Performance Of Maximum Likelihood Estimator Research Articles

Bayesian Estimations of Shannon Entropy and Rényi Entropy of Inverse Weibull Distribution

In this paper, under the symmetric entropy and the scale squared error loss functions, we consider the maximum likelihood (ML) estimation and Bayesian estimation of the Shannon entropy and Rényi entropy of the two-parameter inverse Weibull distribution. In the ML estimation, the dichotomy is used to solve the likelihood equation. In addition, the approximation confidence interval is given by the Delta method. Because the form of estimation results is more complex in the Bayesian estimation, the Lindley approximation method is used to achieve the numerical calculation. Finally, Monte Carlo simulations and a real dataset are used to illustrate the results derived. By comparing the mean square error between the estimated value and the real value, it can be found that the performance of ML estimation of Shannon entropy is better than that of Bayesian estimation, and there is no significant difference between the performance of ML estimation of Rényi entropy and that of Bayesian estimation.

Extended Half-Power Exponential Distribution with Applications to COVID-19 Data

In this paper, the Extended Half-Power Exponential (EHPE) distribution is built on the basis of the Power Exponential model. The properties of the EHPE model are discussed: the cumulative distribution function, the hazard function, moments, and the skewness and kurtosis coefficients. Estimation is carried out by applying maximum likelihood (ML) methods. A Monte Carlo simulation study is carried out to assess the performance of ML estimates. To illustrate the usefulness and applicability of EHPE distribution, two real applications to COVID-19 data in Chile are discussed.

Accuracy and Bias of the Rasch Rating Scale Person Estimates using Maximum Likelihood Approach: A Comparative Study of Various Sample Sizes

The focus of this article is to evaluate the maximum likelihood estimation (MLE) performance in estimating the person parameters in the Rasch rating scale model (RRSM). For that purpose, 1000 iterations of the Markov Chain Monte Carlo (MCMC) simulation technique were performed based on a different number of sample sizes and several number of items. The performance of MLE in estimating the person parameters according to the different number of sample sizes was compared through accuracy and bias measures. Root mean square error (RMSE) and mean absolute error (MAE) were used to examine the accuracy of the estimates, while bias in estimation was assessed through the mean difference of estimates and true values of the person parameters. The simulated survey data sets in this study were generated according to the RRSM under the assumption of normality was satisfied. Results from the simulation analysis showed that in comparison to the larger sample sizes, smaller sample sizes tend to produce higher RMSE and MAE. In addition, the maximum likelihood estimates of the person parameters in smaller sample sizes also recorded a higher value of the mean difference of the person estimates and its true values compared to larger sample sizes. Findings from this study imply that the use of the MLE approach in small sample sizes results in less accurate and highly biased person estimates across the number of items.

Robust Estimation for Proportional Odds Model through Monte Carlo Simulation

Ordinal regression is used to model the ordinal response variable as functions of several explanatory variables. The most commonly used model for ordinal regression is the proportional odds model (POM). The classical technique for estimating the unknown parameters of this model is the maximum likelihood (ML) estimator. However, this method is not suitable for solving problems with extreme observations. A robust regression method is needed to handle the problem of extreme points in the data. This study proposes Huber M-estimator as a robust method to estimate the parameters of the POM with a logistic link function and polytomous explanatory variables. This study assesses ML estimator performance and the robust method proposed through an extensive Monte Carlo simulation study conducted using statistical software, R. Measurement for comparisons are bias, RMSE, and Lipsitzs' goodness of fit test. Various sample sizes, percentages of contamination, and residual standard deviations are considered in the simulation study. Preliminary results show that Huber estimates provide the best results for parameter estimation and overall model fitting. Huber's estimator has reached a 50% breakdown point for data containing extreme points that are quite far from most points. In addition, the presence of extreme points that have only a distance of two times far from most points has no major impact on ML estimates. This means that the estimates for ML and Huber may yield the same results if the model's residual values are between -2 and 2. This situation may also occur for data with a percentage of contamination below 5%.

A new adjusted Liu estimator for the Poisson regression model

SummaryThe Poisson regression model (PRM) is usually applied in the situations when the dependent variable is in the form of count data. For estimating the unknown parameters of the PRM, maximum likelihood estimator (MLE) is commonly used. However, its performance is suspected when the regressors are multicollinear. The performance of MLE is not satisfactory in the presence of multicollinearity. To mitigate this problem, different biased estimators are discussed in the literature, that is, ridge and Liu. However, the drawback of using the traditional Liu estimator is that in most of the times, the shrinkage parameter d, attains a negative value which is the major disadvantage of traditional Liu estimator. So, to overcome this problem, we propose a new adjusted Poisson Liu estimator (APLE) for the PRM which is the robust solution to the problem of multicollinear explanatory variables. For assessment purpose, we perform a theoretical comparison with other competitive estimators. In addition, a Monte Carlo simulation study is conducted to show the superiority of the new estimator. At the end, two real life applications are also considered. From the findings of simulation study and two empirical applications, it is observed that the APLE is the most robust and consistent estimation method as compared to the MLE and other competitive estimators.

