Minimum Variance Unbiased Estimator Research Articles

Estimation of stress-strength reliability in s-out-of-k system for new flexible exponential distribution under progressive type-II censoring

This article addresses stress-strength reliability estimation in s-out-of-k systems under progressive type-II censoring.  The system consists of components facing common stress, where stress and strength variable follows independent new flexible exponential (NFE) distributions, sharing a common parameter. The estimation procedure is carried out using MLE, Bayesian estimation and uniformly minimum variance unbiased estimation (UMVUE) techniques. MLE is derived for known and unknown common parameters. Bayesian estimates are obtained using Lindley’s approximation and Markov chain Monte Carlo techniques due to the complexity of obtaining the exact Bayesian estimates, when the common parameter is unknown. Moreover, UMVUE is derived when the common parameter is known. The 95% asymptotic confidence interval, highest posterior density credible intervals, and Bayesian credible intervals, are constructed. An extensive simulation study is carried out to compare the performance of various censoring schemes and estimation methods. Finally, two progressively censored real datasets are analyzed for illustration.

Covariance Matrix Estimation for High-Throughput Biomedical Data with Interconnected Communities

Estimating a covariance matrix is central to high-dimensional data analysis. Empirical analyses of high-dimensional biomedical data, including genomics, proteomics, microbiome, and neuroimaging, among others, consistently reveal strong modularity in the dependence patterns. In these analyses, intercorrelated high-dimensional biomedical features often form communities or modules that can be interconnected with others. While the interconnected community structure has been extensively studied in biomedical research (e.g., gene co-expression networks), its potential to assist in the estimation of covariance matrices remains largely unexplored. To address this gap, we propose a procedure that leverages the commonly observed interconnected community structure in high-dimensional biomedical data to estimate large covariance and precision matrices. We derive the uniformly minimum variance unbiased estimators for covariance and precision matrices in closed forms and provide theoretical results on their asymptotic properties. Our proposed method enhances the accuracy of covariance- and precision-matrix estimation and demonstrates superior performance compared to the competing methods in both simulations and real data analyses.

Family of Kumaraswamy generalized distribution in estimation of multi-component reliability under the adaptive hybrid progressive censoring schemes

ABSTRACT In this paper, the statistical inference on multi-component stress-strength parameter with non-identical-component strengths, based on Kumaraswamy generalized distribution under adaptive hybrid progressive censoring samples, is considered. The problem is solved in three cases. First, when one parameter is unknown, the maximum likelihood estimation (MLE), Bayes approximations, asymptotic and highest posterior density intervals are obtained. Second, when the common parameter is known, MLE, approximation Bayes estimations, uniformly minimum variance unbiased estimator and different confidence intervals are provided. Third, when all parameters are different and unknown, MLE and Bayesian estimation are studied. The Monte Carlo simulation is employed to compare the estimations. Based on the simulation results, it is observed that the Bayesian estimates perform better than MLEs. Also, the highest posterior density intervals have better performance than asymptotic intervals. Moreover, it is observed that the performance of uniformly minimum variance unbiased estimators is worse than MLEs. To implement the theoretical method, two real data sets are analyzed.

Improved Square-Root Cubature Kalman Filtering Algorithm for Nonlinear Systems with Dual Unknown Inputs

For nonlinear discrete systems with dual unknown inputs, there are many limitations regarding previous nonlinear filters. This paper proposes two new, improved square-root cubature Kalman filtering (ISRCKF) algorithms to estimate system states and dual unknown inputs. Improved square-root cubature Kalman filtering 1 (ISRCKF1) introduces an innovation that first obtains the unknown input estimates from the measurement equation, then updates the innovation to derive the unknown input estimates from the state equation, then uses the already obtained estimates of the dual unknown inputs to correct the one-step estimate of the state, and finally the minimum variance unbiased estimate of the state is obtained. Improved square-root cubature Kalman filtering 2 (ISRCKF2) builds a unified innovation feedback model, then applies the minimum variance unbiased estimation (MVUE) criterion to obtain the estimates of system states and dual unknown inputs, refining a more concise recursive filter but requiring stronger assumptions. Finally, simulation results demonstrate that the above two algorithms can achieve the optimal estimates of system states and dual unknown inputs simultaneously, and ISRCKF2 further enhances the accuracy of both state and dual unknown inputs estimation, which verifies the validity of the proposed algorithms.

