Maximum Likelihood Estimation Estimation Research Articles

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.

Real Estate Return Distributions with New NCREIF Data Series

The accuracy of real estate return distribution parameter estimation is affected by the tools used to do the work as well as the data sets employed. Consistent with previous studies, investment risk models with infinite variance describe distributions of individual property returns in the new NCREIF Indicators: Capital Performance and Property Operations individual property database over the period 1990–2021. Applying Maximum Likelihood Estimation (MLE) to historic data shows real estate investment risk to be heteroscedastic, but the Characteristic Exponent of the investment risk function varies more among property types than previously reported whether computed by MLE or other estimation techniques.

On a Mixed Poisson Liu Regression Estimator for Overdispersed and Multicollinear Count Data.

The mixed Poisson regression models are commonly employed to analyze the overdispersed count data. However, multicollinearity is a common issue when estimating the regression coefficients by using the maximum likelihood estimator (MLE) in such regression models. To deal with the multicollinearity, a Liu estimator was proposed by Liu (1993). The Poisson-Modification of the Quasi Lindley (PMQL) regression model is a mixed Poisson regression model introduced recently. The primary interest of this paper is to introduce the Liu estimator for the PMQL regression model to mitigate the multicollinearity issue. To estimate the Liu parameter, some exiting methods are used, and the superiority conditions of the new estimator over the MLE and PMQL ridge regression estimator are obtained based on the mean square error (MSE) criterion. A Monte Carlo simulation study and applications are used to assess the performance of the new estimator in the scalar mean square error (SMSE) sense. Based on the simulation study and the results of the applications, it is shown that the PMQL Liu estimator performs better than the MLE and some other existing biased estimators in the presence of multicollinearity.

Robust Interval Estimation Using Density Power Divergence

The maximum likelihood estimator (MLE) is known to be asymptotically efficient but it is not robust with respect to outliers and model misspecification. Basu et al. (1998) suggest a family of divergence measures called density power divergences , which is based on density. Each measure in this family is indexed by a single tuning parameter a. This parameter controls the trade-off between asymptotic efficiency and robustness of the estimators. The L₂-distance and the Kullback-Leibler divergence belong to this family. With a appropriately chosen tuning parameter, one can get a minimum density power divergence estimator (MDPDE). For 0 < a < 1, the estimator is in between MLE and minimum L₂-distance estimator L₂E. Note that MLE is efficient-but-nonrobust while L₂E is robust-butinefficient. Hong, Kim (2001) suggest an automatic selection of a. In this paper we will suggest a confidence interval using MDPDE when the data set is contaminated. Bootstrap resampling will be used to obtain the confidence interval. It is expected that the resulting confidence intervals (called MDPD bootstrap confidence intervals) are robust with respect to the outliers. The performances of the MDPDE bootstrap confidence intervals are investigated via simulation study.

A comparison of estimation methods adjusting for selection bias in adaptive enrichment designs with time-to-event endpoints.

Adaptive enrichment designs in clinical trials have been developed to enhance drug developments. They permit, at interim analyses during the trial, to select the sub-populations that benefits the most from the treatment. Because of this selection, the naive maximum likelihood estimation of the treatment effect, commonly used in classical randomized controlled trials, is biased. In the literature, several methods have been proposed to obtain a better estimation of the treatments' effects in such contexts. To date, most of the works have focused on normally distributed endpoints, and some estimators have been proposed for time-to-event endpoints but they have not all been compared side-by-side. In this work, we conduct an extensive simulation study, inspired by a real case-study in heart failure, to compare the maximum-likelihood estimator (MLE) with an unbiased estimator, shrinkage estimators, and bias-adjusted estimators for the estimation of the treatment effect with time-to-event data. The performances of the estimators are evaluated in terms of bias, variance, and mean squared error. Based on the results, along with the MLE, we recommend to provide the unbiased estimator and the single-iteration bias-adjusted estimator: the former completely eradicates the selection bias, but is highly variable with respect to a naive estimator; the latter is less biased than the MLE estimator and only slightly more variable.

Multi-component stress–strength parameter estimation of a non-identical-component strengths system under the adaptive hybrid progressive censoring samples

To produce more flexible models in reliability theory field, statistical inference of stress–strength parameter was studied in a multi-component system with the non-identical-component strengths based on bathtub-shaped distribution under adaptive Type-II hybrid progressive censoring samples. The problem is considered in three cases. First, with assuming that common first parameter is unknown for strengths and stress variables, the maximum likelihood estimation (MLE) and two Bayes approximations are obtained. Also, asymptotic, two bootstrap, and highest posterior density (HPD) intervals are derived. Second, with assuming that common first parameter is known, MLE, exact Bayes estimation, uniformly minimum variance unbiased estimator (UMVUE), and different confidence intervals are provided. Finally, in the third case, with assuming that all parameters are different and unknown, MLE and Bayesian estimation are studied. For comparing different estimates, the Monte Carlo simulation is used and one real data set is employed to implement theoretical methods.

