This study develops an asymptotic theory for estimating the time-varying characteristics of locally stationary functional time series. We investigate a kernel-based method to estimate the time-varying covariance operator and the time-varying mean function of a locally stationary functional time series. In particular, we derive the convergence rate of the kernel estimator of the covariance operator and associated eigenvalue and eigenfunctions and establish a central limit theorem for the kernel-based locally weighted sample mean. As applications of our results, we discuss the prediction of locally stationary functional time series and methods for testing the equality of time-varying mean functions in two functional samples.
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This work considers Gaussian process interpolation with a periodized version of the Mat{\'e}rn covariance function (Stein, 1999, Section 6.7) with Fourier coefficients $\phi$($\alpha$^2 + j^2)^(--$\nu$--1/2). Convergence rates are studied for the joint maximum likelihood estimation of $\nu$ and $\phi$ when the data is sampled according to the model. The mean integrated squared error is also analyzed with fixed and estimated parameters, showing that maximum likelihood estimation yields asymptotically the same error as if the ground truth was known. Finally, the case where the observed function is a ''deterministic'' element of a continuous Sobolev space is also considered, suggesting that bounding assumptions on some parameters can lead to different estimates.
Positive and unlabelled learning is an important problem which arises naturally in many applications. The significant limitation of almost all existing methods lies in assuming that the propensity score function is constant (SCAR assumption), which is unrealistic in many practical situations. Avoiding this assumption, we consider parametric approach to the problem of joint estimation of posterior probability and propensity score functions. We show that under mild assumptions when both functions have the same parametric form (e.g. logistic with different parameters) the corresponding parameters are identifiable. Motivated by this, we propose two approaches to their estimation: joint maximum likelihood method and the second approach based on alternating maximization of two Fisher consistent expressions. Our experimental results show that the proposed methods are comparable or better than the existing methods based on Expectation-Maximisation scheme.
In this paper, we investigate the matrix estimation problem in the multi-response regression model with measurement errors. A nonconvex error-corrected estimator based on a combination of the amended loss function and the nuclear norm regularizer is proposed to estimate the matrix parameter. Then under the (near) low-rank assumption, we analyse statistical and computational theoretical properties of global solutions of the nonconvex regularized estimator from a general point of view. In the statistical aspect, we establish the nonasymptotic recovery bound for any global solution of the nonconvex estimator, under restricted strong convexity on the loss function. In the computational aspect, we solve the nonconvex optimization problem via the proximal gradient method. The algorithm is proved to converge to a near-global solution and achieve a linear convergence rate. In addition, we also verify sufficient conditions for the general results to be held, in order to obtain probabilistic consequences for specific types of measurement errors, including the additive noise and missing data. Finally, theoretical consequences are demonstrated by several numerical experiments on corrupted errors-in-variables multi-response regression models. Simulation results reveal excellent consistency with our theory under high-dimensional scaling.
The ultimate purpose of the statistical analysis of ordinal patterns is to characterize the distribution of the features they induce. In particular, knowing the joint distribution of the pair Entropy-Statistical Complexity for a large class of time series models would allow statistical tests that are unavailable to date. Working in this direction, we characterize the asymptotic distribution of the empirical Shannon's Entropy for any model under which the true normalized Entropy is neither zero nor one. We obtain the asymptotic distribution from the Central Limit Theorem (assuming large time series), the Multivariate Delta Method, and a third-order correction of its mean value. We discuss the applicability of other results (exact, first-, and second-order corrections) regarding their accuracy and numerical stability. Within a general framework for building test statistics about Shannon's Entropy, we present a bilateral test that verifies if there is enough evidence to reject the hypothesis that two signals produce ordinal patterns with the same Shannon's Entropy. We applied this bilateral test to the daily maximum temperature time series from three cities (Dublin, Edinburgh, and Miami) and obtained sensible results.
The precision matrix that encodes conditional linear dependency relations among a set of variables forms an important object of interest in multivariate analysis. Sparse estimation procedures for precision matrices such as the graphical lasso (Glasso) gained popularity as they facilitate interpretability, thereby separating pairs of variables that are conditionally dependent from those that are independent (given all other variables). Glasso lacks, however, robustness to outliers. To overcome this problem, one typically applies a robust plug-in procedure where the Glasso is computed from a robust covariance estimate instead of the sample covariance, thereby providing protection against outliers. In this paper, we study such estimators theoretically, by deriving and comparing their influence function, sensitivity curves and asymptotic variances.