Estimation of a New Stress Strength Index for One Parameter Exponential Family

Stress-strength reliability is widely used in manufacturing industry for producing good quality equipment. A new stress-strength index has been introduced when strengths of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> independent components follow exponential distributions with different scale parameters. We obtain the maximum likelihood estimator, a uniformly minimum variance unbiased estimator, a Bayes estimator, and an analogue of the best scale equivariant estimator for the new index. It is shown that the Bayes estimators' limit is a generalized Bayes estimator under the squared error loss function. The asymptotic distribution of the MLE is derived using the multivariate delta method. A detailed comparison of the risk performance of all these estimators is done numerically. The results are illustrated by two real data analyses.

Minimum variance unbiased Bayesian smoothing for input and state estimation of systems without direct Feedthrough: Mitigating Ill-Posedness of online load identification

In this paper, the problem of moving load identification using pure displacement measurements is addressed. It is known that when no assumptions are made on the statistics of the unknown loads and a minimum variance unbiased (MVU) estimation approach is adopted, the existing methods in the literature suffer from a very elevated load estimation uncertainty. This elevated uncertainty is due to ill-posedness of the problem. In this paper a new method is proposed that addresses this issue via an MVU smoothing approach. To alleviate this problem, a MVU smoothing algorithm is proposed in this study, via modification of a MVU smoothing Bayesian estimator proposed by some of the co-authors of this paper hence referred to as MSBE, which leads to substantial decrease in the moving load estimation uncertainties using pure displacement measurements. The efficacy of the MSBE is studied through a simulated experiment corresponding to a numerical model of an operating steel railway bridge with riveted connections and a multi-axle load. The selection of the hyperparameters of the smoothing and filtering techniques are discussed, and the optimal values are presented. The parametric studies show that the proposed method can yield highly accurate results, and substantially outperforms a celebrated MVU filter proposed by Gillijns and De Moor.

A Better Alternative to the Generalized Bilal Distribution: A New Model and Applications

In 2017, A. M. Abd-Elrahman [A new two-parameter lifetime distribution with decreasing, increasing or upside-down bathtub-shaped failure rate, Commun. Stat. - Theory Methods 46 (2017) 8865–8880, doi:10.1080/03610926.2016.1193198] introduced a generalization of the Bilal distribution, where a new two-parameter distribution, generalized Bilal distribution (GBD), was presented. He showed that its failure rate function can be upside-down bathtub-shaped. The failure rate can either be decreasing or increasing due to some mathematical and statistical reasons, which will be given below. In this paper, we introduce a simple and better alternative to the GBD, which will be denoted by WMD. We show that the WMD is a two-parameter distribution which can fit five different types of data sets with respect to their empirical hazard rate functions. Most properties of the WMD are investigated. Point and interval estimation procedures for the two unknown parameters are presented. The existence and uniqueness of the maximum likelihood estimates are proved. The moment estimates are obtained and we showed that one of these estimates is the minimum variance unbiased estimate (MVUE) for its corresponding parameter. A simulation study is provided and the paper is motivated by applications to four different real data sets. A detailed analysis for the Meeker and Escobar data is provided by the book of Meeker and Escobar [Statistical Methods for Reliability Data, 2nd edn. (John Wiley, 1998)]. The results may show that the new distribution provides a better fit than some other most recent existing and already known distributions in the literature. Finally, some concluding remarks are presented.

Deep Neural Networks Guided Ensemble Learning for Point Estimation

In modern statistics, interests shift from pursuing the uniformly minimum variance unbiased estimator to reducing mean squared error (MSE) or residual squared error. Shrinkage-based estimation and regression methods offer better prediction accuracy and improved interpretation. However, the characterization of such optimal statistics in terms of minimizing MSE remains open and challenging in many problems, for example, estimating the treatment effect in adaptive clinical trials with pre-planned modifications to design aspects based on accumulated data. From an alternative perspective, we propose a deep neural network based automatic method to construct an improved estimator from existing ones. Theoretical properties are studied to provide guidance on applicability of our estimator to seek potential improvement. Simulation studies demonstrate that the proposed method has considerable finite-sample efficiency gain compared to several common estimators. In the Adaptive COVID-19 Treatment Trial (ACTT) as a motivating example, our ensemble estimator essentially contributes to a more ethical and efficient adaptive clinical trial with fewer patients enrolled. The proposed framework can be generally applied to various statistical problems, and can serve as a reference measure to guide statistical research.

Estimation of the Density and Cumulative Distribution Functions of the Topp-Leone Distribution

In this paper, we estimate the probability density function and the cumulative distribution function of the Topp-Leone distribution. We use the maximum likelihood estimation, uniformly minimum variance unbiased estimation, least squares estimation, weighted least squares estimation, Cramér-von Mises estimation, Anderson-Darling estimation and method of percentile estimation. The consistency of these methods is illustrated in a simulation study. Finally, a real data set is given to assess the performance of proposed methods.

