wed, 27 Dec 2017 22:07 ESTMultiscale inference for multivariate deconvolutionhttps://projecteuclid.org/euclid.ejs/1508983572<strong>Konstantin Eckle</strong>, <strong>Nicolai Bissantz</strong>, <strong>Holger Dette</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4179--4219.</p><p><strong>Abstract:</strong><br/>
In this paper we provide new methodology for inference of the geometric features of a multivariate density in deconvolution. Our approach is based on multiscale tests to detect significant directional derivatives of the unknown density at arbitrary points in arbitrary directions. The multiscale method is used to identify regions of monotonicity and to construct a general procedure for the detection of modes of the multivariate density. Moreover, as an important application a significance test for the presence of a local maximum at a pre-specified point is proposed. The performance of the new methods is investigated from a theoretical point of view and the finite sample properties are illustrated by means of a small simulation study.
</p>projecteuclid.org/euclid.ejs/1508983572_20171227220736Wed, 27 Dec 2017 22:07 ESTWithin group variable selection through the Exclusive Lassohttps://projecteuclid.org/euclid.ejs/1509004863<strong>Frederick Campbell</strong>, <strong>Genevera I. Allen</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4220--4257.</p><p><strong>Abstract:</strong><br/>
Many data sets consist of variables with an inherent group structure. The problem of group selection has been well studied, but in this paper, we seek to do the opposite: our goal is to select at least one variable from each group in the context of predictive regression modeling. This problem is NP-hard, but we propose the tightest convex relaxation: a composite penalty that is a combination of the $\ell_{1}$ and $\ell_{2}$ norms. Our so-called Exclusive Lasso method performs structured variable selection by ensuring that at least one variable is selected from each group. We study our method’s statistical properties and develop computationally scalable algorithms for fitting the Exclusive Lasso. We study the effectiveness of our method via simulations as well as using NMR spectroscopy data. Here, we use the Exclusive Lasso to select the appropriate chemical shift from a dictionary of possible chemical shifts for each molecule in the biological sample.
</p>projecteuclid.org/euclid.ejs/1509004863_20171227220736Wed, 27 Dec 2017 22:07 ESTA variational Bayes approach to a semiparametric regression using Gaussian process priorshttps://projecteuclid.org/euclid.ejs/1510111112<strong>Victor M. H. Ong</strong>, <strong>David K. Mensah</strong>, <strong>David J. Nott</strong>, <strong>Seongil Jo</strong>, <strong>Beomjo Park</strong>, <strong>Taeryon Choi</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4258--4296.</p><p><strong>Abstract:</strong><br/>
This paper presents a variational Bayes approach to a semiparametric regression model that consists of parametric and nonparametric components. The assumed univariate nonparametric component is represented with a cosine series based on a spectral analysis of Gaussian process priors. Here, we develop fast variational methods for fitting the semiparametric regression model that reduce the computation time by an order of magnitude over Markov chain Monte Carlo methods. Further, we explore the possible use of the variational lower bound and variational information criteria for model choice of a parametric regression model against a semiparametric alternative. In addition, variational methods are developed for estimating univariate shape-restricted regression functions that are monotonic, monotonic convex or monotonic concave. Since these variational methods are approximate, we explore some of the trade-offs involved in using them in terms of speed, accuracy and automation of the implementation in comparison with Markov chain Monte Carlo methods and discuss their potential and limitations.
</p>projecteuclid.org/euclid.ejs/1510111112_20171227220736Wed, 27 Dec 2017 22:07 ESTAn asymptotic theory for spectral analysis of random fieldshttps://projecteuclid.org/euclid.ejs/1510563632<strong>Soudeep Deb</strong>, <strong>Mohsen Pourahmadi</strong>, <strong>Wei Biao Wu</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4297--4322.</p><p><strong>Abstract:</strong><br/>
For a general class of stationary random fields we study asymptotic properties of the discrete Fourier transform (DFT), periodogram, parametric and nonparametric spectral density estimators under an easily verifiable short-range dependence condition expressed in terms of functional dependence measures. We allow irregularly spaced data which is indexed by a subset $\Gamma $ of $\mathbb{Z}^{d}$. Our asymptotic theory requires minimal restriction on the index set $\Gamma $. Asymptotic normality is derived for kernel spectral density estimators and the Whittle estimator of a parameterized spectral density function. We also develop asymptotic results for a covariance matrix estimate.
