Convergence Rate Of Estimator Research Articles

Pointwise Wavelet Estimations for a Regression Model in Local Hölder Space

This paper considers an unknown functional estimation problem in a regression model with multiplicative and additive noise. A linear wavelet estimator is first constructed by a wavelet projection operator. The convergence rate under the pointwise error of linear wavelet estimators is studied in local Hölder space. A nonlinear wavelet estimator is provided by the hard thresholding method in order to obtain an adaptive estimator. The convergence rate of the nonlinear estimator is the same as the linear estimator up to a logarithmic term. Finally, it should be pointed out that the convergence rates of two wavelet estimators are consistent with the optimal convergence rate on pointwise nonparametric estimation.

Convergence rate of plugin estimates for functional parameters with applications to locally-stationary time-series

Some distribution parameter estimates depend on nuisance parameters, and it is a common practice to plug in their estimates. The convergence properties of the resulting estimate are of interest. We consider the case where both estimators – the one plugged in, and its recipient – are random quantities, while literature covers mostly plugin to a deterministic function. We present a way of computing this final convergence rate, based on the convergence rate of a nonrandom, but perturbed quantity plugged in. The theoretical result is applied to the construction of simultaneous confidence band for auto-correlation functions of a locally-stationary time series.

Distributed Localization for Dynamic Multiagent Systems With Randomly Varying Trajectory Lengths

This article considered the persistent real-time localization problem of dynamic multiagent systems with repetitive running characteristics under directed graph. The trajectory length of the agent is randomly varying caused by system constraints and external environment. A novel distributed iterative learning localization estimation method with full historical average data compensation is designed. The average value of all available historical data in the previous operations is used to compensate the incomplete trajectory. The designed diagnosis mechanism and search mechanism are used to determine whether the agent has stopped running and to screen out the available historical data. In order to reduce the computation burden, an improved distributed localization algorithm with receding horizon historical average data compensation is proposed. The asymptotic convergence of the estimation algorithms in the sense of mathematical expectation is derived through the rigorous analysis. Meanwhile, the influence of the incomplete repetitive trajectory of the agent on estimation error and convergence rate is also analyzed. The experimental result based on QBot-2e robot platform verifies the realizability of the proposed method.

Convergence rate analysis for fixed-point iterations of generalized averaged nonexpansive operators

We estimate convergence rates for fixed-point iterations of a class of nonlinear operators which are partially motivated by convex optimization problems. We introduce the notion of the generalized averaged nonexpansive (GAN) operator with a positive exponent, and provide convergence rate analysis of the fixed-point iteration of the GAN operator. The proposed generalized averaged nonexpansiveness is weaker than averaged nonexpansiveness while stronger than nonexpansiveness. We show that the fixed-point iteration of a GAN operator with a positive exponent converges to its fixed-point and estimate the local convergence rate (the convergence rate in terms of the distance between consecutive iterates) depending on the range of the exponent. We prove that the fixed-point iteration of a GAN operator with a positive exponent strictly smaller than 1 can achieve an exponential global convergence rate (the convergence rate in terms of the distance between an iterate and the solution). Furthermore, we establish the global convergence rate of the fixed-point iteration of a GAN operator, depending on both the exponent of generalized averaged nonexpansiveness and the exponent of the H $$\ddot{\text {o}}$$ lder regularity, if the GAN operator is also H $$\ddot{\text {o}}$$ lder regular. We then apply the established theory to three types of convex optimization problems that appear often in data science to design fixed-point iterative algorithms for solving these optimization problems and to analyze their convergence properties.

Rates of convergence in the two-island and isolation-with-migration models

A number of powerful demographic inference methods have been developed in recent years, with the goal of fitting rich evolutionary models to genetic data obtained from many populations. In this paper we investigate the statistical performance of these methods in the specific case where there is continuous migration between populations. Compared with earlier work, migration significantly complicates the theoretical analysis and requires new techniques. We employ the theories of phase-type distributions and concentration of measure in order to study the two-island and isolation-with-migration models, resulting in both upper and lower bounds on rates of convergence for parametric estimators in migration models. For the upper bounds, we consider inferring rates of coalescent and migration on the basis of directly observing pairwise coalescent times, and, more realistically, when (conditionally) Poisson-distributed mutations dropped on latent trees are observed. We complement these upper bounds with information-theoretic lower bounds which establish a limit, in terms of sample size, below which inference is effectively impossible.

Local linear approach: Conditional density estimate for functional and censored data

Abstract Let Y Y be a random real response, which is subject to right censoring by another random variable C C . In this paper, we study the nonparametric local linear estimation of the conditional density of a scalar response variable and when the covariable takes values in a semi-metric space. Our main aim is to prove under some regularity conditions both the pointwise and the uniform almost-sure consistencies with convergence rates of the conditional density estimator related by this estimation procedure.