Generalized ridge estimator in negative binomial regression model

The ridge estimator has been consistently demonstrated to be an attractive shrinkage method to reduce the effects of multicollinearity. The negative binomial regression model (NBRM) is a well-known model in application when the response variable is a count data with overdispersion. However, it is known that the variance of maximum likelihood estimator (MLE) of the NBRM coefficients can negatively affected in the presence of multicollinearity. In this paper, the generalized ridge estimator is proposed to overcome the limitation of ridge estimator. Several methods for estimating the shrinkage matrix have been adapted. Our Monte Carlo simulation results suggest that the proposed estimator, regardless the type of estimating method of shrinkage matrix is better than the MLE estimator and ridge estimator, in terms of MSE. In addition, some estimating method of shrinkage matrix can bring significant improvement relative to others.

Estimation of $$\alpha -\kappa -\mu $$ mobile fading channel parameters using evolutionary algorithms

This paper proposes the use of evolutionary algorithms (EAs) to estimate the physical parameters of the generalized $$\alpha -\kappa -\mu $$ mobile fading channel model. The estimation of parameters is a fundamental step that allows for the statistical model to adjust to the real experimental data. The The maximum likelihood estimation (MLE) method that is traditionally used for estimating parameters of the $$\alpha -\kappa -\mu $$ channel uses nonlinear numerical methods. In some cases, the use of nonlinear numerical methods may lead the MLE to make physically unacceptable estimations, or even to not be able to obtain a result. Our proposal is to innovate the existing EAs by incorporating an adaptive approach, a new mutation strategy and an adequate fitness function for the estimation of $$\alpha -\kappa -\mu $$ parameters. Experimental results are presented to confirm that parameters estimated by the EAs (genetic algorithms, differential evolution algorithms, and differential evolution algorithms with an adaptive guiding mechanism) are all physically acceptable. These experiments show that the EAs outperform MLE estimation results.

The Weibull log‐logistic mixture distributions: Model, theory and application to lifetime data

AbstractLifetime data collected from reliability tests are among data that often exhibit significant heterogeneity caused by variations in manufacturing, which makes standard lifetime models inadequate. Finite mixture models provide more flexibility for modeling such data. In this paper, the Weibull‐log‐logistic mixture distributions model is introduced as a new class of flexible models for heterogeneous lifetime data. Some statistical properties of the model are presented including the failure rate function, moments generating function, and characteristic function. The identifiability property of the class of all finite mixtures of Weibull‐log‐logistic distributions is proved. The maximum likelihood estimation (MLE) of model parameters under the Type I and Type II censoring schemes is derived. Some numerical illustrations are performed to study the behavior of the obtained estimators. The model is applied to the hard drive failure data made by the Backblaze data center, where it is found that the proposed model provides more flexibility than the univariate life distributions (Weibull, Exponential, logistic, log‐logistic, Frechet). The failure rate of hard disk drives (HDDs) is obtained based on MLE estimates. The analysis of the failure rate function on the basis of SMART attributes shows that the failure of HDDs can have different causes and mechanisms.

Systematic comparison of epidemic growth patterns using two different estimation approaches.

BackgroundDifferent estimation approaches are frequently used to calibrate mathematical models to epidemiological data, particularly for analyzing infectious disease outbreaks. Here, we use two common methods to estimate parameters that characterize growth patterns using the generalized growth model (GGM) calibrated to real outbreak datasets.Materials and methodsData from 31 outbreaks are used to fit the GGM to the ascending phase of each outbreak and estimate the parameters using both least squares (LSQ) and maximum likelihood estimation (MLE) methods. We utilize parametric bootstrapping to construct confidence intervals for parameter estimates. We compare the results including RMSE, Anscombe residual, and 95% prediction interval coverage. We also evaluate the correlation between the estimates from both methods.ResultsComparing LSQ and MLE estimates, most outbreaks have similar parameter estimates, RMSE, Anscombe, and 95% prediction interval coverage. Parameter estimates do not differ across methods when the model yields a good fit to the early growth phase. However, for two outbreaks, there are systematic deviations in model fit to the data that explain differences in parameter estimates (e.g., residuals represent random error rather than systematic deviation).ConclusionOur findings indicate that utilizing LSQ and MLE methods produce similar results in the context of characterizing epidemic growth patterns with the GGM, provided that the model yields a good fit to the data.

