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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 EDTEstimation from nonlinear observations via convex programming with application to bilinear regressionhttps://projecteuclid.org/euclid.ejs/1560909647<strong>Sohail Bahmani</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 1978--2011.</p><p><strong>Abstract:</strong><br/>
We propose a computationally efficient estimator, formulated as a convex program, for a broad class of nonlinear regression problems that involve difference of convex (DC) nonlinearities. The proposed method can be viewed as a significant extension of the “anchored regression” method formulated and analyzed in [10] for regression with convex nonlinearities. Our main assumption, in addition to other mild statistical and computational assumptions, is availability of a certain approximation oracle for the average of the gradients of the observation functions at a ground truth. Under this assumption and using a PAC-Bayesian analysis we show that the proposed estimator produces an accurate estimate with high probability. As a concrete example, we study the proposed framework in the bilinear regression problem with Gaussian factors and quantify a sufficient sample complexity for exact recovery. Furthermore, we describe a computationally tractable scheme that provably produces the required approximation oracle in the considered bilinear regression problem.
</p>projecteuclid.org/euclid.ejs/1560909647_20190618220141Tue, 18 Jun 2019 22:01 EDTLinear regression with sparsely permuted datahttps://projecteuclid.org/euclid.ejs/1546570940<strong>Martin Slawski</strong>, <strong>Emanuel Ben-David</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 1--36.</p><p><strong>Abstract:</strong><br/>
In regression analysis of multivariate data, it is tacitly assumed that response and predictor variables in each observed response-predictor pair correspond to the same entity or unit. In this paper, we consider the situation of “permuted data” in which this basic correspondence has been lost. Several recent papers have considered this situation without further assumptions on the underlying permutation. In applications, the latter is often to known to have additional structure that can be leveraged. Specifically, we herein consider the common scenario of “sparsely permuted data” in which only a small fraction of the data is affected by a mismatch between response and predictors. However, an adverse effect already observed for sparsely permuted data is that the least squares estimator as well as other estimators not accounting for such partial mismatch are inconsistent. One approach studied in detail herein is to treat permuted data as outliers which motivates the use of robust regression formulations to estimate the regression parameter. The resulting estimate can subsequently be used to recover the permutation. A notable benefit of the proposed approach is its computational simplicity given the general lack of procedures for the above problem that are both statistically sound and computationally appealing.
</p>projecteuclid.org/euclid.ejs/1546570940_20190621220139Fri, 21 Jun 2019 22:01 EDTConvergence rates of latent topic models under relaxed identifiability conditionshttps://projecteuclid.org/euclid.ejs/1546570941<strong>Yining Wang</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 37--66.</p><p><strong>Abstract:</strong><br/>
In this paper we study the frequentist convergence rate for the Latent Dirichlet Allocation (Blei, Ng and Jordan, 2003) topic models. We show that the maximum likelihood estimator converges to one of the finitely many equivalent parameters in Wasserstein’s distance metric at a rate of $n^{-1/4}$ without assuming separability or non-degeneracy of the underlying topics and/or the existence of more than three words per document, thus generalizing the previous works of Anandkumar et al. (2012, 2014) from an information-theoretical perspective. We also show that the $n^{-1/4}$ convergence rate is optimal in the worst case.
</p>projecteuclid.org/euclid.ejs/1546570941_20190621220139Fri, 21 Jun 2019 22:01 EDTGeneralised additive dependency inflated models including aggregated covariateshttps://projecteuclid.org/euclid.ejs/1546570942<strong>Young K. Lee</strong>, <strong>Enno Mammen</strong>, <strong>Jens P. Nielsen</strong>, <strong>Byeong U. Park</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 67--93.</p><p><strong>Abstract:</strong><br/>
Let us assume that $X$, $Y$ and $U$ are observed and that the conditional mean of $U$ given $X$ and $Y$ can be expressed via an additive dependency of $X$, $\lambda(X)Y$ and $X+Y$ for some unspecified function $\lambda$. This structured regression model can be transferred to a hazard model or a density model when applied on some appropriate grid, and has important forecasting applications via structured marker dependent hazards models or structured density models including age-period-cohort relationships. The structured regression model is also important when the severity of the dependent variable has a complicated dependency on waiting times $X$, $Y$ and the total waiting time $X+Y$. In case the conditional mean of $U$ approximates a density, the regression model can be used to analyse the age-period-cohort model, also when exposure data are not available. In case the conditional mean of $U$ approximates a marker dependent hazard, the regression model introduces new relevant age-period-cohort time scale interdependencies in understanding longevity. A direct use of the regression relationship introduced in this paper is the estimation of the severity of outstanding liabilities in non-life insurance companies. The technical approach taken is to use B-splines to capture the underlying one-dimensional unspecified functions. It is shown via finite sample simulation studies and an application for forecasting future asbestos related deaths in the UK that the B-spline approach works well in practice. Special consideration has been given to ensure identifiability of all models considered.
</p>projecteuclid.org/euclid.ejs/1546570942_20190621220139Fri, 21 Jun 2019 22:01 EDTEmpirical likelihood inference for non-randomized pretest-posttest studies with missing datahttps://projecteuclid.org/euclid.ejs/1561168837<strong>Shixiao Zhang</strong>, <strong>Peisong Han</strong>, <strong>Changbao Wu</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 2012--2042.</p><p><strong>Abstract:</strong><br/>
Pretest-posttest studies are commonly used for assessing the effect of a treatment or an intervention. We propose an empirical likelihood based approach to both testing and estimation of the treatment effect in non-randomized pretest-posttest studies where the posttest outcomes are subject to missingness. The proposed empirical likelihood ratio test and the estimation procedure are multiply robust in the sense that multiple working models are allowed for the propensity score of treatment assignment, the missingness probability and the outcome regression, and the validity of the test and the estimation requires only a certain combination of those multiple working models to be correctly specified. An empirical likelihood ratio confidence interval can be constructed for the treatment effect and has better coverage probabilities than confidence intervals based on the Wald statistic. Simulations are conducted to demonstrate the finite-sample performances of the proposed methods.
</p>projecteuclid.org/euclid.ejs/1561168837_20190621220139Fri, 21 Jun 2019 22:01 EDTGeneralized threshold latent variable modelhttps://projecteuclid.org/euclid.ejs/1561168838<strong>Yuanbo Li</strong>, <strong>Xunze Zheng</strong>, <strong>Chun Yip Yau</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 2043--2092.</p><p><strong>Abstract:</strong><br/>
This article proposes a generalized threshold latent variable model for flexible threshold modeling of time series. The proposed model encompasses several existing models, and allows a discrete valued threshold variable. Sufficient conditions for stationarity and ergodicity are investigated. The minimum description length principle is applied to formulate a criterion function for parameter estimation and model selection. A computationally efficient procedure for optimizing the criterion function is developed based on a genetic algorithm. Consistency and weak convergence of the parameter estimates are established. Moreover, simulation studies and an application for initial public offering data are presented to illustrate the proposed methodology.