Bias-Correction and Test for Mark-Point Dependence with Replicated Marked Point Processes

Mark-point dependence plays a critical role in research problems that can be fitted into the general framework of marked point processes. In this work, we focus on adjusting for mark-point dependence when estimating the mean and covariance functions of the mark process, given independent replicates of the marked point process. We assume that the mark process is a Gaussian process and the point process is a log-Gaussian Cox process, where the mark-point dependence is generated through the dependence between two latent Gaussian processes. Under this framework, naive local linear estimators ignoring the mark-point dependence can be severely biased. We show that this bias can be corrected using a local linear estimator of the cross-covariance function and establish uniform convergence rates of the bias-corrected estimators. Furthermore, we propose a test statistic based on local linear estimators for mark-point independence, which is shown to converge to an asymptotic normal distribution in a parametric n -convergence rate. Model diagnostics tools are developed for key model assumptions and a robust functional permutation test is proposed for a more general class of mark-point processes. The effectiveness of the proposed methods is demonstrated using extensive simulations and applications to two real data examples. Supplementary materials for this article are available online.

Determining the number of factors in constrained factor models via Bayesian information criterion

This paper estimates the number of factors in constrained and partially constrained factor models (Tsai and Tsay, 2010) based on constrained Bayesian information criterion (CBIC). Following Bai and Ng (2002), the estimation of the number of factors depends on the tradeoff between good fit and parsimony, so we first derive the convergence rate of constrained factor estimates under the framework of large cross-sections (N) and large time dimensions (T). Furthermore, we demonstrate that the penalty for overfitting can be a function of N alone, so the BIC form, which does not work in the case of (unconstrained) approximate factor models, consistently estimates the number of factors in constrained factor models. We then conduct Monte Carlo simulations to show that our proposed CBIC has good finite sample performance and outperforms competing methods.

Finite-difference time-domain simulations: Verification on head-related transfer functions of a rigid sphere model.

This paper presents a verification procedure for finite-difference time-domain-simulated head-related transfer functions (HRTFs) from a simplified model of a human head, a sphere. The analytic solution required by the code verification is computed with the multipole reexpansion technique and used to estimate convergence rates. A solution verification process follows in which asymptotic predictions are computed. For the HRTFs considered and employed grids, results show that the convergence rates attain the expected first-order accuracy at lower frequencies, after which scattered estimates are observed. Results also reveal that the asymptotic predictions are accurate up to 10 kHz, after which bias is observed.

Empirical likelihood and uniform convergence rates for dyadic kernel density estimation

This paper studies the asymptotic properties of and alternative inference methods for kernel density estimation (KDE) for dyadic data. We first establish uniform convergence rates for dyadic KDE. Secondly, we propose a modified jackknife empirical likelihood procedure for inference. The proposed test statistic is asymptotically pivotal regardless of presence of dyadic clustering. The results are further extended to cover the practically relevant case of incomplete dyadic data. Simulations show that this modified jackknife empirical likelihood-based inference procedure delivers precise coverage probabilities even with modest sample sizes and with incomplete dyadic data. Finally, we illustrate the method by studying airport congestion in the United States.

Threshold estimation for continuous three‐phase polynomial regression models with constant mean in the middle regime

AbstractThis paper considers a continuous three‐phase polynomial regression model with two threshold points for dependent data with heteroscedasticity. We assume the model is polynomial of order zero in the middle regime, and is polynomial of higher orders elsewhere. We denote this model by , which includes models with one or no threshold points, denoted by and , respectively, as special cases. We provide an ordered iterative least squares (OiLS) method when estimating and establish the consistency of the OiLS estimators under mild conditions. When the underlying model is and is th‐order differentiable but not th‐order differentiable at the threshold point, we further show the convergence rate of the OiLS estimators, which can be faster than the convergence rate given in Feder when . We also apply a model‐selection procedure for selecting ; . When the underlying model exists, we establish the selection consistency under the aforementioned conditions. Finally, we conduct simulation experiments to demonstrate the finite‐sample performance of our asymptotic results.

Optimal convergence rates of Bayesian wavelet estimation with a novel empirical prior in nonparametric regression model

This paper provides a novel empirical prior for Bayesian wavelet in the nonparametric regression model. The prior centres the wavelet coefficient at the individual sample observation, and we make a modification on the empirical prior by introducing a fractional power. The optimal rates of the posterior contraction and posterior mean estimator are established by the fractional likelihood method over Besov spaces. Comparison with some methods in the literature indicates that the modified empirical method is superior in terms of the mean squared error.

Nonparametric regression for locally stationary random fields under stochastic sampling design

In this study, we develop an asymptotic theory of nonparametric regression for locally stationary random fields (LSRFs) {Xs,An:s∈Rn} in Rp observed at irregularly spaced locations in Rn=[0,An]d⊂Rd. We first derive the uniform convergence rate of general kernel estimators, followed by the asymptotic normality of an estimator for the mean function of the model. Moreover, we consider additive models to avoid the curse of dimensionality arising from the dependence of the convergence rate of estimators on the number of covariates. Subsequently, we derive the uniform convergence rate and joint asymptotic normality of the estimators for additive functions. We also introduce approximately mn-dependent RFs to provide examples of LSRFs. We find that these RFs include a wide class of Lévy-driven moving average RFs.

