Mostrando recursos 1 - 20 de 899

  1. Divide and conquer local average regression

    Chang, Xiangyu; Lin, Shao-Bo; Wang, Yao
    The divide and conquer strategy, which breaks a massive data set into a series of manageable data blocks, and combines the independent results of data blocks to obtain a final decision, has been recognized as a state-of-the-art method to overcome challenges of massive data analysis. In this paper, we equip the classical local average regression with some divide and conquer strategies to infer the regressive relationship of input-output pairs from a massive data set. When the average mixture, a widely used divide and conquer approach, is adopted, we prove that the optimal learning rate can be achieved under some restrictive...

  2. Recursive partitioning and multi-scale modeling on conditional densities

    Ma, Li
    We introduce a nonparametric prior on the conditional distribution of a (univariate or multivariate) response given a set of predictors. The prior is constructed in the form of a two-stage generative procedure, which in the first stage recursively partitions the predictor space, and then in the second stage generates the conditional distribution by a multi-scale nonparametric density model on each predictor partition block generated in the first stage. This design allows adaptive smoothing on both the predictor space and the response space, and it results in the full posterior conjugacy of the model, allowing exact Bayesian inference to be completed...

  3. Efficient moment calculations for variance components in large unbalanced crossed random effects models

    Gao, Katelyn; Owen, Art
    Large crossed data sets, often modeled by generalized linear mixed models, have become increasingly common and provide challenges for statistical analysis. At very large sizes it becomes desirable to have the computational costs of estimation, inference and prediction (both space and time) grow at most linearly with sample size. ¶ Both traditional maximum likelihood estimation and numerous Markov chain Monte Carlo Bayesian algorithms take superlinear time in order to obtain good parameter estimates in the simple two-factor crossed random effects model. We propose moment based algorithms that, with at most linear cost, estimate variance components, measure the uncertainties of those estimates, and...

  4. Nearly assumptionless screening for the mutually-exciting multivariate Hawkes process

    Chen, Shizhe; Witten, Daniela; Shojaie, Ali
    We consider the task of learning the structure of the graph underlying a mutually-exciting multivariate Hawkes process in the high-dimensional setting. We propose a simple and computationally inexpensive edge screening approach. Under a subset of the assumptions required for penalized estimation approaches to recover the graph, this edge screening approach has the sure screening property: with high probability, the screened edge set is a superset of the true edge set. Furthermore, the screened edge set is relatively small. We illustrate the performance of this new edge screening approach in simulation studies.

  5. Estimation accuracy under covariate-adaptive randomization procedures

    Baldi Antognini, Alessandro; Zagoraiou, Maroussa
    In this paper we provide some general asymptotic properties of covariate-adaptive (CA) randomized designs aimed at balancing the allocations of two treatments across a set of chosen covariates. In particular, we establish the central limit theorem for a vast class of covariate-adaptive procedures characterized by i) a different allocation function for each covariate profile and ii) sequences of allocation rules instead of a pre-fixed one. This result allows one to derive theoretically the asymptotic expressions of the loss of information induced by imbalance and the selection bias due to the lack of randomness, that are the fundamental properties for estimation...

  6. Optimal-order uniform and nonuniform bounds on the rate of convergence to normality for maximum likelihood estimators

    Pinelis, Iosif
    It is well known that, under general regularity conditions, the distribution of the maximum likelihood estimator (MLE) is asymptotically normal. Very recently, bounds of the optimal order $O(1/\sqrt{n})$ on the closeness of the distribution of the MLE to normality in the so-called bounded Wasserstein distance were obtained [2, 1], where $n$ is the sample size. However, the corresponding bounds on the Kolmogorov distance were only of the order $O(1/n^{1/4})$. In this paper, bounds of the optimal order $O(1/\sqrt{n})$ on the closeness of the distribution of the MLE to normality in the Kolmogorov distance are given, as well as their nonuniform...

