Convergence Rate Of Estimator Research Articles

A modified Salp Swarm Algorithm for parameter estimation of fractional-order chaotic systems

For the parameter estimation problem in research related to the fractional-order chaotic systems (FOCSs), a modified optimization algorithm based on Salp Swarm Algorithm (SSA) was developed in this paper. The proposed algorithm introduced several improvements on SSA: adding a grouping step, introducing “betrayal” behavior, and improving the update method of the followers. We applied multiple classical optimization algorithms to conduct the parameter estimation experiments on the fractional-order Lorenz chaotic system (Lorenz-FOCS) and the fractional-order Financial chaotic system (Financial-FOCS). In addition, we explored the impact of searching space on parameters estimation through experiments. The experimental results confirmed the feasibility of the modified Salp Swarm Algorithm (MSSA). The MSSA performed better than the SSA and other classical optimization algorithms in terms of the estimation accuracy and convergence rate.

Robust transfer learning of high-dimensional generalized linear model

This paper studies transfer learning of a high-dimensional generalized linear model with the target model as well as source data from different but possibly related models. Both known and unknown transferable domain settings are considered. On the one hand, an improved two-step transfer learning algorithm is proposed and the optimal rate of convergence for estimation is proved when the set of transferable domain is known. On the other hand, when the set of transferable domain is unknown, we propose a data-driven procedure for transfer learning, called Stepwise Selection algorithm, and investigate its finite-sample performance through simulations studies. Experimental results on six datasets demonstrate that the proposed method can perform better.

Estimation and Inference for High-Dimensional Generalized Linear Models with Knowledge Transfer

Transfer learning provides a powerful tool for incorporating data from related studies into a target study of interest. In epidemiology and medical studies, the classification of a target disease could borrow information across other related diseases and populations. In this work, we consider transfer learning for high-dimensional generalized linear models (GLMs). A novel algorithm, TransHDGLM, that integrates data from the target study and the source studies is proposed. Minimax rate of convergence for estimation is established and the proposed estimator is shown to be rate-optimal. Statistical inference for the target regression coefficients is also studied. Asymptotic normality for a debiased estimator is established, which can be used for constructing coordinate-wise confidence intervals of the regression coefficients. Numerical studies show significant improvement in estimation and inference accuracy over GLMs that only use the target data. The proposed methods are applied to a real data study concerning the classification of colorectal cancer using gut microbiomes, and are shown to enhance the classification accuracy in comparison to methods that only use the target data.

Estimation of the invariant density for discretely observed diffusion processes: impact of the sampling and of the asynchronicity

We aim at estimating in a non-parametric way the density π of the stationary distribution of a d-dimensional stochastic differential equation , for , from the discrete observations of a finite sample ,…, with . We propose a kernel density estimator and we study its convergence rates for the pointwise estimation of the invariant density under anisotropic Hölder smoothness constraints. First of all, we find some conditions on the discretization step that ensures it is possible to recover the same rates as if the continuous trajectory of the process was available. As proven in the recent work [Amorino C, Gloter A. Minimax rate of estimation for invariant densities associated to continuous stochastic differential equations over anisotropic Holder classes; 2021. arXiv preprint arXiv:2110.02774], such rates are optimal and new in the context of density estimator. Then we deal with the case where such a condition on the discretization step is not satisfied, which we refer to as the intermediate regime. In this new regime we identify the convergence rate for the estimation of the invariant density over anisotropic Hölder classes, which is the same convergence rate as for the estimation of a probability density belonging to an anisotropic Hölder class, associated to n iid random variables . After that we focus on the asynchronous case, in which each component can be observed at different time points. Even if the asynchronicity of the observations complexifies the computation of the variance of the estimator, we are able to find conditions ensuring that this variance is comparable to the one of the continuous case. We also exhibit that the non-synchronicity of the data introduces additional bias terms in the study of the estimator.

Uniform convergence rates for spot volatility estimation

Uniform convergence rates for spot volatility estimation

Moment convergence rate of estimators in partially linear models under AANA errors

Moment convergence rate of estimators in partially linear models under AANA errors

Multi‐objective optimization‐based robust fault estimation observer design for fuzzy singularly perturbed systems with time‐varying state delays

SummaryIn this paper, the problem of multi‐objective robust fault estimation based on observer is addressed for T‐S fuzzy nonlinear singularly perturbed systems with time‐varying state delay and external disturbance. Under performance constraint, a fuzzy augmented fault estimation observer is introduced to improve the performance of actuator and sensor fault estimation simultaneously. The sufficient conditions for the existence of fault estimator are given in the form of LMIs, where ‐dependent Lyapunov function is considered. The given multi‐objective fault estimator design strategy has a good ability to suppress external disturbance and ensure the estimation accuracy and convergence rate work well for all . Then the largest stability bound and the best attenuation capacity are guaranteed by deriving a modified optimal algorithm. Finally, two examples are given to illustrate the feasibility and correctness of the proposed design method.