Genetic algorithm approach with an adaptive search space based on EM algorithm in two-component mixture Weibull parameter estimation

A mixture of two Weibull distributions (WW) has a variety of usage area from reliability analysis to wind speed modeling. Maximum likelihood (ML) method is the most frequently used method in parameter estimation of WW. Due to the nonlinear nature of the log-likelihood function of WW, usage of iterative techniques is a necessary process. Conventional iterative techniques such as Newton Raphson (NR) require considerable analytical preparatory to work to obtain gradient and may lead to numerical difficulties such as convergence problems. The aim of this paper is to present a genetic algorithm (GA) with an adaptive search space based on the Expectation–Maximization (EM) algorithm to obtain the ML estimators of the parameters of WW. The simulation study is conducted to compare the performances of ML estimators obtained using NR algorithm, EM algorithm, simulated annealing algorithm, and the proposed GA. Furthermore, real data examples are used to compare the efficiency of proposed GA with the existing methods in the literature. Simulation results and real data examples show that the proposed GA has superiority over other techniques in terms of efficiency.

On the James-Stein estimator for the poisson regression model

The Poisson regression model (PRM) aims to model a counting variable y, which is usually estimated by using maximum likelihood estimation (MLE) method. The performance of MLE is not satisfactory in the presence of multicollinearity. Therefore, we propose a Poisson James-Stein estimator (PJSE) as a solution to the problems of inflated variance and standard error of MLE with multicollinear explanatory variables. For assessing the superiority of proposed estimator, we present a theoretical comparison based on the matrix mean squared error (MMSE) and scalar mean squared error (MSE) criterions. A Monte Carlo simulation study is performed under different conditions in order to investigate the performance of the proposed estimator where MSE is considered as an evaluation criterion. In addition, an aircraft damage data is also considered to assess the superiority of proposed estimator. Based on the results of simulation and real data application, it is shown that the PJSE outperforms the classical MLE and other biased estimation methods in a sense of minimum MSE criterion.

New EM-type algorithms for the Heckman selection model

The Heckman selection model is widely used to analyse data for which the outcome is partially observable, and the missing part is not random. The 2-step method, maximum likelihood estimation (MLE), and EM algorithms have been developed to analyse this model; however, they have certain limitations. Three new algorithms (ECM, ECM(NR), and ECME) will be proposed with the advantages of the EM algorithm: easy implementation and numerical stability. Considering bias and mean squared error (MSE), simulations with different correlation values suggest that MLE performs similarly to the proposed algorithms; however, MLE as well as the proposed algorithms yield better estimations than the 2-step method. A simulation study in which standard error is also considered demonstrates that the new algorithms are more robust than MLE, and yield slightly better estimations than the 2-step and the robust 2-stage methods. Real data analyses are also provided to discuss the performance of MLE, 2-step, and the proposed algorithms. A real data analysis concerning the robustness issue further illustrates that, under certain conditions, the proposed algorithms are more efficient and stable.

Location Based Generalized Akash Distribution: Properties and Applications

In recent years, several statistical finite mixture models have been proposed to model the lifetime data with heterogeneity. The Lindley distribution has been highlighted by many authors for these types of lifetime data analysis. This paper introduces a new Lindley family distribution called location-based generalized Akash distribution (NGAD) with monotonic increasing and bathtub failure rates. The density function of NGAD is flexible to cover the left-skewed, right-skewed and symmetrical shapes with different tail-weights. Its fundamental structural properties and its ability to provide a suitable statistical model for various types of data sets are studied. The maximum likelihood (ML) method is used to estimate its unknown parameters and the performance of ML estimates are examined by a simulation study. Finally, several real-data sets with different characteristics are used to illustrate its flexibility. It is observed that NGAD provides a better fit than some other existing modified Lindley distributions.

A Method for Reducing the Performance Gap Between Non-Coherent and Coherent Sub-Arrays

We consider the problem of estimating the angles of arrival (AOAs) of multiple sources from a single snapshot obtained by a set of non-coherent sub-arrays, i.e., while the antenna elements in each sub-array are coherent, each sub-array observes a different unknown phase offset. Previous relevant works are based on eigendecomposition of the sample covariance, which requires a large number of snapshots, or on combining the sub-arrays using non-coherent processing. In this paper, we propose a technique to estimate the sub-arrays phase offsets for a given AOAs hypothesis, which facilitates approximate maximum likelihood estimation (MLE) of the AOAs from a single snapshot. We derive the Cramér-Rao lower bound (CRLB) for the problem at hand, and analytically show that for a single source it may suffice to use a simple non-coherent AOA estimation. However, as we demonstrate by computer simulations, for multiple sources the proposed approach clearly outperforms the non-coherent estimator, and even attains the CRLB in various scenarios. Furthermore, the performance of the proposed method is often close to the performance of MLE in the coherent case, and the gap between the estimators is unavoidable, as implied by the gap between the CRLB for the coherent and non-coherent cases.