</p>projecteuclid.org/euclid.ejs/1561168838_20190621220139Fri, 21 Jun 2019 22:01 EDTRegularised forecasting via smooth-rough partitioning of the regression coefficientshttps://projecteuclid.org/euclid.ejs/1561168839<strong>Hyeyoung Maeng</strong>, <strong>Piotr Fryzlewicz</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 2093--2120.</p><p><strong>Abstract:</strong><br/>
We introduce a way of modelling temporal dependence in random functions $X(t)$ in the framework of linear regression. Based on discretised curves $\{X_{i}(t_{0}),X_{i}(t_{1}),\ldots ,X_{i}(t_{T})\}$, the final point $X_{i}(t_{T})$ is predicted from $\{X_{i}(t_{0}),X_{i}(t_{1}),\ldots ,X_{i}(t_{T-1})\}$. The proposed model flexibly reflects the relative importance of predictors by partitioning the regression parameters into a smooth and a rough regime. Specifically, unconstrained (rough) regression parameters are used for observations located close to $X_{i}(t_{T})$, while the set of regression coefficients for the predictors positioned far from $X_{i}(t_{T})$ are assumed to be sampled from a smooth function. This both regularises the prediction problem and reflects the ‘decaying memory’ structure of the time series. The point at which the change in smoothness occurs is estimated from the data via a technique akin to change-point detection. The joint estimation procedure for the smoothness change-point and the regression parameters is presented, and the asymptotic behaviour of the estimated change-point is analysed. The usefulness of the new model is demonstrated through simulations and four real data examples, involving country fertility data, pollution data, stock volatility series and sunspot number data. Our methodology is implemented in the R package srp, available from CRAN.
</p>projecteuclid.org/euclid.ejs/1561168839_20190621220139Fri, 21 Jun 2019 22:01 EDTExact adaptive confidence intervals for linear regression coefficientshttps://projecteuclid.org/euclid.ejs/1546570943<strong>Peter Hoff</strong>, <strong>Chaoyu Yu</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 94--119.</p><p><strong>Abstract:</strong><br/>
We propose an adaptive confidence interval procedure (CIP) for the coefficients in the normal linear regression model. This procedure has a frequentist coverage rate that is constant as a function of the model parameters, yet provides smaller intervals than the usual interval procedure, on average across regression coefficients. The proposed procedure is obtained by defining a class of CIPs that all have exact $1-\alpha $ frequentist coverage, and then selecting from this class the procedure that minimizes a prior expected interval width. We describe an adaptive approach for estimating the prior distribution from the data, so that the potential risk of a poorly specified prior is reduced. The resulting adaptive confidence intervals maintain exact non-asymptotic $1-\alpha $ coverage if two conditions are met - that the design matrix is full rank (which will be known) and that the errors are normally distributed (which can be checked empirically). No assumptions on the unknown parameters are necessary to maintain exact coverage. Additionally, in a “$p$ growing with $n$” asymptotic scenario, this adaptive FAB procedure is asymptotically Bayes-optimal among $1-\alpha $ frequentist CIPs.
</p>projecteuclid.org/euclid.ejs/1546570943_20190627220410Thu, 27 Jun 2019 22:04 EDTAuxiliary information: the raking-ratio empirical processhttps://projecteuclid.org/euclid.ejs/1546570944<strong>Mickael Albertus</strong>, <strong>Philippe Berthet</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 120--165.</p><p><strong>Abstract:</strong><br/>
We study the empirical measure associated to a sample of size $n$ and modified by $N$ iterations of the raking-ratio method. This empirical measure is adjusted to match the true probability of sets in a finite partition which changes each step. We establish asymptotic properties of the raking-ratio empirical process indexed by functions as $n\rightarrow +\infty $, for $N$ fixed. We study nonasymptotic properties by using a Gaussian approximation which yields uniform Berry-Esseen type bounds depending on $n,N$ and provides estimates of the uniform quadratic risk reduction. A closed-form expression of the limiting covariance matrices is derived as $N\rightarrow +\infty $. In the two-way contingency table case the limiting process has a simple explicit formula.
</p>projecteuclid.org/euclid.ejs/1546570944_20190627220410Thu, 27 Jun 2019 22:04 EDTTrace class Markov chains for the Normal-Gamma Bayesian shrinkage modelhttps://projecteuclid.org/euclid.ejs/1547607848<strong>Liyuan Zhang</strong>, <strong>Kshitij Khare</strong>, <strong>Zeren Xing</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 166--207.</p><p><strong>Abstract:</strong><br/>
High-dimensional data, where the number of variables exceeds or is comparable to the sample size, is now pervasive in many scientific applications. In recent years, Bayesian shrinkage models have been developed as effective and computationally feasible tools to analyze such data, especially in the context of linear regression. In this paper, we focus on the Normal-Gamma shrinkage model developed by Griffin and Brown [7]. This model subsumes the popular Bayesian lasso model, and a three-block Gibbs sampling algorithm to sample from the resulting intractable posterior distribution has been developed in [7]. We consider an alternative two-block Gibbs sampling algorithm, and rigorously demonstrate its advantage over the three-block sampler by comparing specific spectral properties. In particular, we show that the Markov operator corresponding to the two-block sampler is trace class (and hence Hilbert-Schmidt), whereas the operator corresponding to the three-block sampler is not even Hilbert-Schmidt. The trace class property for the two-block sampler implies geometric convergence for the associated Markov chain, which justifies the use of Markov chain CLT’s to obtain practical error bounds for MCMC based estimates. Additionally, it facilitates theoretical comparisons of the two-block sampler with sandwich algorithms which aim to improve performance by inserting inexpensive extra steps in between the two conditional draws of the two-block sampler.
</p>projecteuclid.org/euclid.ejs/1547607848_20190627220410Thu, 27 Jun 2019 22:04 EDTDetection of sparse mixtures: higher criticism and scan statistichttps://projecteuclid.org/euclid.ejs/1547607852<strong>Ery Arias-Castro</strong>, <strong>Andrew Ying</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 208--230.</p><p><strong>Abstract:</strong><br/>
We consider the problem of detecting a sparse mixture as studied by Ingster (1997) and Donoho and Jin (2004). We consider a wide array of base distributions. In particular, we study the situation when the base distribution has polynomial tails, a situation that has not received much attention in the literature. Perhaps surprisingly, we find that in the context of such a power-law distribution, the higher criticism does not achieve the detection boundary. However, the scan statistic does.
</p>projecteuclid.org/euclid.ejs/1547607852_20190627220410Thu, 27 Jun 2019 22:04 EDTImportance sampling the union of rare events with an application to power systems analysishttps://projecteuclid.org/euclid.ejs/1548817590<strong>Art B. Owen</strong>, <strong>Yury Maximov</strong>, <strong>Michael Chertkov</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 231--254.</p><p><strong>Abstract:</strong><br/>
We consider importance sampling to estimate the probability $\mu$ of a union of $J$ rare events $H_{j}$ defined by a random variable $\boldsymbol{x}$. The sampler we study has been used in spatial statistics, genomics and combinatorics going back at least to Karp and Luby (1983). It works by sampling one event at random, then sampling $\boldsymbol{x}$ conditionally on that event happening and it constructs an unbiased estimate of $\mu$ by multiplying an inverse moment of the number of occuring events by the union bound. We prove some variance bounds for this sampler. For a sample size of $n$, it has a variance no larger than $\mu(\bar{\mu}-\mu)/n$ where $\bar{\mu}$ is the union bound. It also has a coefficient of variation no larger than $\sqrt{(J+J^{-1}-2)/(4n)}$ regardless of the overlap pattern among the $J$ events. Our motivating problem comes from power system reliability, where the phase differences between connected nodes have a joint Gaussian distribution and the $J$ rare events arise from unacceptably large phase differences. In the grid reliability problems even some events defined by $5772$ constraints in $326$ dimensions, with probability below $10^{-22}$, are estimated with a coefficient of variation of about $0.0024$ with only $n=10{,}000$ sample values.