</p>projecteuclid.org/euclid.ejs/1510563632_20171227220736Wed, 27 Dec 2017 22:07 ESTNonparametric estimating equations for circular probability density functions and their derivativeshttps://projecteuclid.org/euclid.ejs/1510563633<strong>Marco Di Marzio</strong>, <strong>Stefania Fensore</strong>, <strong>Agnese Panzera</strong>, <strong>Charles C. Taylor</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4323--4346.</p><p><strong>Abstract:</strong><br/>
We propose estimating equations whose unknown parameters are the values taken by a circular density and its derivatives at a point. Specifically, we solve equations which relate local versions of population trigonometric moments with their sample counterparts. Major advantages of our approach are: higher order bias without asymptotic variance inflation, closed form for the estimators, and absence of numerical tasks. We also investigate situations where the observed data are dependent. Theoretical results along with simulation experiments are provided.
</p>projecteuclid.org/euclid.ejs/1510563633_20171227220736Wed, 27 Dec 2017 22:07 ESTA provable smoothing approach for high dimensional generalized regression with applications in genomicshttps://projecteuclid.org/euclid.ejs/1510801790<strong>Fang Han</strong>, <strong>Hongkai Ji</strong>, <strong>Zhicheng Ji</strong>, <strong>Honglang Wang</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4347--4403.</p><p><strong>Abstract:</strong><br/>
In many applications, linear models fit the data poorly. This article studies an appealing alternative, the generalized regression model. This model only assumes that there exists an unknown monotonically increasing link function connecting the response $Y$ to a single index $\boldsymbol{X} ^{\mathsf{T}}\boldsymbol{\beta } ^{*}$ of explanatory variables $\boldsymbol{X} \in{\mathbb{R}} ^{d}$. The generalized regression model is flexible and covers many widely used statistical models. It fits the data generating mechanisms well in many real problems, which makes it useful in a variety of applications where regression models are regularly employed. In low dimensions, rank-based M-estimators are recommended to deal with the generalized regression model, giving root-$n$ consistent estimators of $\boldsymbol{\beta } ^{*}$. Applications of these estimators to high dimensional data, however, are questionable. This article studies, both theoretically and practically, a simple yet powerful smoothing approach to handle the high dimensional generalized regression model. Theoretically, a family of smoothing functions is provided, and the amount of smoothing necessary for efficient inference is carefully calculated. Practically, our study is motivated by an important and challenging scientific problem: decoding gene regulation by predicting transcription factors that bind to cis-regulatory elements. Applying our proposed method to this problem shows substantial improvement over the state-of-the-art alternative in real data.
</p>projecteuclid.org/euclid.ejs/1510801790_20171227220736Wed, 27 Dec 2017 22:07 ESTApproximate likelihood inference in generalized linear latent variable models based on the dimension-wise quadraturehttps://projecteuclid.org/euclid.ejs/1510801791<strong>Silvia Bianconcini</strong>, <strong>Silvia Cagnone</strong>, <strong>Dimitris Rizopoulos</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4404--4423.</p><p><strong>Abstract:</strong><br/>
We propose a new method to perform approximate likelihood inference in latent variable models. Our approach provides an approximation of the integrals involved in the likelihood function through a reduction of their dimension that makes the computation feasible in situations in which classical and adaptive quadrature based methods are not applicable. We derive new theoretical results on the accuracy of the obtained estimators. We show that the proposed approximation outperforms several existing methods in simulations, and it can be successfully applied in presence of multidimensional longitudinal data when standard techniques are not applicable or feasible.
</p>projecteuclid.org/euclid.ejs/1510801791_20171227220736Wed, 27 Dec 2017 22:07 ESTGradient angle-based analysis for spatiotemporal point processeshttps://projecteuclid.org/euclid.ejs/1510887942<strong>Tonglin Zhang</strong>, <strong>Yen-Ning Huang</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4424--4451.</p><p><strong>Abstract:</strong><br/>
Spatiotemporal point processes (STPPs) are important in modeling randomly appeared events developed in space and time. Statistical methods of STPPs have been widely used in applications. In all of these methods, evaluations and inferences of intensity functions are the primary issues. The present article proposes a new method, which attempts to evaluate angles of gradient vectors of intensity functions rather than the intensity functions themselves. According to the nature of many natural and human phenomena, the evaluation of angle patterns of the gradient vectors is more important than the evaluation of their magnitude patterns because changes of angle patterns often indicate global changes of these phenomena. This issue is investigated by simulation studies, where significant variations of gradient angle patterns are identified only when modes of intensity functions are changed. To study these phenomena, the article proposes an analysis method for gradient angles of the first-order intensity function of STPPs. The proposed method is used to analyze aftershock earthquake activities caused by great mainshock earthquakes occurred in Japan 2011 and Indian Ocean 2004, respectively, where a significant global change in the second case is identified.
