A key tool to carry out inference on the unknown copula when modeling a continuous multivariate distribution is a nonparametric estimator known as the empirical copula. One popular way of approximating its sampling distribution consists of using the multiplier bootstrap. The latter is however characterized by a high implementation cost. Given the rank-based nature of the empirical copula, the classical empirical bootstrap of Efron does not appear to be a natural alternative, as it relies on resamples which contain ties. The aim of this work is to investigate the use of subsampling in the aforementioned framework. The latter consists of basing the inference on statistic values computed from subsamples of the initial data. One of its advantages in the rank-based context under consideration is that the formed subsamples do not contain ties. Another advantage is its asymptotic validity under minimalistic conditions. In this work, we show the asymptotic validity of subsampling for several (weighted, smooth) empirical copula processes both in the case of serially independent observations and time series. In the former case, subsampling is observed to be substantially better than the empirical bootstrap and equivalent, overall, to the multiplier bootstrap in terms of finite-sample performance.
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The empirical measure resulting from the nearest neighbors to a given point - \textit{the nearest neighbor measure} - is introduced and studied as a central statistical quantity. First, the associated empirical process is shown to satisfy a uniform central limit theorem under a (local) bracketing entropy condition on the underlying class of functions (reflecting the localizing nature of the nearest neighbor algorithm). Second a uniform non-asymptotic bound is established under a well-known condition, often referred to as Vapnik-Chervonenkis, on the uniform entropy numbers. The covariance of the Gaussian limit obtained in the uniform central limit theorem is equal to the conditional covariance operator (given the point of interest). This suggests the possibility of extending standard approaches - non local - replacing simply the standard empirical measure by the nearest neighbor measure while using the same way of making inference but with the nearest neighbors only instead of the full data.
We consider Bayesian analysis on high-dimensional spheres with angular central Gaussian priors. These priors model antipodally symmetric directional data, are easily defined in Hilbert spaces and occur, for instance, in Bayesian binary classification and level set inversion. In this paper we derive efficient Markov chain Monte Carlo methods for approximate sampling of posteriors with respect to these priors. Our approaches rely on lifting the sampling problem to the ambient Hilbert space and exploit existing dimension-independent samplers in linear spaces. By a push-forward Markov kernel construction we then obtain Markov chains on the sphere, which inherit reversibility and spectral gap properties from samplers in linear spaces. Moreover, our proposed algorithms show dimension-independent efficiency in numerical experiments.
We propose two robust methods for testing hypotheses on unknown parameters of predictive regression models under heterogeneous and persistent volatility as well as endogenous, persistent and/or fat-tailed regressors and errors. The proposed robust testing approaches are applicable both in the case of discrete and continuous time models. Both of the methods use the Cauchy estimator to effectively handle the problems of endogeneity, persistence and/or fat-tailedness in regressors and errors. The difference between our two methods is how the heterogeneous volatility is controlled. The first method relies on robust t-statistic inference using group estimators of a regression parameter of interest proposed in Ibragimov and Muller, 2010. It is simple to implement, but requires the exogenous volatility assumption. To relax the exogenous volatility assumption, we propose another method which relies on the nonparametric correction of volatility. The proposed methods perform well compared with widely used alternative inference procedures in terms of their finite sample properties.
This paper considers estimating functional-coefficient models in panel quantile regression with individual effects, allowing the cross-sectional and temporal dependence for large panel observations. A latent group structure is imposed on the heterogenous quantile regression models so that the number of nonparametric functional coefficients to be estimated can be reduced considerably. With the preliminary local linear quantile estimates of the subject-specific functional coefficients, a classic agglomerative clustering algorithm is used to estimate the unknown group structure and an easy-to-implement ratio criterion is proposed to determine the group number. The estimated group number and structure are shown to be consistent. Furthermore, a post-grouping local linear smoothing method is introduced to estimate the group-specific functional coefficients, and the relevant asymptotic normal distribution theory is derived with a normalisation rate comparable to that in the literature. The developed methodologies and theory are verified through a simulation study and showcased with an application to house price data from UK local authority districts, which reveals different homogeneity structures at different quantile levels.