</p>projecteuclid.org/euclid.ejs/1548817590_20190627220410Thu, 27 Jun 2019 22:04 EDTEstimation of spectral functionals for Levy-driven continuous-time linear models with tapered datahttps://projecteuclid.org/euclid.ejs/1548817591<strong>Mamikon S. Ginovyan</strong>, <strong>Artur A. Sahakyan</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 255--283.</p><p><strong>Abstract:</strong><br/>
The paper is concerned with the nonparametric statistical estimation of linear spectral functionals for Lévy-driven continuous-time stationary linear models with tapered data. As an estimator for unknown functional we consider the averaged tapered periodogram. We analyze the bias of the estimator and obtain sufficient conditions assuring the proper rate of convergence of the bias to zero, necessary for asymptotic normality of the estimator. We prove a a central limit theorem for a suitable normalized stochastic process generated by a tapered Toeplitz type quadratic functional of the model. As a consequence of these results we obtain the asymptotic normality of our estimator.
</p>projecteuclid.org/euclid.ejs/1548817591_20190627220410Thu, 27 Jun 2019 22:04 EDTDesign-based mapping for finite populations of marked pointshttps://projecteuclid.org/euclid.ejs/1561687406<strong>Lorenzo Fattorini</strong>, <strong>Marzia Marcheselli</strong>, <strong>Caterina Pisani</strong>, <strong>Luca Pratelli</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 2121--2149.</p><p><strong>Abstract:</strong><br/>
The estimation of marks for a finite population of points scattered onto a study region is considered when a sample of these points is selected by a probabilistic sampling scheme. At each point, the mark is estimated by means of an inverse distance weighting interpolator. The design-based asymptotic properties of the resulting maps are derived when the study area remains fixed, a sequence of nested populations with increasing size is considered and samples of increasing size are selected. Conditions ensuring design-based asymptotic unbiasedness and consistency are given. They essentially require that marks are the values of a pointwise or uniformly continuous deterministic function, the enlargement of the populations is rather regular and the sequence of sampling designs ensures an asymptotic spatial balance. A computationally simple mean squared error estimator is proposed. A simulation study is performed to assess the theoretical results on artificial populations. Finally, an application for mapping the values of the height of trees in a forest stand located in North Italy is reported.
</p>projecteuclid.org/euclid.ejs/1561687406_20190627220410Thu, 27 Jun 2019 22:04 EDTHypothesis testing near singularities and boundarieshttps://projecteuclid.org/euclid.ejs/1561687407<strong>Jonathan D. Mitchell</strong>, <strong>Elizabeth S. Allman</strong>, <strong>John A. Rhodes</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 1250--1293.</p><p><strong>Abstract:</strong><br/>
The likelihood ratio statistic, with its asymptotic $\chi ^{2}$ distribution at regular model points, is often used for hypothesis testing. However, the asymptotic distribution can differ at model singularities and boundaries, suggesting the use of a $\chi ^{2}$ might be problematic nearby. Indeed, its poor behavior for testing near singularities and boundaries is apparent in simulations, and can lead to conservative or anti-conservative tests. Here we develop a new distribution designed for use in hypothesis testing near singularities and boundaries, which asymptotically agrees with that of the likelihood ratio statistic. For two example trinomial models, arising in the context of inference of evolutionary trees, we show the new distributions outperform a $\chi ^{2}$.
</p>projecteuclid.org/euclid.ejs/1561687407_20190627220410Thu, 27 Jun 2019 22:04 EDTNonparametric inference via bootstrapping the debiased estimatorhttps://projecteuclid.org/euclid.ejs/1561687408<strong>Gang Cheng</strong>, <strong>Yen-Chi Chen</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 2194--2256.</p><p><strong>Abstract:</strong><br/>
In this paper, we propose to construct confidence bands by bootstrapping the debiased kernel density estimator (for density estimation) and the debiased local polynomial regression estimator (for regression analysis). The idea of using a debiased estimator was recently employed by Calonico et al. (2018b) to construct a confidence interval of the density function (and regression function) at a given point by explicitly estimating stochastic variations. We extend their ideas of using the debiased estimator and further propose a bootstrap approach for constructing simultaneous confidence bands. This modified method has an advantage that we can easily choose the smoothing bandwidth from conventional bandwidth selectors and the confidence band will be asymptotically valid. We prove the validity of the bootstrap confidence band and generalize it to density level sets and inverse regression problems. Simulation studies confirm the validity of the proposed confidence bands/sets. We apply our approach to an Astronomy dataset to show its applicability.
</p>projecteuclid.org/euclid.ejs/1561687408_20190627220410Thu, 27 Jun 2019 22:04 EDTTests for qualitative features in the random coefficients modelhttps://projecteuclid.org/euclid.ejs/1562140822<strong>Fabian Dunker</strong>, <strong>Konstantin Eckle</strong>, <strong>Katharina Proksch</strong>, <strong>Johannes Schmidt-Hieber</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2257--2306.</p><p><strong>Abstract:</strong><br/>
The random coefficients model is an extension of the linear regression model that allows for unobserved heterogeneity in the population by modeling the regression coefficients as random variables. Given data from this model, the statistical challenge is to recover information about the joint density of the random coefficients which is a multivariate and ill-posed problem. Because of the curse of dimensionality and the ill-posedness, nonparametric estimation of the joint density is difficult and suffers from slow convergence rates. Larger features, such as an increase of the density along some direction or a well-accentuated mode can, however, be much easier detected from data by means of statistical tests. In this article, we follow this strategy and construct tests and confidence statements for qualitative features of the joint density, such as increases, decreases and modes. We propose a multiple testing approach based on aggregating single tests which are designed to extract shape information on fixed scales and directions. Using recent tools for Gaussian approximations of multivariate empirical processes, we derive expressions for the critical value. We apply our method to simulated and real data.
</p>projecteuclid.org/euclid.ejs/1562140822_20190703040026Wed, 03 Jul 2019 04:00 EDTThe Generalized Lasso Problem and Uniquenesshttps://projecteuclid.org/euclid.ejs/1562637626<strong>Alnur Ali</strong>, <strong>Ryan J. Tibshirani</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2307--2347.</p><p><strong>Abstract:</strong><br/>
We study uniqueness in the generalized lasso problem, where the penalty is the $\ell _{1}$ norm of a matrix $D$ times the coefficient vector. We derive a broad result on uniqueness that places weak assumptions on the predictor matrix $X$ and penalty matrix $D$; the implication is that, if $D$ is fixed and its null space is not too large (the dimension of its null space is at most the number of samples), and $X$ and response vector $y$ jointly follow an absolutely continuous distribution, then the generalized lasso problem has a unique solution almost surely, regardless of the number of predictors relative to the number of samples. This effectively generalizes previous uniqueness results for the lasso problem [32] (which corresponds to the special case $D=I$). Further, we extend our study to the case in which the loss is given by the negative log-likelihood from a generalized linear model. In addition to uniqueness results, we derive results on the local stability of generalized lasso solutions that might be of interest in their own right.