  7. Geometric foundations for scaling-rotation statistics on symmetric positive definite matrices: Minimal smooth scaling-rotation curves in low dimensions

    Groisser, David; Jung, Sungkyu; Schwartzman, Armin
    We investigate a geometric computational framework, called the “scaling-rotation framework”, on $\mathrm{Sym}^{+}(p)$, the set of $p\times p$ symmetric positive-definite (SPD) matrices. The purpose of our study is to lay geometric foundations for statistical analysis of SPD matrices, in situations in which eigenstructure is of fundamental importance, for example diffusion-tensor imaging (DTI). Eigen-decomposition, upon which the scaling-rotation framework is based, determines both a stratification of $\mathrm{Sym}^{+}(p)$, defined by eigenvalue multiplicities, and fibers of the “eigen-composition” map $SO(p)\times\mathrm{Diag}^{+}(p)\to\mathrm{Sym}^{+}(p)$. This leads to the notion of scaling-rotation distance [Jung et al. (2015)], a measure of the minimal amount of scaling and rotation needed to...

  8. Estimation of false discovery proportion in multiple testing: From normal to chi-squared test statistics

    Du, Lilun; Zhang, Chunming
    Multiple testing based on chi-squared test statistics is common in many scientific fields such as genomics research and brain imaging studies. However, the challenges of designing a formal testing procedure when there exists a general dependence structure across the chi-squared test statistics have not been well addressed. To address this gap, we first adopt a latent factor structure ([14]) to construct a testing framework for approximating the false discovery proportion ($\mathrm{FDP}$) for a large number of highly correlated chi-squared test statistics with a finite number of degrees of freedom $k$. The testing framework is then used to simultaneously test $k$...

  9. Kernel ridge vs. principal component regression: Minimax bounds and the qualification of regularization operators

    Dicker, Lee H.; Foster, Dean P.; Hsu, Daniel
    Regularization is an essential element of virtually all kernel methods for nonparametric regression problems. A critical factor in the effectiveness of a given kernel method is the type of regularization that is employed. This article compares and contrasts members from a general class of regularization techniques, which notably includes ridge regression and principal component regression. We derive an explicit finite-sample risk bound for regularization-based estimators that simultaneously accounts for (i) the structure of the ambient function space, (ii) the regularity of the true regression function, and (iii) the adaptability (or qualification) of the regularization. A simple consequence of this upper...

  10. Density estimation for $\tilde{\beta}$-dependent sequences

    Dedecker, Jérôme; Merlevède, Florence
    We study the ${\mathbb{L}}^{p}$-integrated risk of some classical estimators of the density, when the observations are drawn from a strictly stationary sequence. The results apply to a large class of sequences, which can be non-mixing in the sense of Rosenblatt and long-range dependent. The main probabilistic tool is a new Rosenthal-type inequality for partial sums of $BV$ functions of the variables. As an application, we give the rates of convergence of regular Histograms, when estimating the invariant density of a class of expanding maps of the unit interval with a neutral fixed point at zero. These Histograms are plotted in...

  11. A note on central limit theorems for quadratic variation in case of endogenous observation times

    Vetter, Mathias; Zwingmann, Tobias
    This paper is concerned with a central limit theorem for quadratic variation when observations come as exit times from a regular grid. We discuss the special case of a semimartingale with deterministic characteristics and finite activity jumps in detail and illustrate technical issues in more general situations.

  12. Adaptive density estimation based on a mixture of Gammas

    Bochkina, Natalia; Rousseau, Judith
    We consider the problem of Bayesian density estimation on the positive semiline for possibly unbounded densities. We propose a hierarchical Bayesian estimator based on the gamma mixture prior which can be viewed as a location mixture. We study convergence rates of Bayesian density estimators based on such mixtures. We construct approximations of the local Hölder densities, and of their extension to unbounded densities, to be continuous mixtures of gamma distributions, leading to approximations of such densities by finite mixtures. These results are then used to derive posterior concentration rates, with priors based on these mixture models. The rates are minimax...

  13. Estimating a smooth function on a large graph by Bayesian Laplacian regularisation

    Kirichenko, Alisa; van Zanten, Harry
    We study a Bayesian approach to estimating a smooth function in the context of regression or classification problems on large graphs. We derive theoretical results that show how asymptotically optimal Bayesian regularisation can be achieved under an asymptotic shape assumption on the underlying graph and a smoothness condition on the target function, both formulated in terms of the graph Laplacian. The priors we study are randomly scaled Gaussians with precision operators involving the Laplacian of the graph.

  14. A test of Gaussianity based on the Euler characteristic of excursion sets

    Di Bernardino, Elena; Estrade, Anne; León, José R.
    In the present paper, we deal with a stationary isotropic random field $X:{\mathbb{R}}^{d}\to{\mathbb{R}}$ and we assume it is partially observed through some level functionals. We aim at providing a methodology for a test of Gaussianity based on this information. More precisely, the level functionals are given by the Euler characteristic of the excursion sets above a finite number of levels. On the one hand, we study the properties of these level functionals under the hypothesis that the random field $X$ is Gaussian. In particular, we focus on the mapping that associates to any level $u$ the expected Euler characteristic of...

