Convergence Rate Of Estimator Research Articles

Convergence rate of sieves estimates for an autoregressive Hilbert space

In this article, we provide a convergence rate of sieves estimates of the Hilbert autoregressive process kernel operator presented by Berhoune and Bensmain (2018). We found that the convergence rate is governed by the local expected values, variances, L 2 Bracketing entropy, and the approximation error of the sieve.

Calibrated Equilibrium Estimation and Double Selection for High-dimensional Partially Linear Measurement Error Models

In practice, measurement error data is frequently encountered and needs to be handled appropriately. As a result of additional bias induced by measurement error, many existing estimation methods fail to achieve satisfactory performances. This article studies high-dimensional partially linear measurement error models. It proposes a calibrated equilibrium (CARE) estimation method to calibrate the bias caused by measurement error and overcomes the technical difficulty of the objective function unbounded from below in high-dimensional cases due to non-convexity. To facilitate the applications of the CARE estimation method, a bootstrap approach for approximating the covariance matrix of measurement errors is introduced. For the high-dimensional or ultra-high dimensional partially linear measurement error models, a novel multiple testing method, the calibrated equilibrium multiple double selection (CARE–MUSE) algorithm, is proposed to control the false discovery rate (FDR). Under certain regularity conditions, we derive the oracle inequalities for estimation error and prediction risk, along with an upper bound on the number of falsely discovered signs for the CARE estimator. We further establish the convergence rate of the nonparametric function estimator. In addition, FDR and power guarantee for the CARE–MUSE algorithm are investigated under a weaker minimum signal condition, which is insufficient for the CARE estimator to achieve sign consistency. Extensive simulation studies and a real data application demonstrate the satisfactory finite sample performance of the proposed methods.

Analysis of Variance of Tensor Product Reproducing Kernel Hilbert Spaces on Metric Spaces

Many methods have been developed to analyze complex data, such as non-Euclidean shape, network, and manifold data. However, there is a lack of methods for studying interactions among complex data. In this paper, we first propose a novel kernel function for a metric space and construct its associated reproducing kernel Hilbert space. The new nonstationary kernel function provides a flexible and powerful tool for learning complex structures in non-Euclidean data. We then construct an analysis of variance (ANOVA) decomposition of the nonparametric regression function defined on metric space, which provides a hierarchical structure for investigating the main effects and interactions. We develop estimation and computational methods for a semi-parametric model with a multivariate function on a product of metric spaces modeled by the ANOVA decomposition. We establish the convergence rates of parameter and nonparametric function estimates. The application of the proposed methods to the Alzheimer’s Disease Neuroimaging Initiative hippocampus shape data confirms some existing and suggests some new interactions among hippocampal regions. Simulations indicate that the proposed methods work well.

A unique dielectric constant estimation for lunar surface through PolSAR model-based decomposition

Dielectric constant for the earth and planetary surfaces has been estimated using reflection coefficients in the past. A recent trend is to use model-based decomposition for dielectric constant retrieval from polarimetric synthetic aperture radar (polSAR) data. We examine the reported literature in this regard and propose a unique dielectric constant estimation (UDCE) algorithm using three-component decomposition technique. In UDCE, the dielectric constant is obtained directly from one of the elements of the measured coherency matrix in a single step. The dielectric constant estimate from the UDCE is independent of the volume scattering model when single-bounce or double-bounce scattering is dominant. This avoids error propagation from overestimation of volume scattering to the copolarization ratios, and in turn, to the dielectric constant, inherent in reported algorithms that use model-based decomposition. Consequently, a unique solution is obtained. We also demonstrate that the solution from the UDCE is unaffected by using a higher-order model-based decomposition. We evaluate the performance of the proposed UDCE algorithm over three Apollo 12, Apollo 15, and Apollo 17 landing sites on the lunar surface using Chandrayaan- 2 dual-frequency synthetic aperture radar (DFSAR) datasets. An excellent convergence rate for dielectric constant estimation is maintained over all three test sites. Using the proposed UDCE algorithm, the dielectric constant maps are produced for the lunar surface using full polSAR data for the first time. We observe that the generated dielectric constant maps capture all the ground truth features, previously unseen with such clarity.