</p>projecteuclid.org/euclid.ejs/1562637626_20190708220038Mon, 08 Jul 2019 22:00 EDTMedian confidence regions in a nonparametric modelhttps://projecteuclid.org/euclid.ejs/1563436820<strong>Edsel A. Peña</strong>, <strong>Taeho Kim</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2348--2390.</p><p><strong>Abstract:</strong><br/>
The nonparametric measurement error model (NMEM) postulates that $X_{i}=\Delta +\epsilon _{i},i=1,2,\ldots ,n;\Delta \in \Re $ with $\epsilon _{i},i=1,2,\ldots ,n$, IID from $F(\cdot )\in\mathfrak{F}_{c,0}$, where $\mathfrak{F}_{c,0}$ is the class of all continuous distributions with median $0$, so $\Delta $ is the median parameter of $X$. This paper deals with the problem of constructing a confidence region (CR) for $\Delta $ under the NMEM. Aside from the NMEM, the problem setting also arises in a variety of situations, including inference about the median lifetime of a complex system arising in engineering, reliability, biomedical, and public health settings, as well as in the economic arena such as when dealing with household income. Current methods of constructing CRs for $\Delta $ are discussed, including the $T$-statistic based CR and the Wilcoxon signed-rank statistic based CR, arguably the two default methods in applied work when a confidence interval about the center of a distribution is desired. A ‘bottom-to-top’ approach for constructing CRs is implemented, which starts by imposing reasonable invariance or equivariance conditions on the desired CRs, and then optimizing with respect to their mean contents on subclasses of $\mathfrak{F}_{c,0}$. This contrasts with the usual approach of using a pivotal quantity constructed from test statistics and/or estimators and then ‘pivoting’ to obtain the CR. Applications to a real car mileage data set and to Proschan’s famous air-conditioning data set are illustrated. Simulation studies to compare performances of the different CR methods were performed. Results of these studies indicate that the sign-statistic based CR and the optimal CR focused on symmetric distributions satisfy the confidence level requirement, though they tended to have higher contents; while three of the bootstrap-based CR procedures and one of the newly-developed adaptive CR tended to be a tad more liberal, but with smaller contents. A critical recommendation for practitioners is that, under the NMEM , the $T$-statistic based and Wilcoxon signed-rank statistic based CRs should not be used since they either have very degraded coverage probabilities or inflated contents under some of the allowable error distributions under the NMEM.
</p>projecteuclid.org/euclid.ejs/1563436820_20190718040029Thu, 18 Jul 2019 04:00 EDTUnivariate log-concave density estimation with symmetry or modal constraintshttps://projecteuclid.org/euclid.ejs/1563868824<strong>Charles R. Doss</strong>, <strong>Jon A. Wellner</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2391--2461.</p><p><strong>Abstract:</strong><br/>
We study nonparametric maximum likelihood estimation of a log-concave density function $f_{0}$ which is known to satisfy further constraints, where either (a) the mode $m$ of $f_{0}$ is known, or (b) $f_{0}$ is known to be symmetric about a fixed point $m$. We develop asymptotic theory for both constrained log-concave maximum likelihood estimators (MLE’s), including consistency, global rates of convergence, and local limit distribution theory. In both cases, we find the MLE’s pointwise limit distribution at $m$ (either the known mode or the known center of symmetry) and at a point $x_{0}\ne m$. Software to compute the constrained estimators is available in the R package logcondens.mode.
The symmetry-constrained MLE is particularly useful in contexts of location estimation. The mode-constrained MLE is useful for mode-regression. The mode-constrained MLE can also be used to form a likelihood ratio test for the location of the mode of $f_{0}$. These problems are studied in separate papers. In particular, in a separate paper we show that, under a curvature assumption, the likelihood ratio statistic for the location of the mode can be used for hypothesis tests or confidence intervals that do not depend on either tuning parameters or nuisance parameters.
</p>projecteuclid.org/euclid.ejs/1563868824_20190723040028Tue, 23 Jul 2019 04:00 EDTMultiple changepoint detection with partial information on changepoint timeshttps://projecteuclid.org/euclid.ejs/1564041629<strong>Yingbo Li</strong>, <strong>Robert Lund</strong>, <strong>Anuradha Hewaarachchi</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2462--2520.</p><p><strong>Abstract:</strong><br/>
This paper proposes a new minimum description length procedure to detect multiple changepoints in time series data when some times are a priori thought more likely to be changepoints. This scenario arises with temperature time series homogenization pursuits, our focus here. Our Bayesian procedure constructs a natural prior distribution for the situation, and is shown to estimate the changepoint locations consistently, with an optimal convergence rate. Our methods substantially improve changepoint detection power when prior information is available. The methods are also tailored to bivariate data, allowing changes to occur in one or both component series.
</p>projecteuclid.org/euclid.ejs/1564041629_20190725040035Thu, 25 Jul 2019 04:00 EDTNonparametric inference on Lévy measures of compound Poisson-driven Ornstein-Uhlenbeck processes under macroscopic discrete observationshttps://projecteuclid.org/euclid.ejs/1564041630<strong>Daisuke Kurisu</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2521--2565.</p><p><strong>Abstract:</strong><br/>
This study examines a nonparametric inference on a stationary Lévy-driven Ornstein-Uhlenbeck (OU) process $X=(X_{t})_{t\geq 0}$ with a compound Poisson subordinator. We propose a new spectral estimator for the Lévy measure of the Lévy-driven OU process $X$ under macroscopic observations. We also derive, for the estimator, multivariate central limit theorems over a finite number of design points, and high-dimensional central limit theorems in the case wherein the number of design points increases with an increase in the sample size. Built on these asymptotic results, we develop methods to construct confidence bands for the Lévy measure and propose a practical method for bandwidth selection.
</p>projecteuclid.org/euclid.ejs/1564041630_20190725040035Thu, 25 Jul 2019 04:00 EDTFast Bayesian variable selection for high dimensional linear models: Marginal solo spike and slab priorshttps://projecteuclid.org/euclid.ejs/1549335678<strong>Su Chen</strong>, <strong>Stephen G. Walker</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 284--309.</p><p><strong>Abstract:</strong><br/>
This paper presents a method for fast Bayesian variable selection in the normal linear regression model with high dimensional data. A novel approach is adopted in which an explicit posterior probability for including a covariate is obtained. The method is sequential but not order dependent, one deals with each covariate one by one, and a spike and slab prior is only assigned to the coefficient under investigation. We adopt the well-known spike and slab Gaussian priors with a sample size dependent variance, which achieves strong selection consistency for marginal posterior probabilities even when the number of covariates grows almost exponentially with sample size. Numerical illustrations are presented where it is shown that the new approach provides essentially equivalent results to the standard spike and slab priors, i.e. the same marginal posterior probabilities of the coefficients being nonzero, which are estimated via Gibbs sampling. Hence, we obtain the same results via the direct calculation of $p$ probabilities, compared to a stochastic search over a space of $2^{p}$ elements. Our procedure only requires $p$ probabilities to be calculated, which can be done exactly, hence parallel computation when $p$ is large is feasible.
</p>projecteuclid.org/euclid.ejs/1549335678_20190730040138Tue, 30 Jul 2019 04:01 EDTWeak dependence and GMM estimation of supOU and mixed moving average processeshttps://projecteuclid.org/euclid.ejs/1549681240<strong>Imma Valentina Curato</strong>, <strong>Robert Stelzer</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 310--360.</p><p><strong>Abstract:</strong><br/>
We consider a mixed moving average (MMA) process $X$ driven by a Lévy basis and prove that it is weakly dependent with rates computable in terms of the moving average kernel and the characteristic quadruple of the Lévy basis. Using this property, we show conditions ensuring that sample mean and autocovariances of $X$ have a limiting normal distribution. We extend these results to stochastic volatility models and then investigate a Generalized Method of Moments estimator for the supOU process and the supOU stochastic volatility model after choosing a suitable distribution for the mean reversion parameter. For these estimators, we analyze the asymptotic behavior in detail.
</p>projecteuclid.org/euclid.ejs/1549681240_20190730040138Tue, 30 Jul 2019 04:01 EDTOptimal designs for regression with spherical datahttps://projecteuclid.org/euclid.ejs/1549681241<strong>Holger Dette</strong>, <strong>Maria Konstantinou</strong>, <strong>Kirsten Schorning</strong>, <strong>Josua Gösmann</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 361--390.</p><p><strong>Abstract:</strong><br/>
In this paper optimal designs for regression problems with spherical predictors of arbitrary dimension are considered. Our work is motivated by applications in material sciences, where crystallographic textures such as the misorientation distribution or the grain boundary distribution (depending on a four dimensional spherical predictor) are represented by series of hyperspherical harmonics, which are estimated from experimental or simulated data.
