Modified Maximum Likelihood Estimator Research Articles

EWMA control charts based on robust estimators: A powerful tool for monitoring a process with a non-normal distribution

This paper presents a study on using memory-type control charts to detect small to moderate shifts in the process mean, particularly when the underlying distribution deviates from normal assumptions. To address this, robust estimators based on modified maximum likelihood estimation (MMLE) and Lloyd’s BLUE estimators using generalized least square estimation (GLSE) are proposed. The proposed estimators utilize auxiliary information to enhance sensitivity against abrupt changes in the mean. Additionally, they play a crucial role in refining the precision of the estimated mean, contributing to a more accurate and nuanced estimation of the true mean. The efficiency and robustness of these estimators are compared with conventional ones from the literature. The proposed robust estimators are integrated into the design of the exponentially weighted moving average (EWMA) control chart. Performance evaluation is based on average run lengths (ARLs), standard deviation, and percentiles of the run lengths. A real-life example illustrates the significance of the proposed estimators over conventional ones for non-normal distributions. The results demonstrate that the EWMA control chart with proposed estimators outperforms competitors in terms of ARLs, SDRLs, and percentiles. This study offers insights into using robust estimators to enhance memory-type control charts' performance for non-normal distributions.

The impact of natural constraints in linear regression of log transformed response variables

Abstract In linear regression, log transforming the response variable is the usual workaround regarding departures from the assumption of normality. However, the response variable is often subject to natural constraints, which can result in a truncated distribution of the residual errors on the log scale. In forestry, allometric relationships and tree growth are two typical examples a natural constraint; the response variable cannot be negative. Traditional least squares estimators do not account for constrained response variables. For this study, a modified maximum likelihood (MML) estimator that takes natural constraints into account was developed. This estimator was tested through a simulation study and showcased with black spruce tree diameter increment data. Results show that the ordinary least squares estimator underestimated large conditional expectations of the response variable on the original scale. In contrast, the MML estimator showed no evidence of bias for large sample sizes. Departures from distributional assumptions cannot be overlooked when the model is used for predictive purposes. Both Monte Carlo error propagation and prediction intervals rely on these assumptions. In this context, the MML estimator developed for this study can be used to properly propagate the errors and produce reliable prediction intervals.

Assessing earthquake rates and b-value given spatiotemporal variation in catalog completeness: Application to Atlantic Canada

Spatiotemporal variations in the magnitude of completeness Mc make it challenging to confidently assess seismic hazard or even to simply compare earthquake rates between regions. In this study, we introduce new techniques to correct for heterogeneous Mc in a treatment of the eastern and Atlantic Canada earthquake catalog (1985--2022). We first introduce new methodology to predict Mc(x,t) based on the distribution of seismometers. Second, we introduce a modified maximum-likelihood estimator (MLE) for b (the b-value) that accounts for spatiotemporal Mc variation, allowing the inclusion of more earthquakes. Third, we compute the ratio of detected/predicted M>1 earthquakes as a function of Mc and apply it to create a calibrated M>1 event-rate map. The resulting map has advantages over a moment-rate map, which is effectively sensitive only to the very largest earthquakes in the dataset. The new MLE results in a modestly more precise b when applied to the Charlevoix Seismic Zone, and a substantial increase in precision when applied to the full Atlantic Canada region. It may prove useful in future hazard assessments, particularly of regions with highly heterogeneous Mc and relatively sparse catalogs.

Modified Maximum Likelihood Estimation Metal Magnetic Memory Quantitative Identifying Model of Weld Defect Levels Based on Dempster–Shafer Theory

Metal magnetic memory (MMM) is a nondestructive testing technology based on the magnetomechanical effect, which is widely used in the qualitative detection of stress concentration zones for welded joints. However, there is inevitable residual stress after welding, which brings the bottleneck of quantitative identification between the weld residual stress concentration and the early hidden damage. In order to overcome the bottleneck of quantitative identification of weld defect levels with MMM technology, a modified maximum likelihood estimation (MLE) MMM quantitative identifying model is first proposed. The experimental materials are Q235B welded plate specimens. Fatigue tension experiments were operated to find the MMM feature laws of critical hidden crack by comparing with synchronous X-ray detection results. Six MMM characteristic parameters, which are, ΔHp(x), Gxmax, Zxmax, ΔHp(y), Gymax and Zymax, are extracted corresponding to the normal state, the hidden crack state and the macroscopic crack, respectively. The MLE values of the six parameters are obtained by the kernel density functions with optimized bandwidth from the view of mathematical statistics. Furthermore, the modified MLE MMM quantitative identifying model is established based on D–S theory to overcome the partial overlap of MLE values among different defect levels, of which the uncertainty is as low as 0.3%. The verification result from scanning electron microscopy (SEM) is consistent with the prediction of the modified MLE MMM model, which provides a new method for quantitative identification of weld defect levels.