Wasserstein-Kaplan-Meier Survival Regression

Survival analysis plays a pivotal role in medical research, offering valuable insights into the timing of events such as survival time. One common challenge in survival analysis is the necessity to adjust the survival function to account for additional factors, such as age, gender, and ethnicity. We propose an innovative regression model for right-censored survival data across heterogeneous populations, leveraging the Wasserstein space of probability measures. Our approach models the probability measure of survival time and the corresponding nonparametric Kaplan-Meier estimator for each subgroup as elements of the Wasserstein space. The Wasserstein space provides a flexible framework for modeling heterogeneous populations, allowing us to capture complex relationships between covariates and survival times. We address an underexplored aspect by deriving the non-asymptotic convergence rate of the Kaplan-Meier estimator to the underlying probability measure in terms of the Wasserstein metric. The proposed model is supported with a solid theoretical foundation including pointwise and uniform convergence rates, along with an efficient algorithm for model fitting. The proposed model effectively accommodates random variation that may exist in the probability measures across different subgroups, demonstrating superior performance in both simulations and two case studies compared to the Cox proportional hazards model and other alternative models. Supplementary materials for this article are available online.

A Nonlinear Suspension Road Roughness Recognition Method Based on NARX-PASCKF.

Road roughness significantly impacts vehicle safety and dynamic responses. For nonlinear suspension systems, the nonlinear characteristics often make it challenging for estimators to identify the actual road roughness accurately. This paper proposes a hybrid road roughness identification algorithm based on nonlinear auto-regressive with exogenous inputs (NARX) and a process noise adaptive square root cubature Kalman filter (PASCKF) to address this issue. Driven by vehicle acceleration data, an NARX-based road roughness identification system is constructed to mitigate the model uncertainties. Furthermore, a hybrid strategy is proposed. On the one hand, the accurate road roughness estimated by the NARX is converted into process noise covariance, enhancing the estimator's accuracy and convergence rate. Another switching strategy is proposed to optimize the non-convergence issues of the PASCKF. Finally, simulation and actual vehicle experiment data demonstrate that this approach offers superior identification accuracy and adaptability compared to the standalone SCKF algorithm.

Statistical inference of Poisson censored δ-shock model

In this article, the sufficient statistics, complete statistics, and a uniformly minimum variance unbiased estimate (UMVUE) of the failure parameters of the Poisson censored δ -shock model are obtained, which are based on the sample data of shock arrival time. The maximum likelihood estimates of the shock parameters are also obtained by using the EM algorithm, which is based on the total number of observed shock samples. Finally, the estimation errors and convergence rate of the EM iteration are simulated numerically using Matlab R2016b.

Robust causal inference for point exposures with missing confounders

AbstractLarge observational databases are often subject to missing data. As such, methods for causal inference must simultaneously handle confounding and missingness; surprisingly little work has been done at this intersection. Motivated by this, we propose an efficient and robust estimator of the causal average treatment effect from cohort studies when confounders are missing at random. The approach is based on a novel factorization of the likelihood that, unlike alternative methods, facilitates flexible modelling of nuisance functions (e.g., with state‐of‐the‐art machine learning methods) while maintaining nominal convergence rates of the final estimators. Simulated data, derived from an electronic health record‐based study of the long‐term effects of bariatric surgery on weight outcomes, verify the robustness properties of the proposed estimators in finite samples. Our approach may serve as a theoretical benchmark against which ad hoc methods may be assessed.

Estimation in functional partially linear spatial autoregressive model

Functional regression has been a hot topic in statistical research. However, not much work has been done when response variables are cross-sectionally dependent variables and explanatory variables contain a real-valued scalar variable and a functional-valued random variable. In this paper, we consider a new functional partially linear spatial autoregressive model. Based on the functional principal components analysis and basis function approximation, we obtain the estimators of the unknown parameter and functions through the instrumental variables estimation method. The asymptotic normality and convergence rates of estimators are proved under some mild conditions. In addition, we illustrate the finite sample performance of the proposed estimation method through simulation study and a real data analysis.