For this type of estimation problems we explicitly determine optimal designs with respect to the $\Phi _{p}$-criteria introduced by Kiefer (1974) and a class of orthogonally invariant information criteria recently introduced in the literature. In particular, we show that the uniform distribution on the $m$-dimensional sphere is optimal and construct discrete and implementable designs with the same information matrices as the continuous optimal designs. Finally, we illustrate the advantages of the new designs for series estimation by hyperspherical harmonics, which are symmetric with respect to the first and second crystallographic point group.
</p>projecteuclid.org/euclid.ejs/1549681241_20190730040138Tue, 30 Jul 2019 04:01 EDTAdditive partially linear models for massive heterogeneous datahttps://projecteuclid.org/euclid.ejs/1549681242<strong>Binhuan Wang</strong>, <strong>Yixin Fang</strong>, <strong>Heng Lian</strong>, <strong>Hua Liang</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 391--431.</p><p><strong>Abstract:</strong><br/>
We consider an additive partially linear framework for modelling massive heterogeneous data. The major goal is to extract multiple common features simultaneously across all sub-populations while exploring heterogeneity of each sub-population. We propose an aggregation type of estimators for the commonality parameters that possess the asymptotic optimal bounds and the asymptotic distributions as if there were no heterogeneity. This oracle result holds when the number of sub-populations does not grow too fast and the tuning parameters are selected carefully. A plug-in estimator for the heterogeneity parameter is further constructed, and shown to possess the asymptotic distribution as if the commonality information were available. Furthermore, we develop a heterogeneity test for the linear components and a homogeneity test for the non-linear components accordingly. The performance of the proposed methods is evaluated via simulation studies and an application to the Medicare Provider Utilization and Payment data.
</p>projecteuclid.org/euclid.ejs/1549681242_20190730040138Tue, 30 Jul 2019 04:01 EDTMonte Carlo modified profile likelihood in models for clustered datahttps://projecteuclid.org/euclid.ejs/1549962031<strong>Claudia Di Caterina</strong>, <strong>Giuliana Cortese</strong>, <strong>Nicola Sartori</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 432--464.</p><p><strong>Abstract:</strong><br/>
The main focus of the analysts who deal with clustered data is usually not on the clustering variables, and hence the group-specific parameters are treated as nuisance. If a fixed effects formulation is preferred and the total number of clusters is large relative to the single-group sizes, classical frequentist techniques relying on the profile likelihood are often misleading. The use of alternative tools, such as modifications to the profile likelihood or integrated likelihoods, for making accurate inference on a parameter of interest can be complicated by the presence of nonstandard modelling and/or sampling assumptions. We show here how to employ Monte Carlo simulation in order to approximate the modified profile likelihood in some of these unconventional frameworks. The proposed solution is widely applicable and is shown to retain the usual properties of the modified profile likelihood. The approach is examined in two instances particularly relevant in applications, i.e. missing-data models and survival models with unspecified censoring distribution. The effectiveness of the proposed solution is validated via simulation studies and two clinical trial applications.
</p>projecteuclid.org/euclid.ejs/1549962031_20190730040138Tue, 30 Jul 2019 04:01 EDTQuery-dependent ranking and its asymptotic propertieshttps://projecteuclid.org/euclid.ejs/1549962032<strong>Ben Dai</strong>, <strong>Junhui Wang</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 465--488.</p><p><strong>Abstract:</strong><br/>
Ranking, also known as learning to rank in machine learning community, is to rank a number of items based on their relevance to a specific query. In literature, most ranking methods use a uniform ranking function to evaluate the relevance, which completely ignores the heterogeneity among queries. To admit different ranking functions for various queries, a general $U$-process formulation for query-dependent ranking is developed. It allows to incorporate neighborhood structure among queries via various forms of smoothing weights to improve the ranking performance. One of its salient features is its capability of producing reasonable rankings for novel queries that are absent in the training set, which is commonly encountered in practice but often neglected in the literature. The proposed method is implemented via an inexact alternating direction method of multipliers (ADMM) for each query parallelly. Its asymptotic risk bound is established, showing that it achieves desirable ranking accuracy at a fast rate for any query including the novel ones. Furthermore, simulated examples and a real application to the Yahoo! challenge dataset also support the advantage of the query-dependent ranking method against existing competitors.
</p>projecteuclid.org/euclid.ejs/1549962032_20190730040138Tue, 30 Jul 2019 04:01 EDTNon-marginal decisions: A novel Bayesian multiple testing procedurehttps://projecteuclid.org/euclid.ejs/1550134833<strong>Noirrit Kiran Chandra</strong>, <strong>Sourabh Bhattacharya</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 489--535.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the problem of multiple testing where the hypotheses are dependent. In most of the existing literature, either Bayesian or non-Bayesian, the decision rules mainly focus on the validity of the test procedure rather than actually utilizing the dependency to increase efficiency. Moreover, the decisions regarding different hypotheses are marginal in the sense that they do not depend upon each other directly. However, in realistic situations, the hypotheses are usually dependent, and hence it is desirable that the decisions regarding the dependent hypotheses are taken jointly.
In this article, we develop a novel Bayesian multiple testing procedure that coherently takes this requirement into consideration. Our method, which is based on new notions of error and non-error terms, substantially enhances efficiency by judicious exploitation of the dependence structure among the hypotheses. We show that our method minimizes the posterior expected loss associated with an additive “0-1” loss function; we also prove theoretical results on the relevant error probabilities, establishing the coherence and usefulness of our method. The optimal decision configuration is not available in closed form and we propose an efficient simulated annealing algorithm for the purpose of optimization, which is also generically applicable to binary optimization problems.
Extensive simulation studies indicate that in dependent situations, our method performs significantly better than some existing popular conventional multiple testing methods, in terms of accuracy and power control. Moreover, application of our ideas to a real, spatial data set associated with radionuclide concentration in Rongelap islands yielded insightful results.
</p>projecteuclid.org/euclid.ejs/1550134833_20190730040138Tue, 30 Jul 2019 04:01 EDTLipschitz-Killing curvatures of excursion sets for two-dimensional random fieldshttps://projecteuclid.org/euclid.ejs/1550134834<strong>Hermine Biermé</strong>, <strong>Elena Di Bernardino</strong>, <strong>Céline Duval</strong>, <strong>Anne Estrade</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 536--581.</p><p><strong>Abstract:</strong><br/>
In the present paper we study three geometrical characteristics for the excursion sets of a two-dimensional stationary isotropic random field. First, we show that these characteristics can be estimated without bias if the considered field satisfies a kinematic formula, this is for instance the case of fields given by a function of smooth Gaussian fields or of some shot noise fields. By using the proposed estimators of these geometric characteristics, we describe some inference procedures for the estimation of the parameters of the field. An extensive simulation study illustrates the performances of each estimator. Then, we use the Euler characteristic estimator to build a test to determine whether a given field is Gaussian or not, when compared to various alternatives. The test is based on a sparse information, i.e. , the excursion sets for two different levels of the field to be tested. Finally, the proposed test is adapted to an applied case, synthesized 2D digital mammograms.