Testing the equality of treatment means in one-way ANOVA: Short-tailed symmetric error terms with heterogeneous variances

We propose two tests based on fiducial and generalized $p$-value approaches for testing the equality of treatment means in one-way analysis of variance (ANOVA). Modified maximum likelihood (MML) estimators are used in the proposed tests. In contrast to least squares (LS) estimators, MML estimators are highly efficient and robust to plausible deviations from an assumed distribution and to mild data anomalies. In this study, error terms are assumed to have short-tailed symmetric (STS) distributions with heterogeneous variances. The performances of the proposed tests are compared with the fiducial based test using bias-corrected LS estimators via an extensive Monte Carlo simulation study. Finally, two real datasets are analyzed for illustrative purposes.

Robust ratio estimators of population mean for skewed and contaminated population

The efficiency of the traditional ratio estimator decreases with the presence of outliers, inliers or when the underlying distribution is not normal. To improve the efficiency, we propose new robust ratio estimators utilizing the Lloyd’s robust estimator and modified maximum likelihood estimator (MMLE), and provide their theoretical properties. We calculate the mean square error and relative efficiency of the proposed estimators and compare its performance with some other existing traditional estimators via a simulation study under various contamination or misspecification models. Further, we evaluate the exact value of the mean square error and to compare the performance of the proposed estimators using the numerical illustrations.

Optimal Event-Based Policy for Remote Parameter Estimation in Wireless Sensing Architectures Under Resource Constraints

Energy is a resource bottleneck in wireless sensing networks (WSNs) relying on energy harvesting for their operations. This is pronounced in WSNs whose data is used for remote parameter estimation because only a subset of the measured information can be transmitted to the estimator. While much attention has been separately paid to communication schemes for energy-aware data transmission in WSNs under resource constraints and controlled parameter estimation, there has yet to emerge a censoring policy that minimizes the variance of a measured process’ estimated component parameters subject to realistic constraints imposed by the WSN. Consequently, this paper presents the derivation of an optimal event-based policy governing data collection and transmission that accounts for energy and data buffer sizes, stochastic models of harvested energy and event arrivals, value of information of measured data, and temporal death. The policy is optimal in the sense that it maximizes the information rate of transmitted data, thereby producing the best possible estimates of the process parameters using the modified maximum likelihood estimation given the system constraints. Experimental and simulation-based results reflect these objectives and illustrate that the framework is robust against significant uncertainty in the initial parameter estimates.

Estimation of the parameters of the gamma geometric process

There is no doubt that finding the estimators of model parameters accurately and efficiently is very important in many fields. In this study, we obtain the explicit estimators of the unknown model parameters in the gamma geometric process (GP) via the modified maximum likelihood (MML) methodology. These estimators are as efficient as maximum likelihood (ML) estimators. The marginal and joint asymptotic distributions of the MML estimators are also derived and efficiency comparisons between ML and MML estimators are made through an extensive Monte Carlo simulations. Moreover, a real data example is considered to illustrate the performances of the MML estimators together with their ML counterparts. According to simulation results, the performances of MML and ML estimators are close to each other even for small sample sizes.

Error Detection and Pattern Prediction Through Phase II Process Monitoring

The continuous monitoring of the machine is beneficial in improving its process reliability through reflected power function distribution. It is substantial for identifying and removing errors at the early stages of production that ultimately benefit the firms in cost-saving and quality improvement. The current study introduces control charts that help the manufacturing concerns to keep the production process in control. It presents an exponentially weighted moving average and extended exponentially weighted moving average and then compared their performance. The percentiles estimator and the modified maximum likelihood estimator are used to constructing the control charts. The findings suggest that an extended exponentially weighted moving average control chart based on the percentiles estimator performs better than exponentially weighted moving average control charts based on the percentiles estimator and modified maximum likelihood estimator. Further, these results will help the firms in the early detection of errors that enhance the process reliability of the telecommunications and financing industry.