Limit behaviors of the estimator of non parametric regression model based on extended negatively dependent errors

In this article, we study the complete consistency for the estimator of non parametric regression model based on extended negatively dependent errors, and obtain the convergence rates of the complete consistency by using the inequalities for extended negatively dependent random variables. As an application, the complete consistency and convergence rate for the nearest neighbor estimator are obtained. Finally, some simulations are illustrated.

Channel Estimation for mmWave Massive MIMO Systems based on Deep Learning

Channel estimation is a key part of communication systems. However, the channel of millimeter-Wave (mmWave) Massive Multiple-Input Multiple-Output (Massive-MIMO) system has sparse characteristics, and the conventional channel estimation method is prone to noise factors and tends to achieve low estimation accuracy. Therefore, in this paper a channel estimation method is proposed for mmWave Massive MIMO systems based on deep learning. Firstly, a dataset to simulate a real-world environment, is generated by setting specific parameters. Furthermore, the generated channel matrix is adopted as the input of the neural network. Secondly, the attention mechanism is integrated into the deep learning method with ResUNet to enhance the ability of feature extraction. Finally, the channel estimation matrix is obtained via the aforementioned network model. The experimental results indicate that the Massive-MIMO method is superior to the conventional channel estimation methods in channel estimation accuracy and convergence rate, and has a good application prospects.

Sparse regression for low-dimensional time-dynamic varying coefficient models with application to air quality data

Time dynamic varying coefficient models play an important role in applications of biology, medicine, environment, finance, etc. Traditional methods, such as kernel smoothing and spline smoothing, are popular. But explicit expressions are unavailable using these methods, and the convergence rate of coefficient function estimators is slow. To address these problems, we expand the varying component with appropriate basis functions. And then we solve a sparse regression problem via a sequential thresholded least-squares estimator. The “parameterization” leads to explicit expressions and fast computation speed. Convergence of the sequential thresholded least squares algorithm is guaranteed. The asymptotic distribution of the coefficient function estimator is derived under certain assumptions. Our simulation shows the proposed method has higher precision and computing speed. Finally, our proposed method is applied to the study of PM concentration in Beijing. We analyze the relationship between PM and other impact factors.

An improved initial alignment method based on adaptive robust CKF algorithm

When the misalignment angle is a large angle in the strapdown inertial navigation system (SINS), it is necessary to establish a nonlinear error model to estimate the error. Hence, an improved initial alignment method based on adaptive robust CKF algorithm is proposed in this paper. Firstly, based on the analysis results, SINS/GPS nonlinear error model is established. Secondly, in the view of observation gross errors and inaccurate noise statistical characteristics, adaptive robust CKF algorithm is designed. Finally, according to simulation analysis and experiment, adaptive robust CKF algorithm can augment the stability, improve filter estimation accuracy and convergence rate, which significantly improves the initial alignment ability of strapdown inertial navigation system at large azimuth misalignment angle.

Uniform convergence rates for nonparametric estimators smoothed by the beta kernel

AbstractThis paper provides a set of uniform consistency results with rates for nonparametric density and regression estimators smoothed by the beta kernel having support on the unit interval. Weak and strong uniform convergence is explored on the basis of expanding compact sets and general sequences of smoothing parameters. The results in this paper are useful for asymptotic analysis of two‐step semiparametric estimation using a first‐step kernel estimate as a plug‐in. We provide simulations and a real data example illustrating attractive properties of the estimators.

Adaptive estimation for functional data: Using a framelet block‐thresholding method

This article considers a framelet block‐thresholding method for estimating mean and covariance functions from discretely sampled noisy observations. Estimated convergence rates are established for all types of sampling schemes. In particular, the results reveal a phase transition phenomenon related to the number of observations on each curve. It is shown that the proposed procedures are adaptive in automatically adjusting the smoothness properties of the underlying mean and covariance functions. In contrast, theoretical results for other smoothing methods hold in the setting where smoothness parameters are assumed to be known, since the regularization parameters of estimators that depend on smoothness properties need to be chosen carefully. Simulation studies are provided to offer empirical support for the theoretical results. A comparison with other methods demonstrates that the proposed method outperforms in adaptivity. An application to a real dataset is also provided to illustrate the proposed estimation procedure.

Truncated estimation in functional generalized linear regression models

Functional generalized linear models investigate the effect of functional predictors on a scalar response. An interesting case is when the functional predictor is thought to exert an influence on the conditional mean of the response only through its values up to a certain point in the domain. In the literature, models with this type of restriction on the functional effect have been termed truncated or historical regression models. A penalized likelihood estimator is formulated by combining a structured variable selection method with a localized B-spline expansion of the regression coefficient function. In addition to a smoothing penalty that is typical for functional regression, a nested group lasso penalty is also included which guarantees the sequential entering of B-splines and thus induces the desired truncation on the estimator. An optimization scheme is developed to compute the solution path efficiently when varying the truncation tuning parameter. The convergence rate of the coefficient function estimator and consistency of the truncation point estimator are given under suitable smoothness assumptions. The proposed method is demonstrated through simulations and an application involving the effects of blood pressure values in patients who suffered a spontaneous intracerebral hemorrhage.