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.

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 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.

scholarly journals Reliability for Lindley Distribution with an Outlier

2013 ◽  
Vol 3 ◽  
pp. 20-23
Author(s):  
Hossein Jabbari Khamnei
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Simulation Studies ◽  
Likelihood Estimator ◽  
Lindley Distribution

In this paper, we consider the problem of estimating R=P(Y<X) , when Y has lindleydistribution with parameter a and x has lindley distribution with presence of one outlier withparameters b and c , such that X and Y are independent. The maximum likelihood estimator of R isderived and some results of simulation studies are presented.

ON MODELING AND ANALYZING CORRUPTED RELIABILITY DATA

1995 ◽  
Vol 02 (03) ◽  
pp. 245-267
Author(s):  
A.J. WATKINS ◽  
R.M. GRIFFIN
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Model Parameters ◽  
Data Recording ◽  
Estimation Of Parameters ◽  
Likelihood Estimator ◽  
Simulation Experiments ◽  
Reliability Data ◽  
Data Set ◽  
Underlying Distribution

This paper is concerned with the effect of various data recording errors on the estimation of parameters in models commonly used in the analysis of reliability data. We start by outlining sources of such errors, and propose some modeling strategies that allow these errors into the framework of our analysis. It is then shown that the estimation of model parameters needs to take into account a mis-specified model, with the consequence that any theoretical advantages nominally enjoyed by estimators are reduced. In particular, the results from a series of simulation experiments show that the maximum likelihood estimator is no longer asymptotically unbiased. We next outline an approach that generalizes the usual asymptotic theory to obtain expressions for both the mean and variance of the maximum likelihood estimator in the present framework; these expressions involve both the underlying distribution and parameters controlling the extent to which recording errors are present in a data set. We then link these expressions to results obtained in a series of simulation experiments, and show that this approach accommodates a general formulation of the effect of data recording errors. We conclude with a discussion of the practical consequences of this work.

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.

scholarly journals Transforming variables to central normality

2021 ◽  
Author(s):  
Jakob Raymaekers ◽  
Peter J. Rousseeuw
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Simulation Study ◽  
Real Data ◽  
Data Sets ◽  
Transformation Parameter ◽  
Likelihood Estimator ◽  
Extensive Simulation ◽  
Highly Sensitive

AbstractMany real data sets contain numerical features (variables) whose distribution is far from normal (Gaussian). Instead, their distribution is often skewed. In order to handle such data it is customary to preprocess the variables to make them more normal. The Box–Cox and Yeo–Johnson transformations are well-known tools for this. However, the standard maximum likelihood estimator of their transformation parameter is highly sensitive to outliers, and will often try to move outliers inward at the expense of the normality of the central part of the data. We propose a modification of these transformations as well as an estimator of the transformation parameter that is robust to outliers, so the transformed data can be approximately normal in the center and a few outliers may deviate from it. It compares favorably to existing techniques in an extensive simulation study and on real data.

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