Unbiased minimum-variance estimation and dynamic event-driven disturbance rejection control for discrete time-varying systems

This paper studies a dynamic event-driven disturbance rejection control problem for a class of linear discrete time-varying systems with unknown external disturbances, random process and measurement noises. To obtain the accurate estimates of both the system states and external disturbances, simultaneously, an unbiased minimum-variance time-varying extended state observer is designed, and the stability of the estimation error system is analyzed. Then, a dynamic event-driven disturbance rejection controller is designed by minimizing the upper bound of the cost function of the closed-loop system over a finite horizon. It can be shown that, by applying the proposed control approach, not only the transmission frequency of the control signal is reduced, but also the external disturbances are compensated actively. Finally, numerical simulations are given to demonstrate the effectiveness of the proposed control approaches.

Inference on the stress-strength reliability of multi-component systems based on progressive first failure censored samples

This paper studies the statistical estimation of the stress-strength reliability of multi-component systems under the progressive first failure censoring samples, where the lifetime distribution of each component follows the modified Kumaraswamy distribution. Both the point and interval estimations of the parameters in the reliability function are considered. To this aim, some estimations such as maximum likelihood estimation (MLE), asymptotic confidence intervals, uniformly minimum variance unbiased estimation (UMVUE), approximate Bayes estimation, and highest posterior density (HPD) intervals are obtained. By employing the Monte Carlo simulation, comparison of the performance between different estimates is provided. The paper then analyzes a case study for illustration of the proposed method.

Inference for the Johnson SB distribution

This article develops inferential procedures for the Johnson SB distribution and the associated stress-strength model. The maximum likelihood estimators, uniformly minimum variance unbiased estimators and generalized confidence intervals for the parameters of the Johnson SB distribution are presented. The objective Bayesian inference methods based on the Jeffreys and reference priors are derived. Moreover, the generalized confidence lower limits and Bayesian credible lower limits for the reliability of the stress-strength model are provided. The performance of the proposed methods is evaluated by Monte Carlo simulation. Finally, an example is used to illustrate the proposed procedures.

Asymptotic UMVUE: Asymptotic Moments Matching the UMVUE under the Ewens Sampling Formula

The Ewens sampling formula is a distribution related to the random partition of a positive integer. In this study, we investigate the issue of non-existent solutions in parameter estimation under the distribution. We derive the first and second moments matching estimators to the uniformly minimum variance unbiased estimator (UMVUE) in the asymptotic sense (hereafter referred to as Asymptotic UMUVE) using the Ewens sampling formula. A Monte Carlo simulation study is performed to evaluate the efficiency of the resulting estimators. AMS Subject Classification: 62F12, 62P25

Estimating reciprocals of scale parameters of two exponential populations with common location and ordered scales using censored samples

The problem of estimating reciprocals of scale parameters (hazard rates) from two exponential populations with a common location and ordered scale parameters have been considered under the progressive type-II censoring scheme from a decision-theoretic viewpoint. The loss function is considered as quadratic. The maximum-likelihood estimators (MLEs) and the uniformly minimum variance unbiased estimators (UMVUEs) are derived. Sufficient conditions are derived for improving estimators in affine and scale equivariant classes. Consequently, improved estimators over the MLEs and the UMVUEs are derived. A numerical comparison among all the proposed estimators is made, and conclusions are drawn regarding their performances with respect to the quadratic loss function.

Pose-Dependent Cutting Force Identification for Robotic Milling

AbstractCutting force identification is critical to improving industrial robot performance and reducing machining vibration. However, most indirect identification methods of cutting force are not applicable since the modal parameters of the robotic milling system vary with the robot pose. This paper presents a novel pose-dependent method to identify the cutting force using the acceleration signal generated by robotic milling. First, the modal parameters at different machining points are employed as a training dataset to develop the Gaussian Process Regression (GPR) model. Next, the modal parameters predicted by the GPR model are employed to optimize the cutting force estimation based on the minimum variance unbiased estimate method. Then, the Kalman filter method is employed to update the covariance matrix of the cutting force identification error and the state estimation error. Lastly, the effectiveness of the proposed method is verified with robotic milling experiments, and the results show that the identification error and time are acceptable under the condition of variable robot pose.