</p>projecteuclid.org/euclid.ejs/1510887942_20171227220736Wed, 27 Dec 2017 22:07 ESTPartition structure and the $A$-hypergeometric distribution associated with the rational normal curvehttps://projecteuclid.org/euclid.ejs/1510887943<strong>Shuhei Mano</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4452--4487.</p><p><strong>Abstract:</strong><br/>
A distribution whose normalization constant is an $A$-hypergeometric polynomial is called an $A$-hypergeometric distribution. Such a distribution is in turn a generalization of the generalized hypergeometric distribution on the contingency tables with fixed marginal sums. In this paper, we will see that an $A$-hypergeometric distribution with a homogeneous matrix of two rows, especially, that associated with the rational normal curve, appears in inferences involving exchangeable partition structures. An exact sampling algorithm is presented for the general (any number of rows) $A$-hypergeometric distributions. Then, the maximum likelihood estimation of the $A$-hypergeometric distribution associated with the rational normal curve, which is an algebraic exponential family, is discussed. The information geometry of the Newton polytope is useful for analyzing the full and the curved exponential family. Algebraic methods are provided for evaluating the $A$-hypergeometric polynomials.
</p>projecteuclid.org/euclid.ejs/1510887943_20171227220736Wed, 27 Dec 2017 22:07 ESTNovel sampling design for respondent-driven samplinghttps://projecteuclid.org/euclid.ejs/1511773484<strong>Mohammad Khabbazian</strong>, <strong>Bret Hanlon</strong>, <strong>Zoe Russek</strong>, <strong>Karl Rohe</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4769--4812.</p><p><strong>Abstract:</strong><br/>
Respondent-driven sampling (RDS) is a method of chain referral sampling popular for sampling hidden and/or marginalized populations. As such, even under the ideal sampling assumptions, the performance of RDS is restricted by the underlying social network: if the network is divided into communities that are weakly connected to each other, then RDS is likely to oversample one of these communities. In order to diminish the “referral bottlenecks” between communities, we propose anti-cluster RDS (AC-RDS), an adjustment to the standard RDS implementation. Using a standard model in the RDS literature, namely, a Markov process on the social network that is indexed by a tree, we construct and study the Markov transition matrix for AC-RDS. We show that if the underlying network is generated from the Stochastic Blockmodel with equal block sizes, then the transition matrix for AC-RDS has a larger spectral gap and consequently faster mixing properties than the standard random walk model for RDS. In addition, we show that AC-RDS reduces the covariance of the samples in the referral tree compared to the standard RDS and consequently leads to a smaller variance and design effect. We confirm the effectiveness of the new design using both the Add-Health networks and simulated networks.
</p>projecteuclid.org/euclid.ejs/1511773484_20171227220736Wed, 27 Dec 2017 22:07 ESTBayesian inference for multivariate extreme value distributionshttps://projecteuclid.org/euclid.ejs/1511773485<strong>Clément Dombry</strong>, <strong>Sebastian Engelke</strong>, <strong>Marco Oesting</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4813--4844.</p><p><strong>Abstract:</strong><br/>
Statistical modeling of multivariate and spatial extreme events has attracted broad attention in various areas of science. Max-stable distributions and processes are the natural class of models for this purpose, and many parametric families have been developed and successfully applied. Due to complicated likelihoods, the efficient statistical inference is still an active area of research, and usually composite likelihood methods based on bivariate densities only are used. Thibaud et al. (2016) use a Bayesian approach to fit a Brown–Resnick process to extreme temperatures. In this paper, we extend this idea to a methodology that is applicable to general max-stable distributions and that uses full likelihoods. We further provide simple conditions for the asymptotic normality of the median of the posterior distribution and verify them for the commonly used models in multivariate and spatial extreme value statistics. A simulation study shows that this point estimator is considerably more efficient than the composite likelihood estimator in a frequentist framework. From a Bayesian perspective, our approach opens the way for new techniques such as Bayesian model comparison in multivariate and spatial extremes.
</p>projecteuclid.org/euclid.ejs/1511773485_20171227220736Wed, 27 Dec 2017 22:07 ESTCorrection to: Nonparametric Laguerre estimation in the multiplicative censoring modelhttps://projecteuclid.org/euclid.ejs/1511773486<strong>Denis Belomestny</strong>, <strong>Fabienne Comte</strong>, <strong>Valentine Genon-Catalot</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4845--4850.</p><p><strong>Abstract:</strong><br/>
The paper “Nonparametric Laguerre estimation in the multiplicative censoring model”, Electronic Journal of Statistics , 2016, 10 , 3114–3152, contains a wrong statement. We localize the place of the error and give a correct proof.