</p>projecteuclid.org/euclid.ejs/1550134834_20190730040138Tue, 30 Jul 2019 04:01 EDTGeneralized M-estimators for high-dimensional Tobit I modelshttps://projecteuclid.org/euclid.ejs/1550286094<strong>Jelena Bradic</strong>, <strong>Jiaqi Guo</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 582--645.</p><p><strong>Abstract:</strong><br/>
This paper develops robust confidence intervals in high-dimensional and left-censored regression. Type-I censored regression models, where a competing event makes the variable of interest unobservable, are extremely common in practice. In this paper, we develop smoothed estimating equations that are adaptive to censoring level and are more robust to the misspecification of the error distribution. We propose a unified class of robust estimators, including one-step Mallow’s, Schweppe’s, and Hill-Ryan’s estimator that are adaptive to the left-censored observations. In the ultra-high-dimensional setting, where the dimensionality can grow exponentially with the sample size, we show that as long as the preliminary estimator converges faster than $n^{-1/4}$, the one-step estimators inherit asymptotic distribution of fully iterated version. Moreover, we show that the size of the residuals of the Bahadur representation matches those of the pure linear models – that is, the effects of censoring disappear asymptotically. Simulation studies demonstrate that our method is adaptive to the censoring level and asymmetry in the error distribution, and does not lose efficiency when the errors are from symmetric distributions.
</p>projecteuclid.org/euclid.ejs/1550286094_20190730040138Tue, 30 Jul 2019 04:01 EDTContraction and uniform convergence of isotonic regressionhttps://projecteuclid.org/euclid.ejs/1550286095<strong>Fan Yang</strong>, <strong>Rina Foygel Barber</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 646--677.</p><p><strong>Abstract:</strong><br/>
We consider the problem of isotonic regression, where the underlying signal $x$ is assumed to satisfy a monotonicity constraint, that is, $x$ lies in the cone $\{x\in \mathbb{R}^{n}:x_{1}\leq \dots\leq x_{n}\}$. We study the isotonic projection operator (projection to this cone), and find a necessary and sufficient condition characterizing all norms with respect to which this projection is contractive. This enables a simple and non-asymptotic analysis of the convergence properties of isotonic regression, yielding uniform confidence bands that adapt to the local Lipschitz properties of the signal.
</p>projecteuclid.org/euclid.ejs/1550286095_20190730040138Tue, 30 Jul 2019 04:01 EDTSpectral clustering in the dynamic stochastic block modelhttps://projecteuclid.org/euclid.ejs/1550286096<strong>Marianna Pensky</strong>, <strong>Teng Zhang</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 678--709.</p><p><strong>Abstract:</strong><br/>
In the present paper, we have studied a Dynamic Stochastic Block Model (DSBM) under the assumptions that the connection probabilities, as functions of time, are smooth and that at most $s$ nodes can switch their class memberships between two consecutive time points. We estimate the edge probability tensor by a kernel-type procedure and extract the group memberships of the nodes by spectral clustering. The procedure is computationally viable, adaptive to the unknown smoothness of the functional connection probabilities, to the rate $s$ of membership switching, and to the unknown number of clusters. In addition, it is accompanied by non-asymptotic guarantees for the precision of estimation and clustering.
</p>projecteuclid.org/euclid.ejs/1550286096_20190730040138Tue, 30 Jul 2019 04:01 EDTIsotonic regression meets LASSOhttps://projecteuclid.org/euclid.ejs/1550632213<strong>Matey Neykov</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 710--746.</p><p><strong>Abstract:</strong><br/>
This paper studies a two step procedure for monotone increasing additive single index models with Gaussian designs. The proposed procedure is simple, easy to implement with existing software, and consists of consecutively applying LASSO and isotonic regression. Aside from formalizing this procedure, we provide theoretical guarantees regarding its performance: 1) we show that our procedure controls the in-sample squared error; 2) we demonstrate that one can use the procedure for predicting new observations, by showing that the absolute prediction error can be controlled with high-probability. Our bounds show a tradeoff of two rates: the minimax rate for estimating high dimensional quadratic loss, and the minimax nonparametric rate for estimating a monotone increasing function.
</p>projecteuclid.org/euclid.ejs/1550632213_20190730040138Tue, 30 Jul 2019 04:01 EDTAsymptotic theory of penalized splineshttps://projecteuclid.org/euclid.ejs/1553133771<strong>Luo Xiao</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 747--794.</p><p><strong>Abstract:</strong><br/>
The paper gives a unified study of the large sample asymptotic theory of penalized splines including the O -splines using B-splines and an integrated squared derivative penalty [22], the P -splines which use B-splines and a discrete difference penalty [13], and the T -splines which use truncated polynomials and a ridge penalty [24]. Extending existing results for O -splines [7], it is shown that, depending on the number of knots and appropriate smoothing parameters, the $L_{2}$ risk bounds of penalized spline estimators are rate-wise similar to either those of regression splines or to those of smoothing splines and could each attain the optimal minimax rate of convergence [32]. In addition, convergence rate of the $L_{\infty }$ risk bound, and local asymptotic bias and variance are derived for all three types of penalized splines.
</p>projecteuclid.org/euclid.ejs/1553133771_20190730040138Tue, 30 Jul 2019 04:01 EDTA statistical test of isomorphism between metric-measure spaces using the distance-to-a-measure signaturehttps://projecteuclid.org/euclid.ejs/1553565705<strong>Claire Brécheteau</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 795--849.</p><p><strong>Abstract:</strong><br/>
We introduce the notion of DTM-signature, a measure on $\mathbb{R}$ that can be associated to any metric-measure space. This signature is based on the function distance to a measure (DTM) introduced in 2009 by Chazal, Cohen-Steiner and Mérigot. It leads to a pseudo-metric between metric-measure spaces, that is bounded above by the Gromov-Wasserstein distance. This pseudo-metric is used to build a statistical test of isomorphism between two metric-measure spaces, from the observation of two $N$-samples.
The test is based on subsampling methods and comes with theoretical guarantees. It is proven to be of the correct level asymptotically. Also, when the measures are supported on compact subsets of $\mathbb{R}^{d}$, rates of convergence are derived for the $L_{1}$-Wasserstein distance between the distribution of the test statistic and its subsampling approximation. These rates depend on some parameter $\rho >1$. In addition, we prove that the power is bounded above by $\exp (-CN^{1/\rho })$, with $C$ proportional to the square of the aforementioned pseudo-metric between the metric-measure spaces. Under some geometrical assumptions, we also derive lower bounds for this pseudo-metric.
An algorithm is proposed for the implementation of this statistical test, and its performance is compared to the performance of other methods through numerical experiments.
</p>projecteuclid.org/euclid.ejs/1553565705_20190730040138Tue, 30 Jul 2019 04:01 EDTInvariant test based on the modified correction to LRT for the equality of two high-dimensional covariance matriceshttps://projecteuclid.org/euclid.ejs/1553565706<strong>Qiuyan Zhang</strong>, <strong>Jiang Hu</strong>, <strong>Zhidong Bai</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 1, 850--881.</p><p><strong>Abstract:</strong><br/>
In this paper, we propose an invariant test based on the modified correction to the likelihood ratio test (LRT) of the equality of two high-dimensional covariance matrices. It is well-known that the classical log-LRT is not well defined when the dimension is larger than or equal to one of the sample sizes. Or even the log-LRT is well-defined, it is usually perceived as a bad statistic in the high-dimensional cases because of their low powers under some alternatives. In this paper, we will justify the usefulness of the modified log-LRT, and an invariant test that works well in cases where the dimension is larger than the sample sizes. Besides, the test is established under the weakest conditions on the dimensions and the moments of the samples. The asymptotic distribution of the proposed test statistic is also obtained under the null hypothesis. What is more, we also propose a lite version of the modified LRT in the paper. A simulation study and a real data analysis show that the performances of the two proposed statistics are invariant under affine transformations.