Effect of dimensionality on convergence rates of kernel ridge regression estimator

Despite the curse of dimensionality, kernel ridge regression often exhibits good performance in practical applications, even when the dimension is moderately large. However, it has been shown that kernel ridge regression cannot be free from the curse of dimensionality. Until now, the literature on kernel ridge regression has suggested that the gap between theory and practice in relation to dimensionality has not narrowed. In this study, we first investigate when the influence of dimensionality does not significantly affect the convergence rate of the kernel ridge regression. Specifically, we study the convergence rate of L2 and L∞ risks for the kernel ridge estimator, with a focus on reproducing kernel Hilbert space (RKHS) generated by a product kernel. We show that the univariate optimal convergence rate up to a logarithmic factor in L2 and L∞ risks can be achieved by controlling the size of the RKHS. The result of a numerical study confirms our theoretical findings.

Mean and covariance estimation for discretely observed high-dimensional functional data: Rates of convergence and division of observational regimes

Estimation of the mean and covariance parameters for functional data is a critical task, with local linear smoothing being a popular choice. In recent years, many scientific domains are producing multivariate functional data for which p, the number of curves per subject, is often much larger than the sample size n. In this setting of high-dimensional functional data, much of developed methodology relies on preliminary estimates of the unknown mean functions and the auto- and cross-covariance functions. This paper investigates the convergence rates of local linear estimators in terms of the maximal error across components and pairs of components for mean and covariance functions, respectively, in both L2 and uniform metrics. The local linear estimators utilize a generic weighting scheme that can adjust for differing numbers of discrete observations Nij across curves j and subjects i, where the Nij vary with n. Particular attention is given to the equal weight per observation (OBS) and equal weight per subject (SUBJ) weighting schemes. The theoretical results utilize novel applications of concentration inequalities for functional data and demonstrate that, similar to univariate functional data, the order of the Nij relative to p and n divides high-dimensional functional data into three regimes (sparse, dense, and ultra-dense), with the high-dimensional parametric convergence rate of log(p)/n1/2 being attainable in the latter two.

Local convergence rates of the nonparametric least squares estimator with applications to transfer learning

Local convergence rates of the nonparametric least squares estimator with applications to transfer learning

Nonparametric causal inference with confounders missing not at random

We consider the estimation and inference of Average Causal Effects (ACE) when confounders are missing not at random. The identification has been discussed in literature; however, limited effort has been devoted into developing feasible nonparametric inference methods. The primary challenge arises from the estimation process of the missingness mechanism, an ill‐posed problem that poses obstacles in establishing asymptotic theory. This paper contributes to filling this gap in the following ways. Firstly, we introduce a weak pseudo‐metric to guarantee a faster convergence rate of the missingness mechanism estimator. Secondly, we employ a representer to derive the explicit expression of the influence function. We also propose a practical and stable approach to estimate the variance and construct the confidence interval. We verify our theoretical results in the simulation studies.

On Stability Estimate of Optimal Transport Maps Using m-th Polynomial Convexity

In 1781, Gaspard Monge first proposed the practical problem of relocating building materials while minimizing workers’ effort. Mathematically, the problem can be reiterated as finding a mapping T0 that transforms a random variable (X) following probability measure (μ) into a random variable (Y) following probability measure (ν), with minimal cost. Afterward, it has been widely studied and applied in statistics, machine learning, and economics, which concern the study of “distance” between usually a pair of probability distributions. The focus of this paper is centered on investigating and generalizing stability estimates for optimal transport plans, particularly through the lens of strong polynomial convexity. Building on previous research using plug-in estimators to strengthen the convergence rate of discrete or semi-discrete estimators for optimal transport plans, this paper introduces a novel stability estimate leveraging L-Lipschitz continuity and a paradigmatic methodology based on polynomial convexity, the understanding of which remains limited.

Minimax estimation of functional principal components from noisy discretized functional data

AbstractFunctional Principal Component Analysis is a reference method for dimension reduction of curve data. Its theoretical properties are now well understood in the simplified case where the sample curves are fully observed without noise. However, functional data are noisy and necessarily observed on a finite discretization grid. Common practice consists in smoothing the data and then to compute the functional estimates, but the impact of this denoising step on the procedure's statistical performance are rarely considered. Here we prove new convergence rates for functional principal component estimators. We introduce a double asymptotic framework: one corresponding to the sampling size and a second to the size of the grid. We prove that estimates based on projection onto histograms show optimal rates in a minimax sense. Theoretical results are illustrated on simulated data and the method is applied to the visualization of genomic data.

