scholarly journals A New Maximum Likelihood Estimator Formulated in Pole-Residue Modal Model

2019 ◽  
Vol 9 (15) ◽  
pp. 3120
Author(s):  
Sandro Amador ◽  
Mahmoud El-Kafafy ◽  
Álvaro Cunha ◽  
Rune Brincker
Keyword(s):  
Maximum Likelihood ◽  
Least Squares ◽  
Maximum Likelihood Estimator ◽  
Confidence Intervals ◽  
Modal Parameters ◽  
Parameter Estimates ◽  
Modal Parameter ◽  
Likelihood Estimator ◽  
Pole Residue ◽  
Modal Model

Recently, a lot of efforts have been devoted to developing more precise Modal Parameter Estimation (MPE) techniques. This is explained by the necessity in civil, mechanical and aerospace engineering of obtaining accurate estimates for the modal parameters of the tested structures, as well as of determining reliable confidence intervals for these estimates. The Non-linear Least Squares (NLS) identification techniques based on Maximum Likelihood (ML) have been increasingly used in modal analysis to improve precision of estimates provided by the Least Squares (LS) based estimators when they are not accurate enough. Apart from providing more accurate estimates, the main advantage of the ML estimators, with regard to their LS counterparts, is that they allow for taking into account not only the measured Frequency Response Functions (FRFs) but also the noise information during the parametric identification process and, therefore, provide the modal parameters estimates together with their uncertainties bounds. In this paper, a new derivation of a Maximum Likelihood Estimator formulated in Pole-residue Modal Model (MLE-PMM) is presented. The proposed formulation is meant to be used in combination with the Least Squares Frequency Domain (LSCF) to improve the precision of the modal parameter estimates and compute their confidence intervals. Aiming at demonstrating the efficiency of the proposed approach, it is applied to two simulated examples in the final part of the paper.

Comparing Single-Equation Estimators in a Simultaneous Equation System

1986 ◽  
Vol 2 (1) ◽  
pp. 1-32 ◽  
Author(s):  
T. W. Anderson ◽  
Naoto Kunitomo ◽  
Kimio Morimune
Keyword(s):  
Maximum Likelihood ◽  
Least Squares ◽  
Maximum Likelihood Estimator ◽  
Equation System ◽  
Single Equation ◽  
Ordinary Least Squares ◽  
Limited Information ◽  
Likelihood Estimator ◽  
Mean Squared Errors ◽  
Simultaneous Equation System

Comparisons of estimators are made on the basis of their mean squared errors and their concentrations of probability computed by means of asymptotic expansions of their distributions when the disturbance variance tends to zero and alternatively when the sample size increases indefinitely. The estimators include k-class estimators (limited information maximum likelihood, two-stage least squares, and ordinary least squares) and linear combinations of them as well as modifications of the limited information maximum likelihood estimator and several Bayes' estimators. Many inequalities between the asymptotic mean squared errors and concentrations of probability are given. Among medianunbiasedestimators, the limited information maximum likelihood estimator dominates the median-unbiased fixed k-class estimator.

scholarly journals Optimality of the maximum likelihood estimator in astrometry

2018 ◽  
Vol 616 ◽  
pp. A95 ◽  
Author(s):  
Sebastian Espinosa ◽  
Jorge F. Silva ◽  
Rene A. Mendez ◽  
Rodrigo Lobos ◽  
Marcos Orchard
Keyword(s):  
Maximum Likelihood ◽  
Least Squares ◽  
Maximum Likelihood Estimator ◽  
Weighted Least Squares ◽  
Least Square ◽  
Performance Limits ◽  
Likelihood Estimator ◽  
Weighted Least Square ◽  
Wide Range ◽  
Close Form

Context. Astrometry relies on the precise measurement of the positions and motions of celestial objects. Driven by the ever-increasing accuracy of astrometric measurements, it is important to critically assess the maximum precision that could be achieved with these observations. Aims. The problem of astrometry is revisited from the perspective of analyzing the attainability of well-known performance limits (the Cramér–Rao bound) for the estimation of the relative position of light-emitting (usually point-like) sources on a charge-coupled device (CCD)-like detector using commonly adopted estimators such as the weighted least squares and the maximum likelihood. Methods. Novel technical results are presented to determine the performance of an estimator that corresponds to the solution of an optimization problem in the context of astrometry. Using these results we are able to place stringent bounds on the bias and the variance of the estimators in close form as a function of the data. We confirm these results through comparisons to numerical simulations under a broad range of realistic observing conditions. Results. The maximum likelihood and the weighted least square estimators are analyzed. We confirm the sub-optimality of the weighted least squares scheme from medium to high signal-to-noise found in an earlier study for the (unweighted) least squares method. We find that the maximum likelihood estimator achieves optimal performance limits across a wide range of relevant observational conditions. Furthermore, from our results, we provide concrete insights for adopting an adaptive weighted least square estimator that can be regarded as a computationally efficient alternative to the optimal maximum likelihood solution. Conclusions. We provide, for the first time, close-form analytical expressions that bound the bias and the variance of the weighted least square and maximum likelihood implicit estimators for astrometry using a Poisson-driven detector. These expressions can be used to formally assess the precision attainable by these estimators in comparison with the minimum variance bound.

