Convergence Rate Of Estimator Research Articles

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.

Least squares estimation for distribution-dependent stochastic differential delay equations

<p style='text-indent:20px;'>The parametric estimation of drift parameter for distribution - dependent stochastic differential delay equations with a small diffusion is presented. The principle technique of our investigation is to construct an appropriate contrast function and carry out a limiting type of argument to show the consistency and convergence rate of the least squares estimator of the drift parameter via interacting particle systems. In addition, two examples are constructed to demonstrate the effectiveness of our work.</p>

Kernel density estimation under widely orthant dependence

The kernel density estimator for widely orthant dependent random variables is studied. The exponential inequalities and the exponential rate for the estimator of a density function with a uniform version over compact sets are investigated. Further, the consistency of the estimator is proved. The results are generalizations of some existing outcomes for both associated and negatively associated samples. The convergence rate of the kernel density estimator is illustrated via a simulation study. Moreover, a real data analysis is presented.

Empirical process theory for nonsmooth functions under functional dependence

We provide an empirical process theory for locally stationary processes over nonsmooth function classes. An important novelty over other approaches is the use of the flexible functional dependence measure to quantify dependence. A functional central limit theorem and nonasymptotic maximal inequalities are provided. The theory is used to prove the functional convergence of the empirical distribution function (EDF) and to derive uniform convergence rates for kernel density estimators both for stationary and locally stationary processes. A comparison with earlier results based on other measures of dependence is carried out.

The hazard level set

The hazard rate function plays a fundamental role in survival analysis. Its statistical inference methods have been systemically and exclusively studied. When does the hazard rate reach a particular warning level? This is a basic question of interest to the investigator but largely left to be explored in practice. We define a level set of hazard rate to address this issue and propose a kernel smoothing estimator for such a level set. In terms of the Hausdorff distance, we establish the consistency, convergence rate and asymptotic distribution of the level set estimator. The validity of the proposed confidence set, based on the bootstrap method, for the level set of hazard rate function is theoretically justified. We conduct comprehensive simulation studies to assess the finite-sample performance of the proposed method, which is further illustrated with a breast cancer study.

Estimating the number of components in finite mixture models via the Group-Sort-Fuse procedure

Estimation of the number of components (or order) of a finite mixture model is a long standing and challenging problem in statistics. We propose the Group-Sort-Fuse (GSF) procedure—a new penalized likelihood approach for simultaneous estimation of the order and mixing measure in multidimensional finite mixture models. Unlike methods which fit and compare mixtures with varying orders using criteria involving model complexity, our approach directly penalizes a continuous function of the model parameters. More specifically, given a conservative upper bound on the order, the GSF groups and sorts mixture component parameters to fuse those which are redundant. For a wide range of finite mixture models, we show that the GSF is consistent in estimating the true mixture order and achieves the n−1/2 convergence rate for parameter estimation up to polylogarithmic factors. The GSF is implemented for several univariate and multivariate mixture models in the R package GroupSortFuse. Its finite sample performance is supported by a thorough simulation study, and its application is illustrated on two real data examples.

Curve fitting and jump detection on nonparametric regression with missing data

In this paper, by virtual of the inverse probability weighted technique, we considered the jump-preserving estimation on the nonparametric regression models with missing data on response variable. First, we used local piecewise-linear expansion respectively with left and right kernel to approximate the unknown regression function. Second, we obtained the left- and right-limit estimation of regression function at each observed points and then determinated the final estimators by residual sums of squares. Third, we presented the convergence rate of estimators and the residual sums of squares. Finally, we illustrated the performance of our proposed method through some simulation studies and a conjunctivitis example from The Affiliated Hospital of Hangzhou Normal University.

Transfer Learning for High-Dimensional Linear Regression: Prediction, Estimation and Minimax Optimality.

This paper considers estimation and prediction of a high-dimensional linear regression in the setting of transfer learning where, in addition to observations from the target model, auxiliary samples from different but possibly related regression models are available. When the set of informative auxiliary studies is known, an estimator and a predictor are proposed and their optimality is established. The optimal rates of convergence for prediction and estimation are faster than the corresponding rates without using the auxiliary samples. This implies that knowledge from the informative auxiliary samples can be transferred to improve the learning performance of the target problem. When the set of informative auxiliary samples is unknown, we propose a data-driven procedure for transfer learning, called Trans-Lasso, and show its robustness to non-informative auxiliary samples and its efficiency in knowledge transfer. The proposed procedures are demonstrated in numerical studies and are applied to a dataset concerning the associations among gene expressions. It is shown that Trans-Lasso leads to improved performance in gene expression prediction in a target tissue by incorporating data from multiple different tissues as auxiliary samples.

Over-parametrized deep neural networks minimizing the empirical risk do not generalize well

Recently it was shown in several papers that backpropagation is able to find the global minimum of the empirical risk on the training data using over-parametrized deep neural networks. In this paper, a similar result is shown for deep neural networks with the sigmoidal squasher activation function in a regression setting, and a lower bound is presented which proves that these networks do not generalize well on a new data in the sense that networks which minimize the empirical risk do not achieve the optimal minimax rate of convergence for estimation of smooth regression functions.