A Smoothing EKF-UI-WDF Method for Simultaneous Identification of Structural Systems and Unknown Seismic Inputs without Direct Feedthrough

It is of great significance to identify structural state-parameters and the unknown seismic inputs using partial measurements of structural acceleration responses for the rapid evaluation of structures after unknown seismic excitations. However, unknown seismic inputs do not directly appear in the observation equations of measured absolute floor accelerations of building structures, i.e., there is no direct feedthrough of unknown seismic inputs in the observation equations. Current methods for the identification of joint structural systems and unknown inputs are either inapplicable or greatly influenced by measurement noises. In this paper, a method so-called smoothing extended Kalman filter with unknown input without direct feedthrough (smoothing EKF-UI-WDF) is proposed. The identification algorithm is derived in the framework of minimum-variance unbiased estimation (MVUE), and the smoothing technique is adopted to introduce subsequent observation steps in the current identification step. Then, structural states, parameters, and unknown seismic excitations without direct feedthrough are simultaneously identified recursively with only a few steps delay, and the identification results are tolerant to measurement noises. The proposed method is verified by a numerical simulation model and a practical engineering case study. Both identification results validate the effectiveness of the proposed method for the simultaneous identification of structural systems and seismic inputs without direct feedthrough.

One-Stage Multiple Comparisons of the Mean Lifetimes of k Treatments with the Average for Exponential Distributions under Heteroscedasticity

Making use of uniformly minimum-variance unbiased estimators for the parameters of two-parameter exponential distributions and the distribution of pivotal quantities, we propose one-stage multiple comparison procedures for k mean lifetimes with the average under heteroscedasticity. The multiple comparison procedures include one-sided and two-sided confidence intervals. These intervals can be applied to identify which treatment’s mean lifetime is better than the average or worse than the average in terms of the mean lifetimes of all treatments. Critical values are obtained in order to assure users that the given confidence coefficient has been reached; they are organized in table format for practical and convenient use. An example is provided to demonstrate the proposed techniques, wherein the mean survival times of four different lung cancer categories are compared with the average.

A linear Bayesian filter for input and state estimation of structural systems

AbstractThis paper proposes a linear recursive Bayesian filter for minimum variance unbiased joint input and state estimation of structural systems. Unlike the augmented Kalman filter (AKF), the proposed filter falls within the category of Bayesian filters in which unknown inputs are estimated without attributing any fictitious input model or statistics. Also, in contrast with the existing algorithms in the latter category, such as the Gillijns and De Moor Filters (GDFs), the developed filter applies to systems with and without direct feedthrough, in particular, systems with a rank‐deficient feedforward matrix. Because of the latter features, the filter is referred to as universal filter (UF) for convenience. The numerical examples show that the UF performs better than the AKF. Due to its structure, the UF does not require the tuning of the hyperparameters for inputs, and therefore the problematic instability of the AKF is not encountered in the case of a large modeling error variance of the input. For systems with direct feedthrough, the error and covariance propagation terms differ due to the distinct state space. Consequently, the UF can enhance estimations due to the well‐conditionedness of the relevant inversion problem. Moreover, the UF can deal with systems with rank‐deficient feedforward matrix where these systems are not covered by GDFs.

Multi-component Reliability Inference in Modified Weibull Extension Distribution and Progressive Censoring Scheme.

The statistical inference of multi-component reliability stress-strength system with nonidentical-component strengths is considered for the modified Weibull extension distribution in the presence of progressive censoring samples. For this aim, we study the estimation of multi-component reliability parameter in classical and Bayesian inference. So we derive some point and interval estimates such as maximum likelihood estimation, asymptotic confidence intervals, uniformly minimum variance unbiased estimation, approximate and exact Bayes estimation and highest posterior density intervals. Comparing of different estimates is provided by employing the Monte Carlo simulation, the mean squared error and coverage probabilities. Finally, one real data is utilized to illustrate the applicability of this new model.

A New Adaptive OMP-MAP Algorithm-based Iterative Sparse Channel Estimation for OFDM Underwater Communication

A sparse channel estimation approach based on doubly spread underwater acoustic (UWA) channels is widely used todetect coherent acoustic orthogonal frequency division multiplexing (OFDM) signals. A new time-domain channelestimation (CE) technique for OFDM based UWA communication with Rician fading is used to exploit the channel sparsity.First, to improve the estimation accuracy in high noise conditions, we have exploited the channel sparsity to generate aclosed-form equation for the termination condition. Then, in low-level noise instances, the additional criterion to balanceestimation accuracy and computing costs has been established. By incorporating these two requirements within theorthogonal-matching-pursuit (OMP) structure, an adaptive-OMP (AOMP) algorithm has been proposed. The AOMP andmaximum a posteriori probability (MAP) techniques are combined to provide a computationally efficient, and a newAOMP-MAP scheme for estimating the sparse complex channel path gain has been proposed. Further, The minimumvariance unbiased estimator is used to improve the proposed CE technique. Exploiting the experimental channel data,computer simulations reveal that the proposed CE technique obtains the outstanding outcomes.