Likelihood estimation of sparse topic distributions in topic models and its applications to Wasserstein document distance calculations

This paper studies the estimation of high-dimensional, discrete, possibly sparse, mixture models in the context of topic models. The data consists of observed multinomial counts of p words across n independent documents. In topic models, the p×n expected word frequency matrix is assumed to be factorized as a p×K word-topic matrix A and a K×n topic-document matrix T. Since columns of both matrices represent conditional probabilities belonging to probability simplices, columns of A are viewed as p-dimensional mixture components that are common to all documents while columns of T are viewed as the K-dimensional mixture weights that are document specific and are allowed to be sparse. The main interest is to provide sharp, finite sample, ℓ1-norm convergence rates for estimators of the mixture weights T when A is either known or unknown. For known A, we suggest MLE estimation of T. Our nonstandard analysis of the MLE not only establishes its ℓ1 convergence rate, but also reveals a remarkable property: the MLE, with no extra regularization, can be exactly sparse and contain the true zero pattern of T. We further show that the MLE is both minimax optimal and adaptive to the unknown sparsity in a large class of sparse topic distributions. When A is unknown, we estimate T by optimizing the likelihood function corresponding to a plug in, generic, estimator Aˆ of A. For any estimator Aˆ that satisfies carefully detailed conditions for proximity to A, we show that the resulting estimator of T retains the properties established for the MLE. Our theoretical results allow the ambient dimensions K and p to grow with the sample sizes. Our main application is to the estimation of 1-Wasserstein distances between document generating distributions. We propose, estimate and analyze new 1-Wasserstein distances between alternative probabilistic document representations, at the word and topic level, respectively. We derive finite sample bounds on the estimated proposed 1-Wasserstein distances. For word level document-distances, we provide contrast with existing rates on the 1-Wasserstein distance between standard empirical frequency estimates. The effectiveness of the proposed 1-Wasserstein distances is illustrated by an analysis of an IMDB movie reviews data set. Finally, our theoretical results are supported by extensive simulation studies.

A stabilizer free weak Galerkin finite element method for second‐order Sobolev equation

AbstractIn this study, we consider a stabilizer free weak Galerkin (SFWG) finite element method to solve a second‐order Sobolev equation. The SFWG method has various assets, including the support for higher order of accuracy and fewer coefficients. We have analyzed error estimates for both semidiscrete and fully discrete schemes using a non‐standard projection operator. The convergence rate of numerical error estimates of in norm and in norm are obtained, which are two order higher than the optimal order associated with WG finite element space At last, several numerical examples are reported to validate the supercloseness property and efficiency of the proposed SFWG method. These experiments confirm the accuracy of the theoretical findings.

Minimax estimation in multi-task regression under low-rank structures

This study investigates the minimaxity of a multi-task nonparametric regression problem. We formulate a simultaneous function estimation problem based on information pooling across multiple experiments under a low-dimensional structure. A nonparametric reduced rank regression estimator based on the nuclear norm penalisation scheme is proposed to incorporate the low-dimensional structure in the estimation process. A rank of a set of functions is defined in terms of their Fourier coefficients to formally characterise the dependence structure among functions. Minimax upper and lower bounds are established under various asymptotic scenarios to examine the role of the low-rank structure in determining optimal rates of convergence. The results confirm that exploiting the low-rank structure can significantly improve the convergence rate for the simultaneous estimation of multiple functions. The results also imply that the proposed estimator is rate optimal in the minimax sense for the rank-constraint Sobolev class of vector-valued functions.

Empirical‐process‐based specification tests for diffusion models

AbstractWe develop two joint tests for the parametric drift and volatility functions of a diffusion model based on empirical processes. One key feature of our joint tests is that they account for different convergence rates of parameter estimators. The tests are of classical Kolmogorov–Smirnov and Cramér–von Mises types, and are asymptotically distribution free. The proposed tests have nontrivial power against a class of local alternatives with different convergence rates for the drift and volatility terms. Monte Carlo simulations show that the tests perform quite well in finite samples and outperform the nonparametric test of Hong and Li. The new tests are applied to EUR/USD exchange rate data and generate some interesting empirical findings that are consistent with our theoretical results and simulation studies.

