Minimum Variance Unbiased Estimator Research Articles

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.

Estimation of Bonferroni Curve and Bonferroni Index of the Pareto Distribution

Abstract In this article, we consider classical and Bayesian estimations of some economic measures, specially Bonferroni Curve and Bonferroni Index of the Pareto distribution. We obtain the Maximum Likelihood Estimator and Uniform Minimum Variance Unbiased Estimator in the classical setup, and their properties are studied. Additionally, we conduct Bayesian estimation procedures based on symmetric loss functions and a truncated gamma prior. The precision of the estimators is evaluated under different sample sizes via Monte Carlo simulation. Furthermore, a real dataset is provided to compute all the estimators.

Saturated load forecasting based on improved logistic regression and affinity propagation

Saturated load forecasting (SLF) is a crucial guarantee for stable operation of power grid. However, insufficient data makes SLF collapse. Although logistic regression (LR) can be employed to tackle this problem, its non-linearity may lead to analytically unsolvable maximum likelihood (ML) function. Due to this dilemma, this paper proposes a sequential LR estimator. It depends on minimum variance unbiased estimator (MVUE) and maximum likelihood estimator (MLE) to form an iterative method: in which a linearized statistical model is established by inverse substitution to reduce complexity. Moreover, a new forecasting framework combining LR estimator with clustering algorithms is constructed. In this framework, load data are clustered based on affinity propagation (AP) before LR. Subsequently, SLF result is obtained from corresponding categories by iteration. Simulation results show that the method proposed in this paper has lower computational complexity and higher prediction accuracy.

Unscented recursive three-step filter based unbiased minimum-variance estimation for a class of nonlinear systems

For the direct feedthrough system where unknown input affects both process equation and measurement equation, an unscented recursive three-step filter based unbiased minimum-variance (URTSF-UMV) estimation algorithm is proposed. The algorithm is based on the unbiased minimum-variance (UMV) method and uses statistical linearisation technology to approximate the nonlinear system and the nonlinear measurement function to linear regression form. The estimation of unknown input is obtained from innovation through weighted least-squares (WLS) estimation. The approximate linear regression form allows us to process the nonlinear system with unknown input direct feedthrough by using the UMV state estimation framework, and derive the filter by minimising the trace of the state error covariance of the measurement update under the unbiased condition. The URTSF-UMV requires that the unknown input distribution matrix is full column rank. We use the full rank decomposition method to deal with the nonlinear system containing the unknown input rank-deficient distribution matrix. The effectiveness of the URTSF-UMV algorithm is demonstrated through an illustrative simulation example.

Classical and Bayesian estimations of performance measures in G e o / G e o / 1 queue

. In this article, we consider an early arrival G e o / G e o / 1 queue from the statistical perspective. The estimations of parameters and system performances under Classical and Bayesian frameworks are considered. Based on queue length data, we attempt to study the uniform minimum-variance unbiased estimators (UMVUEs) and the closed Bayesian estimators for various queueing characteristics. We compare their properties using Monte Carlo (MC) simulations. As a comparison and improvement, based on arrival and departure data over a period, we use a Bivariate Beta distribution as joint prior, and solve the performance estimators using the properties of Gaussian hypergeometric function. We derive Bayesian estimators and conduct numerical simulations using Markov Chain Monte Carlo (MCMC) method.

Maximum correntropy unscented filter based on unbiased minimum-variance estimation for a class of nonlinear systems

Introduction: The unscented Kalman filter based on unbiased minimum-variance (UKF-UMV) estimation is usually used to handle the state estimation problem of nonlinear systems with an unknown input. When the nonlinear system is disturbed by non-Gaussian noise, the performance of UKF-UMV will seriously deteriorate.Methods: A maximum correntropy unscented filter based on the unbiased minimum variance (MCUF-UMV) estimation method is proposed on the basis of the UKF-UMV without the need for estimation of an unknown input and uses the maximum correntropy criterion (MCC) and fixed-point iterative algorithm for state estimation.Results: When the measurement noise of the nonlinear system is non-Gaussian noise, the algorithm performs well.Discussion: Our proposed algorithm also does not require estimation of an unknown input, and there is no prior knowledge available about the unknown input or any prior assumptions. The unknown input can be any signal. Finally, a simulation example is used to demonstrate the effectiveness and reliability of the algorithm.

On Point Estimators for Gamma and Beta Distributions

Let X 1 , … , X n be a random sample from the gamma distribution with density f ( x ) = λ α x α − 1 e − λ x / Γ ( α ) , x > 0, where both α > 0 (the shape parameter) and λ > 0 (the reciprocal scale parameter) are unknown. The main result shows that the uniformly minimum variance unbiased estimator (UMVUE) of the shape parameter, α, exists if and only if n ≥ 4 ; moreover, it has finite variance if and only if n ≥ 6 . More precisely, the form of the UMVUE is given for all parametric functions α, λ, 1 / α , and 1 / λ . Furthermore, a highly efficient estimating procedure for the two-parameter beta distribution is also given. This is based on a Stein-type covariance identity for the beta distribution, followed by an application of the theory of U-statistics and the delta-method.

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