scholarly journals Classical and Bayesian Inference of Conditional Stress-Strength Model under Kumaraswamy Distribution

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Fathy H. Riad ◽  
Mohammad Mehdi Saber ◽  
Mehrdad Taghipour ◽  
M. M. Abd El-Raouf
Keyword(s):  
Bayesian Inference ◽  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Asymptotic Distribution ◽  
Confidence Intervals ◽  
Bootstrap Method ◽  
Strength Model ◽  
Likelihood Estimator ◽  
Kumaraswamy Distribution ◽  
Strength Models

Stress-strength models have been frequently studied in recent years. An applicable extension of these models is conditional stress-strength models. The maximum likelihood estimator of conditional stress-strength models, asymptotic distribution of this estimator, and its confidence intervals are obtained for Kumaraswamy distribution. In addition, Bayesian estimation and bootstrap method are applied to the model.

scholarly journals Statistical Inference for the Inverted Scale Family under General Progressive Type-II Censoring

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 731
Author(s):  
Jing Gao ◽  
Kehan Bai ◽  
Wenhao Gui
Keyword(s):  
Monte Carlo ◽  
Maximum Likelihood ◽  
Monte Carlo Simulations ◽  
Maximum Likelihood Estimator ◽  
Confidence Intervals ◽  
Interval Estimation ◽  
Bayesian Estimator ◽  
Small Samples ◽  
Likelihood Estimator ◽  
Scale Family

Two estimation problems are studied based on the general progressively censored samples, and the distributions from the inverted scale family (ISF) are considered as prospective life distributions. One is the exact interval estimation for the unknown parameter θ , which is achieved by constructing the pivotal quantity. Through Monte Carlo simulations, the average 90 % and 95 % confidence intervals are obtained, and the validity of the above interval estimation is illustrated with a numerical example. The other is the estimation of R = P ( Y < X ) in the case of ISF. The maximum likelihood estimator (MLE) as well as approximate maximum likelihood estimator (AMLE) is obtained, together with the corresponding R-symmetric asymptotic confidence intervals. With Bootstrap methods, we also propose two R-asymmetric confidence intervals, which have a good performance for small samples. Furthermore, assuming the scale parameters follow independent gamma priors, the Bayesian estimator as well as the HPD credible interval of R is thus acquired. Finally, we make an evaluation on the effectiveness of the proposed estimations through Monte Carlo simulations and provide an illustrative example of two real datasets.

scholarly journals Robust Reliability Estimation for Lindley Distribution—A Probability Integral Transform Statistical Approach

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1634
Author(s):  
Muhammad Aslam Mohd Safari ◽  
Nurulkamal Masseran ◽  
Muhammad Hilmi Abdul Majid
Keyword(s):  
Maximum Likelihood ◽  
Least Squares ◽  
Maximum Likelihood Estimator ◽  
Integral Transform ◽  
Ordinary Least Squares ◽  
Likelihood Estimator ◽  
Reliability Data ◽  
Lindley Distribution ◽  
Probability Integral Transform ◽  
Probability Integral

In the modeling and analysis of reliability data via the Lindley distribution, the maximum likelihood estimator is the most commonly used for parameter estimation. However, the maximum likelihood estimator is highly sensitive to the presence of outliers. In this paper, based on the probability integral transform statistic, a robust and efficient estimator of the parameter of the Lindley distribution is proposed. We investigate the relative efficiency of the new estimator compared to that of the maximum likelihood estimator, as well as its robustness based on the breakdown point and influence function. It is found that this new estimator provides reasonable protection against outliers while also being simple to compute. Using a Monte Carlo simulation, we compare the performance of the new estimator and several well-known methods, including the maximum likelihood, ordinary least-squares and weighted least-squares methods in the absence and presence of outliers. The results reveal that the new estimator is highly competitive with the maximum likelihood estimator in the absence of outliers and outperforms the other methods in the presence of outliers. Finally, we conduct a statistical analysis of four reliability data sets, the results of which support the simulation results.

The Comparison Between the Bayes Estimator and the Maximum Likelihood Estimator of the Reliability Function for Negative Exponential Distribution

2018 ◽  
pp. 439
Author(s):  
Hazim Mansour Gorgees ◽  
Bushra Abdualrasool Ali ◽  
Raghad Ibrahim Kathum
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Exponential Distribution ◽  
Reliability Function ◽  
Bayes Estimator ◽  
Likelihood Estimator ◽  
Monte Carlo Simulation Technique ◽  
Negative Exponential Distribution ◽  
Negative Exponential ◽  
Better Than

     In this paper, the maximum likelihood estimator and the Bayes estimator of the reliability function for negative exponential distribution has been derived, then a Monte –Carlo simulation technique was employed to compare the performance of such estimators. The integral mean square error (IMSE) was used as a criterion for this comparison. The simulation results displayed that the Bayes estimator performed better than the maximum likelihood estimator for different samples sizes.

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