</p>projecteuclid.org/euclid.ejs/1553565706_20190730040138Tue, 30 Jul 2019 04:01 EDTTowards a complete picture of stationary covariance functions on spheres cross timehttps://projecteuclid.org/euclid.ejs/1564732820<strong>Philip White</strong>, <strong>Emilio Porcu</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2566--2594.</p><p><strong>Abstract:</strong><br/>
With the advent of wide-spread global and continental-scale spatiotemporal datasets, increased attention has been given to covariance functions on spheres over time. This paper provides results for stationary covariance functions of random fields defined over $d$-dimensional spheres cross time. Specifically, we provide a bridge between the characterization in Berg and Porcu (2017) for covariance functions on spheres cross time and Gneiting’s lemma (Gneiting, 2002) that deals with planar surfaces.
We then prove that there is a valid class of covariance functions similar in form to the Gneiting class of space-time covariance functions (Gneiting, 2002) that replaces the squared Euclidean distance with the great circle distance. Notably, the provided class is shown to be positive definite on every $d$-dimensional sphere cross time, while the Gneiting class is positive definite over $\mathbb{R} ^{d}\times \mathbb{R} $ for fixed $d$ only.
In this context, we illustrate the value of our adapted Gneiting class by comparing examples from this class to currently established nonseparable covariance classes using out-of-sample predictive criteria. These comparisons are carried out on two climate reanalysis datasets from the National Centers for Environmental Prediction and National Center for Atmospheric Research. For these datasets, we show that examples from our covariance class have better predictive performance than competing models.
</p>projecteuclid.org/euclid.ejs/1564732820_20190802040026Fri, 02 Aug 2019 04:00 EDTDiscrete minimax estimation with treeshttps://projecteuclid.org/euclid.ejs/1565748200<strong>Luc Devroye</strong>, <strong>Tommy Reddad</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2595--2623.</p><p><strong>Abstract:</strong><br/>
We propose a simple recursive data-based partitioning scheme which produces piecewise-constant or piecewise-linear density estimates on intervals, and show how this scheme can determine the optimal $L_{1}$ minimax rate for some discrete nonparametric classes.
</p>projecteuclid.org/euclid.ejs/1565748200_20190813220332Tue, 13 Aug 2019 22:03 EDTEfficient methods for the estimation of the multinomial parameter for the two-trait group testing modelhttps://projecteuclid.org/euclid.ejs/1565748203<strong>Gregory Haber</strong>, <strong>Yaakov Malinovsky</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2624--2657.</p><p><strong>Abstract:</strong><br/>
Estimation of a single Bernoulli parameter using pooled sampling is among the oldest problems in the group testing literature. To carry out such estimation, an array of efficient estimators have been introduced covering a wide range of situations routinely encountered in applications. More recently, there has been growing interest in using group testing to simultaneously estimate the joint probabilities of two correlated traits using a multinomial model. Unfortunately, basic estimation results, such as the maximum likelihood estimator (MLE), have not been adequately addressed in the literature for such cases. In this paper, we show that finding the MLE for this problem is equivalent to maximizing a multinomial likelihood with a restricted parameter space. A solution using the EM algorithm is presented which is guaranteed to converge to the global maximizer, even on the boundary of the parameter space. Two additional closed form estimators are presented with the goal of minimizing the bias and/or mean square error. The methods are illustrated by considering an application to the joint estimation of transmission prevalence for two strains of the Potato virus Y by the aphid Myzus persicae .
</p>projecteuclid.org/euclid.ejs/1565748203_20190813220332Tue, 13 Aug 2019 22:03 EDTGoodness-of-fit testing the error distribution in multivariate indirect regressionhttps://projecteuclid.org/euclid.ejs/1565748204<strong>Justin Chown</strong>, <strong>Nicolai Bissantz</strong>, <strong>Holger Dette</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2658--2685.</p><p><strong>Abstract:</strong><br/>
We propose a goodness-of-fit test for the distribution of errors from a multivariate indirect regression model, which we assume belongs to a location-scale family under the null hypothesis. The test statistic is based on the Khmaladze transformation of the empirical process of standardized residuals. This goodness-of-fit test is consistent at the root-$n$ rate of convergence, and the test can maintain power against local alternatives converging to the null at a root-$n$ rate.
</p>projecteuclid.org/euclid.ejs/1565748204_20190813220332Tue, 13 Aug 2019 22:03 EDTConvergence rate for the $\lambda $-Medial-Axis estimation under regularity conditionshttps://projecteuclid.org/euclid.ejs/1566353060<strong>Catherine Aaron</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2686--2716.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{X}_{n}=\{X_{1},\ldots X_{n}\}\subset \mathbb{R}^{d}$ be a iid random sample of observations drawn with a probability distribution supported by $S$ a compact set satisfying that both $S$ and $\overline{S^{c}}$ are $r_{0}$-convex ($r_{0}>0$). In this paper we study some properties of an estimator of the inner medial axis of $S$ based on the $\lambda $-medial axis. The proposed estimator depends on the choices of $\mathcal{Y}\subset \mathcal{X}_{n}$ an estimator of $\partial S$ and $\hat{S}_{n}$ an estimator of $S$. In a first general theorem we prove that our medial axis estimator converges to the medial axis with a rate $O(\max _{y\in \mathcal{Y}}d(y,\partial S),(\max _{y\in \partial S}d(y,\mathcal{Y})^{2})$. A corollary being that the choice of $\mathcal{Y}$ as the intersection of the sample and its $r$-convex hull, $\mathcal{Y}=C_{r}(\mathcal{X}_{n})\cap \mathcal{X}_{n}$, allows to estimate the medial axis with a convergence rate $O((\ln n/n)^{2/(d+1)})$. In a practical point of view, computational aspects are discussed, algorithms are given and a way to tune the parameters is proposed. A small simulation study is performed to illustrate the results.
</p>projecteuclid.org/euclid.ejs/1566353060_20190820220437Tue, 20 Aug 2019 22:04 EDTOn geometric probability distributions on the torus with applications to molecular biologyhttps://projecteuclid.org/euclid.ejs/1566353061<strong>Alessandro Selvitella</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2717--2763.</p><p><strong>Abstract:</strong><br/>
In this paper, we study a family of probability distributions, alternative to the von Mises family, called Inverse Stereographic Normal Distributions . These distributions are counterparts of the Gaussian Distribution on $\mathbb{S}^{1}$ (univariate) and $\mathbb{T}^{n}$ (multivariate). We discuss some key properties of the models, such as unimodality and closure with respect to marginalizing and conditioning. We compare this family of distributions to the von Mises ’ family and the Wrapped Normal Distribution . Then, we discuss some inferential problems, introduce a notion of moments which is natural for inverse stereographic distributions and revisit a version of the CLT in this context. We construct point estimators, confidence intervals and hypothesis tests and discuss briefly sampling methods. Finally, we conclude with some applications to molecular biology and some illustrative examples. This study is motivated by the Protein Folding Problem and by the fact that a large number of proteins involved in the DNA-metabolism assume a toroidal shape with some amorphous regions.
</p>projecteuclid.org/euclid.ejs/1566353061_20190820220437Tue, 20 Aug 2019 22:04 EDTNonparametric inference for continuous-time event counting and link-based dynamic network modelshttps://projecteuclid.org/euclid.ejs/1566353062<strong>Alexander Kreiß</strong>, <strong>Enno Mammen</strong>, <strong>Wolfgang Polonik</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2764--2829.</p><p><strong>Abstract:</strong><br/>
A flexible approach for modeling both dynamic event counting and dynamic link-based networks based on counting processes is proposed, and estimation in these models is studied. We consider nonparametric likelihood based estimation of parameter functions via kernel smoothing. The asymptotic behavior of these estimators is rigorously analyzed in an asymptotic framework where the number of nodes tends to infinity. The finite sample performance of the estimators is illustrated through an empirical analysis of bike share data.