In this paper, we consider function-indexed normalized weighted integrated periodograms for equidistantly sampled multivariate continuous-time state space models which are multivariate continuous-time ARMA processes. Thereby, the sampling distance is fixed and the driving L\'evy process has at least a finite fourth moment. Under different assumptions on the function space and the moments of the driving L\'evy process we derive a central limit theorem for the function-indexed normalized weighted integrated periodogram. Either the assumption on the function space or the assumption on the existence of moments of the L\'evy process is weaker. Furthermore, we show the weak convergence in both the space of continuous functions and in the dual space to a Gaussian process and give an explicit representation of the covariance function. The results can be used to derive the asymptotic behavior of the Whittle estimator and to construct goodness-of-fit test statistics as the Grenander-Rosenblatt statistic and the Cram\'er-von Mises statistic. We present the exact limit distributions of both statistics and show their performance through a simulation study.
The parameters of the log-logistic distribution are generally estimated based on classical methods such as maximum likelihood estimation, whereas these methods usually result in severe biased estimates when the data contain outliers. In this paper, we consider several alternative estimators, which not only have closed-form expressions, but also are quite robust to a certain level of data contamination. We investigate the robustness property of each estimator in terms of the breakdown point. The finite sample performance and effectiveness of these estimators are evaluated through Monte Carlo simulations and a real-data application. Numerical results demonstrate that the proposed estimators perform favorably in a manner that they are comparable with the maximum likelihood estimator for the data without contamination and that they provide superior performance in the presence of data contamination.
A general framework with a series of different methods is proposed to improve the estimate of convex function (or functional) values when only noisy observations of the true input are available. Technically, our methods catch the bias introduced by the convexity and remove this bias from a baseline estimate. Theoretical analysis are conducted to show that the proposed methods can strictly reduce the expected estimate error under mild conditions. When applied, the methods require no specific knowledge about the problem except the convexity and the evaluation of the function. Therefore, they can serve as off-the-shelf tools to obtain good estimate for a wide range of problems, including optimization problems with random objective functions or constraints, and functionals of probability distributions such as the entropy and the Wasserstein distance. Numerical experiments on a wide variety of problems show that our methods can significantly improve the quality of the estimate compared with the baseline method.
Entrywise Estimation of Singular Vectors of Low-Rank Matrices with Heteroskedasticity and Dependence
We propose an estimator for the singular vectors of high-dimensional low-rank matrices corrupted by additive subgaussian noise, where the noise matrix is allowed to have dependence within rows and heteroskedasticity between them. We prove finite-sample $\ell_{2,\infty}$ bounds and a Berry-Esseen theorem for the individual entries of the estimator, and we apply these results to high-dimensional mixture models. Our Berry-Esseen theorem clearly shows the geometric relationship between the signal matrix, the covariance structure of the noise, and the distribution of the errors in the singular vector estimation task. These results are illustrated in numerical simulations. Unlike previous results of this type, which rely on assumptions of gaussianity or independence between the entries of the additive noise, handling the dependence between entries in the proofs of these results requires careful leave-one-out analysis and conditioning arguments. Our results depend only on the signal-to-noise ratio, the sample size, and the spectral properties of the signal matrix.
We study the problem of estimating an unknown parameter in a distributed and online manner. Existing work on distributed online learning typically either focuses on asymptotic analysis, or provides bounds on regret. However, these results may not directly translate into bounds on the error of the learned model after a finite number of time-steps. In this paper, we propose a distributed online estimation algorithm which enables each agent in a network to improve its estimation accuracy by communicating with neighbors. We provide non-asymptotic bounds on the estimation error, leveraging the statistical properties of the underlying model. Our analysis demonstrates a trade-off between estimation error and communication costs. Further, our analysis allows us to determine a time at which the communication can be stopped (due to the costs associated with communications), while meeting a desired estimation accuracy. We also provide a numerical example to validate our results.
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