Further Properties and Estimations of Exponentiated Generalized Linear Exponential Distribution

The recent exponentiated generalized linear exponential distribution is a generalization of the generalized linear exponential distribution and the exponentiated generalized linear exponential distribution. In this paper, we study some statistical properties of this distribution such as negative moments, moments of order statistics, mean residual lifetime, and their asymptotic distributions for sample extreme order statistics. Different estimation procedures include the maximum likelihood estimation, the corrected maximum likelihood estimation, the modified maximum likelihood estimation, the maximum product of spacing estimation, and the least squares estimation are compared via a Monte Carlo simulation study in terms of their biases, mean squared errors, and their rates of obtaining reliable estimates. Recommendations are made from the simulation results and a numerical example is presented to illustrate its use for modeling a rainfall data from Orlando, Florida.

Reliability estimation and parameter estimation for inverse Weibull distribution under different loss functions

In this paper, the classical and Bayesian estimators of the unknown parameters and the reliability function of the inverse Weibull distribution are considered. The maximum likelihood estimators (MLEs) and modified maximum likelihood estimators (MMLEs) are used in the classical parameter estimation. Bayesian estimators of the parameters are obtained by using symmetric and asymmetric loss functions under informative and non-informative priors. Bayesian computations are derived by using Lindley approximation and Markov chain Monte Carlo (MCMC) methods. The asymptotic confidence intervals are constructed based on the maximum likelihood estimators. The Bayesian credible intervals of the parameters are obtained by using the MCMC method. Furthermore, the performances of these estimation methods are compared concerning their biases and mean square errors through a simulation study. It is seen that the Bayes estimators perform better than the classical estimators. Finally, two real-life examples are given for illustrative purposes.

Lifetime-based analysis of binary fluorophores mixtures in the low photon count limit

SummarySingle biomolecule sensing often requires the quantification of multiple fluorescent species. Here, we theoretically and experimentally use time-resolved fluorescence via Time Correlated Single Photon Counting (TCSPC) to accurately quantify fluorescent species with similar chromatic signatures. A modified maximum likelihood estimator is introduced to include two fluorophore species, with compensation of the instrument response function. We apply this algorithm to simulated data of a simplified two-fluorescent species model, as well as to experimental data of fluorophores' mixtures and to a model protein, doubly labeled with different fluorophores' ratio. We show that 100 to 200 photons per fluorophore, in a 10-ms timescale, are sufficient to provide an accurate estimation of the dyes' ratio on the model protein. Our results provide estimation for the desired photon integration time toward implementation of TCSPC in systems with fast occurring events, such as translocation of biomolecules through nanopores or single-molecule burst analyses experiments.

Ranked set sampling on estimation of for inverse Weibull distribution and its applications

PurposeDistribution. The purpose of this study is to obtain the modified maximum likelihood estimator of stress–strength model using the ranked set sampling, to obtain the asymptotic and bootstrap confidence interval of P[Y < X], to compare the performance of author’s estimates with the estimates under simple random sampling and to apply author’s estimates on head and neck cancer.Design/methodology/approachThe maximum likelihood estimator of R = P[Y < X], where X and Y are two independent inverse Weibull random variables common shape parameter that affect the shape of the distribution, and different scale parameters that have an effect on the distribution dispersion are given under ranked set sampling. Together with the asymptotic and bootstrap confidence interval, Monte Carlo simulation shows that this estimator performs better than the estimator under simple random sampling. Also, the asymptotic and bootstrap confidence interval under ranked set sampling is better than these interval estimators under simple random sampling. The application to head and neck cancer disease data shows that the estimator of R = P[Y < X] that shows the treatment with radiotherapy is more efficient than the treatment with a combined radiotherapy and chemotherapy under ranked set sampling that is better than these estimators under simple random sampling.FindingsThe ranked set sampling is more effective than the simple random sampling for the inference of stress-strength model based on inverse Weibull distribution.Originality/valueThis study sheds light on the author’s estimates on head and neck cancer.

Modified Maximum Likelihood Estimation for Generalized Exponential Distribution

For a Modified Maximum Likelihood Estimate of the parameters of generalized exponential distribution (GE), a hyperbolic approximation is used instead of linear approximation for a function which appears in the Maximum Likelihood equation. This estimate is shown to perform better, in accuracy and simplicity of calculation, than the one based on linear approximation for the same function. Numerical computation for random samples of different sizes from generalized exponential distribution (GE), using type II censoring is done and is shown to be better than that obtained by Lee et al. [1].