</p>projecteuclid.org/euclid.ejs/1566353062_20190820220437Tue, 20 Aug 2019 22:04 EDTMultidimensional linear functional estimation in sparse Gaussian models and robust estimation of the meanhttps://projecteuclid.org/euclid.ejs/1567065621<strong>Olivier Collier</strong>, <strong>Arnak S. Dalalyan</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2830--2864.</p><p><strong>Abstract:</strong><br/>
We consider two problems of estimation in high-dimensional Gaussian models. The first problem is that of estimating a linear functional of the means of $n$ independent $p$-dimensional Gaussian vectors, under the assumption that at most $s$ of the means are nonzero. We show that, up to a logarithmic factor, the minimax rate of estimation in squared Euclidean norm is between $(s^{2}\wedge n)+sp$ and $(s^{2}\wedge np)+sp$. The estimator that attains the upper bound being computationally demanding, we investigate suitable versions of group thresholding estimators that are efficiently computable even when the dimension and the sample size are very large. An interesting new phenomenon revealed by this investigation is that the group thresholding leads to a substantial improvement in the rate as compared to the element-wise thresholding. Thus, the rate of the group thresholding is $s^{2}\sqrt{p}+sp$, while the element-wise thresholding has an error of order $s^{2}p+sp$. To the best of our knowledge, this is the first known setting in which leveraging the group structure leads to a polynomial improvement in the rate.
The second problem studied in this work is the estimation of the common $p$-dimensional mean of the inliers among $n$ independent Gaussian vectors. We show that there is a strong analogy between this problem and the first one. Exploiting it, we propose new strategies of robust estimation that are computationally tractable and have better rates of convergence than the other computationally tractable robust (with respect to the presence of the outliers in the data) estimators studied in the literature. However, this tractability comes with a loss of the minimax-rate-optimality in some regimes.
</p>projecteuclid.org/euclid.ejs/1567065621_20190829040038Thu, 29 Aug 2019 04:00 EDTBayesian learning of weakly structural Markov graph laws using sequential Monte Carlo methodshttps://projecteuclid.org/euclid.ejs/1567065622<strong>Jimmy Olsson</strong>, <strong>Tatjana Pavlenko</strong>, <strong>Felix L. Rios</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2865--2897.</p><p><strong>Abstract:</strong><br/>
We present a sequential sampling methodology for weakly structural Markov laws, arising naturally in a Bayesian structure learning context for decomposable graphical models. As a key component of our suggested approach, we show that the problem of graph estimation, which in general lacks natural sequential interpretation, can be recast into a sequential setting by proposing a recursive Feynman-Kac model that generates a flow of junction tree distributions over a space of increasing dimensions. We focus on particle McMC methods to provide samples on this space, in particular on particle Gibbs (PG), as it allows for generating McMC chains with global moves on an underlying space of decomposable graphs. To further improve the PG mixing properties, we incorporate a systematic refreshment step implemented through direct sampling from a backward kernel. The theoretical properties of the algorithm are investigated, showing that the proposed refreshment step improves the performance in terms of asymptotic variance of the estimated distribution. The suggested sampling methodology is illustrated through a collection of numerical examples demonstrating high accuracy in Bayesian graph structure learning in both discrete and continuous graphical models.
</p>projecteuclid.org/euclid.ejs/1567065622_20190829040038Thu, 29 Aug 2019 04:00 EDTSparse Poisson regression with penalized weighted score functionhttps://projecteuclid.org/euclid.ejs/1567065625<strong>Jinzhu Jia</strong>, <strong>Fang Xie</strong>, <strong>Lihu Xu</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2898--2920.</p><p><strong>Abstract:</strong><br/>
By introducing a weighted score function, we propose a new penalized method, similar to square root lasso, to study sparse Poisson regression problems. The corresponding new estimator not only has $\ell _{1}$ consistency but also enjoys the tuning free property. We further verify our theoretical results by numerical simulations and apply them to an image reconstruction problem.
</p>projecteuclid.org/euclid.ejs/1567065625_20190829040038Thu, 29 Aug 2019 04:00 EDTMaximum likelihood estimation for Gaussian processes under inequality constraintshttps://projecteuclid.org/euclid.ejs/1567497627<strong>François Bachoc</strong>, <strong>Agnès Lagnoux</strong>, <strong>Andrés F. López-Lopera</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2921--2969.</p><p><strong>Abstract:</strong><br/>
We consider covariance parameter estimation for a Gaussian process under inequality constraints (boundedness, monotonicity or convexity) in fixed-domain asymptotics. We address the estimation of the variance parameter and the estimation of the microergodic parameter of the Matérn and Wendland covariance functions. First, we show that the (unconstrained) maximum likelihood estimator has the same asymptotic distribution, unconditionally and conditionally to the fact that the Gaussian process satisfies the inequality constraints. Then, we study the recently suggested constrained maximum likelihood estimator. We show that it has the same asymptotic distribution as the (unconstrained) maximum likelihood estimator. In addition, we show in simulations that the constrained maximum likelihood estimator is generally more accurate on finite samples. Finally, we provide extensions to prediction and to noisy observations.
</p>projecteuclid.org/euclid.ejs/1567497627_20190903040055Tue, 03 Sep 2019 04:00 EDTOn the asymptotic variance of the debiased Lassohttps://projecteuclid.org/euclid.ejs/1568794145<strong>Sara van de Geer</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 2970--3008.</p><p><strong>Abstract:</strong><br/>
We consider the high-dimensional linear regression model $Y=X\beta^{0}+\epsilon$ with Gaussian noise $\epsilon$ and Gaussian random design $X$. We assume that $\Sigma:=\mathrm{I\hskip-0.48emE}X^{T}X/n$ is non-singular and write its inverse as $\Theta :=\Sigma^{-1}$. The parameter of interest is the first component $\beta_{1}^{0}$ of $\beta^{0}$. We show that in the high-dimensional case the asymptotic variance of a debiased Lasso estimator can be smaller than $\Theta_{1,1}$. For some special such cases we establish asymptotic efficiency. The conditions include $\beta^{0}$ being sparse and the first column $\Theta_{1}$ of $\Theta$ being not sparse. These sparsity conditions depend on whether $\Sigma$ is known or not.
</p>projecteuclid.org/euclid.ejs/1568794145_20190918040920Wed, 18 Sep 2019 04:09 EDTDynamic adaptive procedures that control the false discovery ratehttps://projecteuclid.org/euclid.ejs/1568794148<strong>Peter W. MacDonald</strong>, <strong>Kun Liang</strong>, <strong>Arnold Janssen</strong>. <p><strong>Source: </strong>Electronic Journal of Statistics, Volume 13, Number 2, 3009--3024.</p><p><strong>Abstract:</strong><br/>
In the multiple testing problem with independent tests, the classical linear step-up procedure controls the false discovery rate (FDR) at level $\pi_{0}\alpha$, where $\pi_{0}$ is the proportion of true null hypotheses and $\alpha$ is the target FDR level. Adaptive procedures can improve power by incorporating estimates of $\pi_{0}$, which typically rely on a tuning parameter. Fixed adaptive procedures set their tuning parameters before seeing the data and can be shown to control the FDR in finite samples. We develop theoretical results for dynamic adaptive procedures whose tuning parameters are determined by the data. We show that, if the tuning parameter is chosen according to a stopping time rule, the corresponding dynamic adaptive procedure controls the FDR in finite samples. Examples include the recently proposed right-boundary procedure and the widely used lowest-slope procedure, among others. Simulation results show that the right-boundary procedure is more powerful than other dynamic adaptive procedures under independence and mild dependence conditions. The right-boundary procedure is implemented in the Bioconductor R package calm.
</p>projecteuclid.org/euclid.ejs/1568794148_20190918040920Wed, 18 Sep 2019 04:09 EDT