</p>projecteuclid.org/euclid.ejs/1511773486_20171227220736Wed, 27 Dec 2017 22:07 ESTA sharp oracle inequality for Graph-Slopehttps://projecteuclid.org/euclid.ejs/1512032447<strong>Pierre C. Bellec</strong>, <strong>Joseph Salmon</strong>, <strong>Samuel Vaiter</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 4851--4870.</p><p><strong>Abstract:</strong><br/>
Following recent success on the analysis of the Slope estimator, we provide a sharp oracle inequality in term of prediction error for Graph-Slope, a generalization of Slope to signals observed over a graph. In addition to improving upon best results obtained so far for the Total Variation denoiser (also referred to as Graph-Lasso or Generalized Lasso), we propose an efficient algorithm to compute Graph-Slope. The proposed algorithm is obtained by applying the forward-backward method to the dual formulation of the Graph-Slope optimization problem. We also provide experiments showing the practical applicability of the method.
</p>projecteuclid.org/euclid.ejs/1512032447_20171227220736Wed, 27 Dec 2017 22:07 ESTDistributional equivalence and structure learning for bow-free acyclic path diagramshttps://projecteuclid.org/euclid.ejs/1514430421<strong>Christopher Nowzohour</strong>, <strong>Marloes H. Maathuis</strong>, <strong>Robin J. Evans</strong>, <strong>Peter Bühlmann</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 5342--5374.</p><p><strong>Abstract:</strong><br/>
We consider the problem of structure learning for bow-free acyclic path diagrams (BAPs). BAPs can be viewed as a generalization of linear Gaussian DAG models that allow for certain hidden variables. We present a first method for this problem using a greedy score-based search algorithm. We also prove some necessary and some sufficient conditions for distributional equivalence of BAPs which are used in an algorithmic approach to compute (nearly) equivalent model structures. This allows us to infer lower bounds of causal effects. We also present applications to real and simulated datasets using our publicly available R-package.
</p>projecteuclid.org/euclid.ejs/1514430421_20171227220736Wed, 27 Dec 2017 22:07 ESTOn the asymptotic efficiency of selection procedures for independent Gaussian populationshttps://projecteuclid.org/euclid.ejs/1514430422<strong>Royi Jacobovic</strong>, <strong>Or Zuk</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 5375--5405.</p><p><strong>Abstract:</strong><br/>
The field of discrete event simulation and optimization techniques motivates researchers to adjust classic ranking and selection (R&S) procedures to the settings where the number of populations is large. We use insights from extreme value theory in order to reveal the asymptotic properties of R&S procedures. Namely, we generalize the asymptotic result of Robbins and Siegmund regarding selection from independent Gaussian populations with known constant variance by their means to the case of selecting a subset of varying size out of a given set of populations. In addition, we revisit the problem of selecting the population with the highest mean among independent Gaussian populations with unknown and possibly different variances. Particularly, we derive the relative asymptotic efficiency of Dudewicz and Dalal’s and Rinott’s procedures, showing that the former can be asymptotically superior by a multiplicative factor which is larger than one, but this factor may be reduced by proper choice of parameters. We also use our asymptotic results to suggest that the sample size in the first stage of the two procedures should be logarithmic in the number of populations.
</p>projecteuclid.org/euclid.ejs/1514430422_20171227220736Wed, 27 Dec 2017 22:07 ESTExchangeable Markov survival processes and weak continuity of predictive distributionshttps://projecteuclid.org/euclid.ejs/1514430423<strong>Walter Dempsey</strong>, <strong>Peter McCullagh</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 11, Number 2, 5406--5451.</p><p><strong>Abstract:</strong><br/>
We study exchangeable, Markov survival processes – stochastic processes giving rise to infinitely exchangeable non-negative sequences $(T_{1},T_{2},\ldots)$. We show how these are determined by their characteristic index $\{\zeta_{n}\}_{n=1}^{\infty}$. We identify the harmonic process as the family of exchangeable, Markov survival processes that compose the natural set of statistical models for time-to-event data. In particular, this two-dimensional family comprises the set of exchangeable, Markov survival processes with weakly continuous predictive distributions. The harmonic process is easy to generate sequentially, and a simple expression exists for both the joint probability distribution and multivariate survivor function. We show a close connection with the Kaplan-Meier estimator of the survival distribution. Embedded within the process is an infinitely exchangeable ordered partition. Aspects of the process, such as the distribution of the number of blocks, are investigated.