Exponential intervened Poisson distribution

We introduce a new family of distributions using intervened power series distribution. It is a rich family containing distributions such as Marshall–Olkin extended families of distributions, families of distributions generated through truncated negative binomial, and families of distributions generated through truncated binomial distribution. Also, the proposed distribution is a generalization of the families of distributions generated through zero truncated Poisson distribution. As a particular case, we study exponential intervened Poisson (EIP) distribution. This is a generalization of exponential-Poisson distribution. The shape properties of the pdf and hazard rate are proved. The model identifiability is established. The expression for moments, mean deviation, distribution of order statistics, and residual life function of EIP distribution are derived. The unknown parameters of the distribution are estimated using the method of maximum likelihood. The existence and uniqueness of maximum likelihood estimates (MLEs) are proved. Simulation study is carried out to show the performance of MLEs. EIP distribution is fitted to two real data sets and it is shown that the distribution is more appropriate than other competitive models. The adequacy of the EIP model for the data set is established using parametric bootstrap approach.

Statistical Inferences of $$R=P(X<Y)$$ for Exponential Distribution Based on Generalized Order Statistics

In this paper, we have derived the classical and Bayesian inferences for stress–strength reliability $$R=P(X<Y)$$ , when the stress–strength data are available in the form of generalized order statistics (gos). It is supposed that the two random samples are mutually independent and obtained from the exponential population. Based on gos, maximum likelihood estimator (MLE) and uniformly minimum variance unbiased estimator (UMVUE) for R of the exponential distribution have been obtained. We have also constructed the exact confidence interval (CI) and asymptotic CI for R. In addition, we have derived the Bayes estimator for R by considering squared error loss function. Simulation study has been performed for comparing the performance of MLE and UMVUE. A Monte–Carlo simulation is also carried out for comparing the performance of Bayes estimator with different priors. For illustrative purposes, a real data analysis is also provided.

Asymptotic and Bootstrap Confidence Intervals for the Process Capability Index cpy Based on Lindley Distributed Quality Characteristic

SYNOPTIC ABSTRACTProcess capability indices (PCIs) have been widely applied in measuring product potential and performance. It is of great significance to quality control engineers, as it quantifies the relation between the actual performance of the process and the preset specifications of the product. Among the plethora of the suggested PCIs, most of them were developed for normally distributed processes. In this article, we consider generalized process capability index Cpy suggested by Maiti et al. (2010), which can be used for normal, non-normal, and continuous as well as discrete random variables. The objective of this article is twofold. First, we obtain maximum likelihood estimator (MLE) and minimum variance unbiased estimator (MVUE) of the PCI Cpy for the Lindley distributed quality characteristics. Second, we compare asymptotic confidence interval (ACI) with four bootstrap confidence intervals (BCIs); namely, standard bootstrap (s-boot), percentile bootstrap (p-boot), Student’s t bootstrap (t-boot), and bias-corrected accelerated bootstrap (BCa-boot) of Cpy based on maximum likelihood method of estimation. Monte Carlo simulations have been carried out to compare the performance of MLEs and MVUEs, and also investigate the average widths, coverage probabilities, and relative coverages of ACI and BCIs of Cpy. Two real data sets have been analyzed for illustrative purposes.

Separate estimation of laser micro-Doppler parameters based on maximum likelihood schemes

Laser micro-Doppler (MD) effect is capable of obtaining obvious modulation in weak vibration detection. It helps to estimate target micro-motion parameters with high precision, which may extend the application field of MD to subtle identification and recognition. In laser detection, the multiple scattering points in the field of view will generate the single-channel multi-component (SCMC) signal. Moreover, the micro-Doppler features of each component will be overlapped in the time-frequency domain because of the similar micro-motion parameters. The overlapped SCMC signal makes the estimation of the MD parameters a very difficult problem, and there has been no good method so far. In this paper, a separate parameter estimator based on the maximum likelihood framework is proposed to deal with this underdetermined problem. First, the detailed period scanning method is presented to improve the estimation accuracy of micro-motion frequency from the singular value ratio (SVR) spectrum. Further, the amplitude ratio information of each component is extracted from the SVR spectrum. Then, the closed-form expressions of the maximum likelihood estimation (MLE) for the remaining micro-motion parameters are derived, where the mean likelihood estimation is used to approximate to the performance of MLE. The high nonlinearity and multi-peak distribution shape of the likelihood function (LF) in laser MD signal will lead to incorrect estimation result. To this end, a new LF based on the energy spectrum characteristics is designed. The new LF acts as a smoothing filter to the probability density function, through which the ideal PDF distribution form that has only one smooth peak is obtained. With this modification, the requirements for the initialization are reduced and the robustness in low SNR situation is increased. The Markov chain Monte Carlo sampling is employed to implement the MLE. The Gibbs method is chosen to solve the multi-dimensional parametric problems, and the detailed process is listed. In the end, the simulation results prove the feasibility and high efficiency of the proposed method. The accuracy of parameter estimation reaches the Cramer-Rao boundary. The inverse Radon transform is used as a comparison with the experiment, and the results show the precise estimation advantage of the presented method.