State-space models (SSMs) are a common tool for modeling multi-variate discrete-time signals. The linear-Gaussian (LG) SSM is widely applied as it allows for a closed-form solution at inference, if the model parameters are known. However, they are rarely available in real-world problems and must be estimated. Promoting sparsity of these parameters favours both interpretability and tractable inference. In this work, we propose GraphIT, a majorization-minimization (MM) algorithm for estimating the linear operator in the state equation of an LG-SSM under sparse prior. A versatile family of non-convex regularization potentials is proposed. The MM method relies on tools inherited from the expectation-maximization methodology and the iterated reweighted-l1 approach. In particular, we derive a suitable convex upper bound for the objective function, that we then minimize using a proximal splitting algorithm. Numerical experiments illustrate the benefits of the proposed inference technique.
The hierarchical small-world network is a real-world network. It models well the benefit transmission web of the pyramid selling in China and many other countries. In this paper, by applying the spectral graph theory, we study three important aspects of the consensus problem in the hierarchical small-world network: convergence speed, communication time-delay robustness, and network coherence. Firstly, we explicitly determine the Laplacian eigenvalues of the hierarchical small-world network by making use of its treelike structure. Secondly, we find that the consensus algorithm on the hierarchical small-world network converges faster than that on some well-studied sparse networks, but is less robust to time delay. The closed-form of the first-order and the second-order network coherence are also derived. Our result shows that the hierarchical small-world network has an optimal structure of noisy consensus dynamics. Therefore, we provide a positive answer to two open questions of Yi \emph{et al}. Finally, we argue that some network structure characteristics, such as large maximum degree, small average path length, and large vertex and edge connectivity, are responsible for the strong robustness with respect to external perturbations.
Generalized linear mixed models are powerful tools for analyzing clustered data, where the unknown parameters are classically (and most commonly) estimated by the maximum likelihood and restricted maximum likelihood procedures. However, since the likelihood based procedures are known to be highly sensitive to outliers, M-estimators have become popular as a means to obtain robust estimates under possible data contamination. In this paper, we prove that, for sufficiently smooth general loss functions defining the M-estimators in generalized linear mixed models, the tail probability of the deviation between the estimated and the true regression coefficients have an exponential bound. This implies an exponential rate of consistency of these M-estimators under appropriate assumptions, generalizing the existing exponential consistency results from univariate to multivariate responses. We have illustrated this theoretical result further for the special examples of the maximum likelihood estimator and the robust minimum density power divergence estimator, a popular example of model-based M-estimators, in the settings of linear and logistic mixed models, comparing it with the empirical rate of convergence through simulation studies.
When data is collected in an adaptive manner, even simple methods like ordinary least squares can exhibit non-normal asymptotic behavior. As an undesirable consequence, hypothesis tests and confidence intervals based on asymptotic normality can lead to erroneous results. We propose a family of online debiasing estimators to correct these distributional anomalies in least squares estimation. Our proposed methods take advantage of the covariance structure present in the dataset and provide sharper estimates in directions for which more information has accrued. We establish an asymptotic normality property for our proposed online debiasing estimators under mild conditions on the data collection process and provide asymptotically exact confidence intervals. We additionally prove a minimax lower bound for the adaptive linear regression problem, thereby providing a baseline by which to compare estimators. There are various conditions under which our proposed estimators achieve the minimax lower bound. We demonstrate the usefulness of our theory via applications to multi-armed bandit, autoregressive time series estimation, and active learning with exploration.
We analyze stochastic gradient descent (SGD) type algorithms on a high-dimensional sphere which is parameterized by a neural network up to a normalization constant. We provide a new algorithm for the setting of supervised learning and show its convergence both theoretically and numerically. We also provide the first proof of convergence for the unsupervised setting, which corresponds to the widely used variational Monte Carlo (VMC) method in quantum physics.
In this article, we propose a new class of consistent tests for $p$-variate normality. These tests are based on the characterization of the standard multivariate normal distribution, that the Hessian of the corresponding cumulant generating function is identical to the $p\times p$ identity matrix and the idea of decomposing the information from the joint distribution into the dependence copula and all marginal distributions. Under the null hypothesis of multivariate normality, our proposed test statistic is independent of the unknown mean vector and covariance matrix so that the distribution-free critical value of the test can be obtained by Monte Carlo simulation. We also derive the asymptotic null distribution of proposed test statistic and establish the consistency of the test against different fixed alternatives. Last but not least, a comprehensive and extensive Monte Carlo study also illustrates that our test is a superb yet computationally convenient competitor to many well-known existing test statistics.
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