  15. Model selection for the segmentation of multiparameter exponential family distributions

    Cleynen, Alice; Lebarbier, Emilie
    We consider the segmentation problem of univariate distributions from the exponential family with multiple parameters. In segmentation, the choice of the number of segments remains a difficult issue due to the discrete nature of the change-points. In this general exponential family distribution framework, we propose a penalized $\log$-likelihood estimator where the penalty is inspired by papers of L. Birgé and P. Massart. The resulting estimator is proved to satisfy some oracle inequalities. We then further study the particular case of categorical variables by comparing the values of the key constants when derived from the specification of our general approach and...

  16. Optimal prediction for sparse linear models? Lower bounds for coordinate-separable M-estimators

    Zhang, Yuchen; Wainwright, Martin J.; Jordan, Michael I.
    For the problem of high-dimensional sparse linear regression, it is known that an $\ell_{0}$-based estimator can achieve a $1/n$ “fast” rate for prediction error without any conditions on the design matrix, whereas in the absence of restrictive conditions on the design matrix, popular polynomial-time methods only guarantee the $1/\sqrt{n}$ “slow” rate. In this paper, we show that the slow rate is intrinsic to a broad class of M-estimators. In particular, for estimators based on minimizing a least-squares cost function together with a (possibly nonconvex) coordinate-wise separable regularizer, there is always a “bad” local optimum such that the associated prediction error...

  17. Some properties of the autoregressive-aided block bootstrap

    Niebuhr, Tobias; Kreiss, Jens-Peter; Paparoditis, Efstathios
    We investigate properties of a hybrid bootstrap procedure for general, strictly stationary sequences, called the autoregressive-aided block bootstrap which combines a parametric autoregressive bootstrap with a nonparametric moving block bootstrap. The autoregressive-aided block bootstrap consists of two main steps, namely an autoregressive model fit and an ensuing (moving) block resampling of residuals. The linear parametric model-fit prewhitenes the time series so that the dependence structure of the remaining residuals gets closer to that of a white noise sequence, while the moving block bootstrap applied to these residuals captures nonlinear features that are not taken into account by the linear autoregressive...

  18. Adaptive wavelet multivariate regression with errors in variables

    Chichignoud, Michaël; Hoang, Van Ha; Pham Ngoc, Thanh Mai; Rivoirard, Vincent
    In the multidimensional setting, we consider the errors-in- variables model. We aim at estimating the unknown nonparametric multivariate regression function with errors in the covariates. We devise an adaptive estimators based on projection kernels on wavelets and a deconvolution operator. We propose an automatic and fully data driven procedure to select the wavelet level resolution. We obtain an oracle inequality and optimal rates of convergence over anisotropic Hölder classes. Our theoretical results are illustrated by some simulations.

  19. Prediction weighted maximum frequency selection

    Liu, Hongmei; Rao, J. Sunil
    Shrinkage estimators that possess the ability to produce sparse solutions have become increasingly important to the analysis of today’s complex datasets. Examples include the LASSO, the Elastic-Net and their adaptive counterparts. Estimation of penalty parameters still presents difficulties however. While variable selection consistent procedures have been developed, their finite sample performance can often be less than satisfactory. We develop a new strategy for variable selection using the adaptive LASSO and adaptive Elastic-Net estimators with $p_{n}$ diverging. The basic idea first involves using the trace paths of their LARS solutions to bootstrap estimates of maximum frequency (MF) models conditioned on dimension....

  20. Cross-calibration of probabilistic forecasts

    Strähl, Christof; Ziegel, Johanna
    When providing probabilistic forecasts for uncertain future events, it is common to strive for calibrated forecasts, that is, the predictive distribution should be compatible with the observed outcomes. Often, there are several competing forecasters of different skill. We extend common notions of calibration where each forecaster is analyzed individually, to stronger notions of cross-calibration where each forecaster is analyzed with respect to the other forecasters. In particular, cross-calibration distinguishes forecasters with respect to increasing information sets. We provide diagnostic tools and statistical tests to assess cross-calibration. The methods are illustrated in simulation examples and applied to probabilistic forecasts for inflation...

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