Parameter Estimation via Unlabeled Sensing Using Distributed Sensors

In this letter, the problem of estimating an unknown deterministic parameter is studied, where each sensor acquires a noisy version of the signal and the data at fusion center are unlabeled. Two scenarios are studied: one in which each sensor uses analog communication to transmit its observations with different channel coefficients, and the other in which each sensor uses a different threshold to quantize the noisy signal with a transition matrix describing the channel. Sufficient conditions are provided, under which maximum likelihood (ML) estimators can be found in polynomial time, and numerical simulations are conducted to evaluate performances of ML estimators.

Parameter Estimation of Shared Frailty Models Based on Particle Swarm Optimization

Standard survival techniques such as proportional hazards model are suffering from the unobserved heterogeneity. Frailty models provide an alternative way in order to account for heterogeneity caused by unobservable risk factors. Although vast studies have been done on estimation procedures, Evolutionary Algorithms (EAs) haven't received much attention in frailty studies. In this paper, we investigate the estimation performance of maximum likelihood estimation (MLE) via Particle Swarm Optimization (PSO) in modelling multivariate survival data with shared gamma frailty. Simulation studies and real data application are performed in order to assess the performance of MLE via PSO, quasi-Newton and conjugate gradient method.

Reliability Estimation For The Generalised Inverted Rayleigh Distribution

In this article, we consider a generalized inverted Rayleigh distribution. The maximum likelihood estimators (MLEs) of scale and shape parameters are obtained. Also, we compute MLE of reliability function. Asymptotic condence interval for the parameters and reliability function is constructed. Simulation study is conducted to investigate performance of MLEs and con- dence intervals. An illustration with real data is provided.

Slashed generalized Rayleigh distribution

ABSTRACTWe introduce a new family of distributions suitable for fitting positive data sets with high kurtosis which is called the Slashed Generalized Rayleigh Distribution. This distribution arises as the quotient of two independent random variables, one being a generalized Rayleigh distribution in the numerator and the other a power of the uniform distribution in the denominator. We present properties and carry out estimation of the model parameters by moment and maximum likelihood (ML) methods. Finally, we conduct a small simulation study to evaluate the performance of ML estimators and analyze real data sets to illustrate the usefulness of the new model.

English

This study presents a probabilistic model for daily extreme rainfall. The Annual Maximum Series (AMS) data of daily rainfall in Makurdi was fitted to Generalized Extreme Value (GEV) distribution using Maximum Likelihood Estimation (MLE) and Bayesian Markov Chain Monte Carlo (Bayesian MCMC) simulations. MLE is a reliable principle to derive an efficient estimator for a model as sample size approaches infinity. Results in this study show that despite the asymptotic requirement of the MLE, its performance can be improved when adopting Bayesian MCMC. The comparison between the performance of MLE and Bayesian MCMC methods using Percent Bias (PBIAS), Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) proved Bayesian MCMC is the better method to estimate the distribution parameters of extreme daily rainfall amount in Makurdi. Based on the 36-year record of rainfall (1979-2014) in Makurdi, return levels for the next 10, 100, 500, 1000 and 10000 years were derived. Key words: Extreme daily rainfall, generalized extreme value distribution, parameter estimation, t-year return level, Makurdi.

A Maximum Likelihood Estimation of the Thermal Resistance of a Cavity Wall from On-site Measurements

The thermal resistances of a building's components are main parameters in assessing the overall thermal performance of the building envelope. In order to assess the actual, as-built thermal performance of building components, a reliable thermal characterization of building elements from on-site measurements is required. In this paper, the thermal resistance of a cavity wall is estimated by means of maximum likelihood estimation (MLE) from on-site meas rements. To evaluate the performance of MLE the method is applied on various data sets of the outdoor test wall. Based on the results for all data sets, the potentials of the dynamic estimation technique are considered. Special attention is paid to the accuracy of the estimation results in function of the measurement length and the measurement period throughout theyear.