Power system abnormal pattern detection for new energy big data

Abstract The energy crisis is a problem that countries all over the world pay more and more attention to, and a series of ecological problems caused by it have become increasingly prominent. It is difficult for traditional fossil fuels to maintain a healthy and coordinated sustainable development of society and economy. The establishment of a sustainable energy system has become the development trend of various countries to solve energy problems. Electric energy is a secondary energy that all primary energy can be converted into, and an irreplaceable consumable for all industrial technologies and people’s lives. Electric power data has the characteristics of large rate span, numerous data sources, complicated interaction methods, and various types of data. The existence of abnormal data in the power system will greatly reduce the accuracy of the system state estimation and the state estimation convergence rate. This paper introduces the power grid industrial control system, combines the data flow of power big data, and analyzes the abnormal information detection process in detail. It takes the data stream acquired by the acquisition unit PMU of the wide area measurement system as the research object. The rapid development of the Hadoop big data platform provides important technical support for the research of power grid big data. Based on the Hadoop platform, the clustering algorithm is used to complete the anomaly detection of real-time data. The LOF algorithm has poor performance when dealing with a large amount of high-dimensional data, and has high time and space complexity. In order to make up for the shortcomings of the LOF algorithm, this paper uses the K-means clustering algorithm to propose an improved algorithm K-LOF of the density-based local abnormal factor detection algorithm LOF, and optimizes the neighborhood query process. It is verified by experiments that the K-LOF algorithm can effectively reduce the time complexity of the anomaly detection algorithm and improve the detection accuracy by 2–4.2%.

Nonparametric inference for reversed mean models with panel count data

Panel count data typically refer to data arising from studies with recurrent events, in which subjects are observed only at discrete time points rather than under continuous observations. We investigate a general situation where a recurrent event process is eventually truncated by an informative terminal event and we are particularly interested in behaviors of the recurrent event process near the terminal event. We propose a reversed mean model for estimating the mean function of the recurrent event process. We develop a two-stage sieve likelihood-based method to estimate the mean function, which overcomes the computational difficulties arising from a nuisance functional parameter involved in the likelihood. The consistency and the convergence rate of the two-stage estimator are established. Allowing for the convergence rate slower than the standard rate, we develop the general weak convergence theory of M-estimators with a nuisance functional parameter, and then apply it to the proposed estimator for deriving the asymptotic normality. Furthermore, a class of two-sample tests is developed. The proposed methods are evaluated with extensive simulation studies and illustrated with panel count data from the Chinese Longitudinal Healthy Longevity Study.

Nonparametric estimation of additive models with errors-in-variables

In the estimation of nonparametric additive models, conventional methods, such as backfitting and series approximation, cannot be applied when measurement error is present in a covariate. This paper proposes a two-stage estimator for such models. In the first stage, to adapt to the additive structure, we use a series approximation together with a ridge approach to deal with the ill-posedness brought by mismeasurement. We derive the uniform convergence rate of this first-stage estimator and characterize how the measurement error slows down the convergence rate for ordinary/super smooth cases. To establish the limiting distribution, we construct a second-stage estimator via one-step backfitting with a deconvolution kernel using the first-stage estimator. The asymptotic normality of the second-stage estimator is established for ordinary/super smooth measurement error cases. Finally, a Monte Carlo study and an empirical application highlight the applicability of the estimator.

Functional linear regression for discretely observed data: from ideal to reality

Summary Despite extensive studies on functional linear regression, there exists a fundamental gap in theory between the ideal estimation from fully observed covariate functions and the reality that one can only observe functional covariates discretely with noise. The challenge arises when deriving a sharp perturbation bound for the estimated eigenfunctions in the latter case, which renders existing techniques for functional linear regression not applicable. We use a pooling method to attain the estimated eigenfunctions and propose a sample-splitting strategy to estimate the principal component scores, which facilitates the theoretical treatment for discretely observed data. The slope function is estimated by approximated least squares, and we show that the resulting estimator attains the optimal convergence rates for both estimation and prediction when the number of measurements per subject reaches a certain magnitude of the sample size. This phase transition phenomenon differs from the known results for the pooled mean and covariance estimation, and reveals the elevated difficulty in estimating the regression function. Numerical experiments, using simulated and real data examples, yield favourable results when compared with existing methods.

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.