</p>projecteuclid.org/euclid.ejs/1514430423_20171227220736Wed, 27 Dec 2017 22:07 ESTChange detection via affine and quadratic detectorshttps://projecteuclid.org/euclid.ejs/1514970025<strong>Yang Cao</strong>, <strong>Arkadi Nemirovski</strong>, <strong>Yao Xie</strong>, <strong>Vincent Guigues</strong>, <strong>Anatoli Juditsky</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 12, Number 1, 1--57.</p><p><strong>Abstract:</strong><br/>
The goal of the paper is to develop a specific application of the convex optimization based hypothesis testing techniques developed in A. Juditsky, A. Nemirovski, “Hypothesis testing via affine detectors,” Electronic Journal of Statistics 10 :2204–2242, 2016. Namely, we consider the Change Detection problem as follows: observing one by one noisy observations of outputs of a discrete-time linear dynamical system, we intend to decide, in a sequential fashion, on the null hypothesis that the input to the system is a nuisance, vs. the alternative that the input is a “nontrivial signal,” with both the nuisances and the nontrivial signals modeled as inputs belonging to finite unions of some given convex sets. Assuming the observation noises are zero mean sub-Gaussian, we develop “computation-friendly” sequential decision rules and demonstrate that in our context these rules are provably near-optimal.
</p>projecteuclid.org/euclid.ejs/1514970025_20180103040028Wed, 03 Jan 2018 04:00 ESTConfidence intervals for the means of the selected populationshttps://projecteuclid.org/euclid.ejs/1515142842<strong>Claudio Fuentes</strong>, <strong>George Casella</strong>, <strong>Martin T. Wells</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 12, Number 1, 58--79.</p><p><strong>Abstract:</strong><br/>
Consider an experiment in which $p$ independent populations $\pi_{i}$ with corresponding unknown means $\theta_{i}$ are available, and suppose that for every $1\leq i\leq p$, we can obtain a sample $X_{i1},\ldots,X_{in}$ from $\pi_{i}$. In this context, researchers are sometimes interested in selecting the populations that yield the largest sample means as a result of the experiment, and then estimate the corresponding population means $\theta_{i}$. In this paper, we present a frequentist approach to the problem and discuss how to construct simultaneous confidence intervals for the means of the $k$ selected populations, assuming that the populations $\pi_{i}$ are independent and normally distributed with a common variance $\sigma^{2}$. The method, based on the minimization of the coverage probability, obtains confidence intervals that attain the nominal coverage probability for any $p$ and $k$, taking into account the selection procedure.
</p>projecteuclid.org/euclid.ejs/1515142842_20180105040046Fri, 05 Jan 2018 04:00 ESTOn misspecifications in regularity and properties of estimatorshttps://projecteuclid.org/euclid.ejs/1515747853<strong>Oleg V. Chernoyarov</strong>, <strong>Yury A. Kutoyants</strong>, <strong>Andrei P. Trifonov</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 12, Number 1, 80--106.</p><p><strong>Abstract:</strong><br/>
The problem of parameter estimation by the continuous time observations of a deterministic signal in white Gaussian noise is considered. The asymptotic properties of the maximum likelihood estimator are described in the asymptotic of small noise (large signal-to-noise ratio). We are interested in the situation when there is a misspecification in the regularity conditions. In particular it is supposed that the statistician uses a discontinuous (change-point type) model of signal, when the true signal is continuously differentiable function of the unknown parameter.
</p>projecteuclid.org/euclid.ejs/1515747853_20180112040419Fri, 12 Jan 2018 04:04 ESTLocally stationary functional time serieshttps://projecteuclid.org/euclid.ejs/1516006818<strong>Anne van Delft</strong>, <strong>Michael Eichler</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 12, Number 1, 107--170.</p><p><strong>Abstract:</strong><br/>
The literature on time series of functional data has focused on processes of which the probabilistic law is either constant over time or constant up to its second-order structure. Especially for long stretches of data it is desirable to be able to weaken this assumption. This paper introduces a framework that will enable meaningful statistical inference of functional data of which the dynamics change over time. We put forward the concept of local stationarity in the functional setting and establish a class of processes that have a functional time-varying spectral representation. Subsequently, we derive conditions that allow for fundamental results from nonstationary multivariate time series to carry over to the function space. In particular, time-varying functional ARMA processes are investigated and shown to be functional locally stationary according to the proposed definition. As a side-result, we establish a Cramér representation for an important class of weakly stationary functional processes. Important in our context is the notion of a time-varying spectral density operator of which the properties are studied and uniqueness is derived. Finally, we provide a consistent nonparametric estimator of this operator and show it is asymptotically Gaussian using a weaker tightness criterion than what is usually deemed necessary.
</p>projecteuclid.org/euclid.ejs/1516006818_20180115040022Mon, 15 Jan 2018 04:00 EST