The Kullback–Leibler Divergence and the Convergence Rate of Fast Covariance Matrix Estimators in Galaxy Clustering Analysis

We present a method to quantify the convergence rate of the fast estimators of the covariance matrices in the large-scale structure analysis. Our method is based on the Kullback–Leibler (KL) divergence, which describes the relative entropy of two probability distributions. As a case study, we analyze the delete-d jackknife estimator for the covariance matrix of the galaxy correlation function. We introduce the information factor or the normalized KL divergence with the help of a set of baseline covariance matrices to diagnose the information contained in the jackknife covariance matrix. Using a set of quick particle mesh mock catalogs designed for the Baryon Oscillation Spectroscopic Survey DR11 CMASS galaxy survey, we find that the jackknife resampling method succeeds in recovering the covariance matrix with 10 times fewer simulation mocks than that of the baseline method at small scales (s ≤ 40 h −1 Mpc). However, the ability to reduce the number of mock catalogs is degraded at larger scales due to the increasing bias on the jackknife covariance matrix. Note that the analysis in this paper can be applied to any fast estimator of the covariance matrix for galaxy clustering measurements.

Partial Derivatives Estimation of Multivariate Variance Function in Heteroscedastic Model via Wavelet Method

For derivative function estimation, conventional research only focuses on the derivative estimation of one-dimensional functions. This paper considers partial derivatives estimation of a multivariate variance function in a heteroscedastic model. A wavelet estimator of partial derivatives of a multivariate variance function is proposed. The convergence rates of a wavelet estimator under different estimation errors are discussed. It turns out that the strong convergence rate of the wavelet estimator is the same as the optimal uniform almost sure convergence rate of nonparametric function problems.

Wavelet estimations of a density function in two-class mixture model

This paper considers nonparametric estimations of a density function in a two-class mixture model. A linear wavelet estimator and an adaptive wavelet estimator are constructed. Upper bound estimations over $ L^{p}\; (1\leq p < +\infty) $ risk of those wavelet estimators are proved in Besov spaces. When $ \tilde{p}\geq p\geq1 $, the convergence rate of adaptive wavelet estimator is the same as the linear estimator up to a $ \ln n $ factor. The adaptive wavelet estimator can get better than the linear estimator in the case of $ 1\leq \tilde{p} < p $. Finally, some numerical experiments are presented to validate the theoretical results.

Adaptively robust high-dimensional matrix factor analysis under Huber loss function

The explosion of data volume and the expansion in data dimensionality have led to a critical challenge in analyzing high-dimensional matrix time series for big data-related applications. In this regard, factor models for matrix-valued high-dimensional time series provide a powerful tool, that reduces the dimensionality of the variables with low-rank structures. However, existing high-dimensional matrix factor models suffer from two limitations in complex scenarios. One is that it is difficult to make robust inferences for datasets with heavy-tailed distributions. The other is that existing models require additional parameters for fine-tuning to guarantee performance. We propose an adaptively robust high-dimensional matrix factor model based on a specified Huber loss function to tackle the challenges mentioned above. An efficient iterative algorithm is provided to consistently determine the additional parameters of our model for robust estimation. The robustness of the model estimation is greatly improved by incorporating the Huber loss. Furthermore, we theoretically investigate the proposed method and derive the convergence rates of the robust estimators to examine its performance. Simulations show that the proposed method outperforms previous models in the estimation of heavy-tailed data. A real-world data analysis on a financial portfolio dataset illustrates that the method can be used to extract useful knowledge from high-dimensional matrix time series.

Regression analysis of longitudinal data with mixed synchronous and asynchronous longitudinal covariates

In linear models, omitting a covariate that is orthogonal to covariates in the model does not result in biased coefficient estimation. This generally does not hold for longitudinal data, where additional assumptions are needed to get an unbiased coefficient estimation in addition to the orthogonality between omitted longitudinal covariates and longitudinal covariates in the model. We propose methods to mitigate the omitted variable bias under weaker assumptions. A two-step estimation procedure is proposed to infer the asynchronous longitudinal covariates when such covariates are observed. For mixed synchronous and asynchronous longitudinal covariates, we get a parametric convergence rate for the coefficient estimation of the synchronous longitudinal covariates by the two-step method. Extensive simulation studies provide numerical support for the theoretical findings. We illustrate the performance of our method on a dataset from the Alzheimer’s Disease Neuroimaging Initiative study.