Single-Molecule, Non-Chromatic Discrimination in Binary Fluorophores Mixtures

SummarySingle biomolecule sensing often involves quantification of multiple fluorescent species, coupled for example to different DNAs or proteins resides. While chromatic separation of the fluorescent species is well-established and broadly used it involves several limitations, particularly when the total photon budget is limited. Here we theoretically and experimentally explore the use of time-resolved fluorescence via Time Correlated Single Photon Counting (TCSPC) to accurately quantify multiple fluorescent species. A modified maximum likelihood estimator (MLE) is introduced to include two fluorophore species, with the compensation of the instrument response. We apply this algorithm to simulated data of a simplified two fluorescent species model, as well as to experimental data of free fluorophores mixtures and doubly labelled proteins in order to estimate the errors in the two species ratios as a function of photon budget available. Moreover, focusing on the case of labeled proteins in which the ratios of the two species per molecule is a discrete variable subject to the chemical labeling numbers of the two dyes, we obtain that between 1,000 to 2,000 photons can already provide an accurate estimation of the dyes’ ratio in labelled Ovalbumin proteins. These values correspond to roughly 100 to 200 photons per fluorophore suggesting a realistic path for dyes ratio determination from a single passage of a protein in the detection volume in roughly 10 ms timescale. Our results highlight the way in which temporal limitations of the TCSPC technique could hinder its implementation in systems with fast occurring events such as translocation of biomolecules through nanopores, or fast diffusion in single-molecule burst analyses experiments and provide estimation for the desired photon integration time.

A new approach using the genetic algorithm for parameter estimation in multiple linear regression with long-tailed symmetric distributed error terms: An application to the Covid-19 data

Maximum likelihood (ML) estimators of the model parameters in multiple linear regression are obtained using genetic algorithm (GA) when the distribution of the error terms is long-tailed symmetric. We compare the efficiencies of the ML estimators obtained using GA with the corresponding ML estimators obtained using other iterative techniques via an extensive Monte Carlo simulation study. Robust confidence intervals based on modified ML estimators are used as the search space in GA. Our simulation study shows that GA outperforms traditional algorithms in most cases. Therefore, we suggest using GA to obtain the ML estimates of the multiple linear regression model parameters when the distribution of the error terms is LTS. Finally, real data of the Covid-19 pandemic, a global health crisis in early 2020, is presented for illustrative purposes.

A Simple Random Sampling Modified Dual to Product Estimator for Estimating Population Mean using Order Statistics

Bandopadhyaya (1980) developed a dual to product estimator using robust modified maximum likelihood estimators (MMLE’s). Their properties were obtained theoretically and supported through simulations studies with generated as well as one real data set. Robustness properties in the presence of outliers and confidence intervals were studied.

The modified control charts for monitoring the shape parameter of weighted power function distribution under classical estimator

AbstractZaka et al provided a new distribution called the Weighted Power function distribution (WPFD), which has application in reliability engineering and survival analysis. They used different estimation methods to estimate the unknown parameters of WPFD and proved that modified maximum likelihood estimator (MMLE) is best to consider for the estimation of parameters. We have constructed the memoryless and memory‐based control charts based on the assumption that the distribution of the underlying process does not follow the normal distribution. In this paper, we provide modified control charts using MMLE of the shape parameter for WPFD. We develop control charts to keep the process in control when the distribution of errors of underlying process follows WPFD. We propose the modified memoryless control chart, that is, Shewhart control chart and modified memory‐based control chart, that is, Exponentially weighted moving average (EWMA) and Hybrid exponentially weighted moving average (HEWMA) control charts. We have made the comparison of the proposed control charts using Monte Carlo simulation and the real‐life application for both and the memoryless control charts and memory‐based control charts. We see that HEWMA based on MMLE performs better as compared to other proposed control charts.

Inference in Multiple Linear Regression Model with Generalized Secant Hyperbolic Distribution Errors

We study multiple linear regression model under non-normally distributed random error by considering the family of generalized secant hyperbolic distributions. We derive the estimators of model parameters by using modified maximum likelihood methodology and explore the properties of the modified maximum likelihood estimators so obtained. We show that the proposed estimators are more efficient and robust than the commonly used least square estimators. We also develop the relevant test of hypothesis procedures and compared the performance of such tests vis-a-vis the classical tests that are based upon the least square approach.

Inference in Multiple Linear Regression Model with Generalized Secant Hyperbolic Distribution Errors

We study multiple linear regression model under non-normally distributed random error by considering the family of generalized secant hyperbolic distributions. We derive the estimators of model parameters by using modified maximum likelihood methodology and explore the properties of the modified maximum likelihood estimators so obtained. We show that the proposed estimators are more efficient and robust than the commonly used least square estimators. We also develop the relevant test of hypothesis procedures and compared the performance of such tests vis-a-vis the classical tests that are based upon the least square approach.