A Short Note on Obtaining Item Parameter Estimates of IRT Models with Bayesian Estimation in Mplus

Parameter estimation of Item Response Theory (IRT) models can be applied using both Bayesian and non-Bayesian methods. Although maximum likelihood estimation (MLE), a non-Bayesian method, has predominated since the 1970s, there is an increasing use of Bayesian methods, due to their capability for estimating complex models and for their implementation in commercially available software. In view of the recent increase in the popularity of these methods, a comparison between model parameter estimates from the two types of methods would be useful for practitioners. In this study, we compare MLE and Bayesian estimation, two popular methods for obtaining parameter estimates for dichotomous IRT models, using the MLE and Bayes estimator options as implemented in the Mplus software package. Results indicated Bayesian and MLE estimates differed only slightly, clearly demonstrating the consistency between estimates from the two methods. Further, Bayes estimator option in Mplus can be a viable and relatively easy to use tool for calibrations of IRT models.

Bayesian estimation of ridge parameter under different loss functions

In linear regression modeling, the presence of multicollinearity among the explanatory variables has undesirable effects on the maximum likelihood estimator (MLE). To overcome this effect, we proposed some new ridge parameters under Bayesian paradigm. Moreover, we also compare these ridge parameters with Bayesian approach under different loss functions. To access the performance of new ridge parameters, we conduct a Monte Carlo simulation study where mean squared error (MSE) is considered as an evaluation criterion. In addition, a real life example is also provided to assess the superiority of the proposed estimators on the basis of MSE and cross-validation approaches. The simulation and real application results show that the Bayesian ridge parameter estimated under Precautionary loss function is better as compared to the other loss functions as well as the MLE and ordinary ridge regression estimator.

Exponential random graph models for little networks

Statistical models for social networks have enabled researchers to study complex social phenomena that give rise to observed patterns of relationships among social actors and to gain a rich understanding of the interdependent nature of social ties and actors. Much of this research has focused on social networks within medium to large social groups. To date, these advances in statistical models for social networks, and in particular, of Exponential-Family Random Graph Models (ERGMS), have rarely been applied to the study of small networks, despite small network data in teams, families, and personal networks being common in many fields. In this paper, we revisit the estimation of ERGMs for small networks and propose using exhaustive enumeration when possible. We developed an R package that implements the estimation of pooled ERGMs for small networks using Maximum Likelihood Estimation (MLE), called “ergmito”. Based on the results of an extensive simulation study to assess the properties of the MLE estimator, we conclude that there are several benefits of direct MLE estimation compared to approximate methods and that this creates opportunities for valuable methodological innovations that can be applied to modeling social networks with ERGMs.

Statistical inference for the lifetime performance index of products with Pareto distribution on basis of general progressive type II censored sample

In this paper, we study the statistical inference for the lifetime performance index of the Pareto distribution based on the general progressive type II censored data. Through data transformation, we derive the maximum likelihood estimator (MLE) of lifetime performance index and confidence interval. Further, the Bayesian estimator and the associated credible interval based on informative and non informative prior functions are also considered under the squared error loss function. Based on the non Bayesian and Bayesian estimators of lifetime performance indicators, the hypothesis testing process is constructed to evaluate the life performance of products. After the Monte Carlo simulation, we find that the Bayesian estimator is far better than MLE, and the Bayesian estimator based on informative prior has the best performance. The estimated mean squared errors for both MLE and Bayesian estimator are small, indicating that our considered method is effective to assess the lifetime performance of the products. Finally, a numerical example is analyzed for illustrative purposes.

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.

A new Liu-type estimator for the Inverse Gaussian Regression Model

ABSTRACTThe Inverse Gaussian Regression Model (IGRM) is used when the response variable is positively skewed and follows the inverse Gaussian distribution. In this article, we propose a Liu-type estimator to combat multicollinearity in the IGRM. The variance of the Maximum Likelihood Estimator (MLE) is overstated due to the presence of severe multicollinearity. Moreover, some estimation methods are suggested to estimate the optimal value of the shrinkage parameter. The performance of the proposed estimator is compared with the MLE and some other existing estimators in the sense of mean squared error through Monte Carlo simulation and different real-life applications. Under certain conditions, it is concluded that the proposed estimator is superior to the MLE, ridge, and Liu estimator.

PCN28 METHOD TO ESTIMATE IMPACT OF PROGNOSTIC FACTORS ON SURVIVAL OUTCOMES USING MAXIMUM LIKELIHOOD ESTIMATION

Intermediate treatment response can result in differences in oncology survival outcomes. Quantifying the impact of this response can be performed with regression models when individual patient data (IPD) of survival times and response status for each patient are known. However, when meta-analyzing data from published sources, only the proportion of responders aggregated over the population is often known. Maximum likelihood estimation (MLE) is a procedure to estimate the hazard ratio (HR) of response based on the distribution of the observed survival data. This work demonstrates the application of MLE to a mixture of data types typically available for meta-analysis: IPD with known response status, IPD with aggregate response (such as reconstructed IPD), and median survival with aggregate response. 10 hypothetical trial populations were randomly generated using the Weibull distribution with shape (0.5), scale (0.2), and HR of response (0.8). MLE estimates of these parameters were generated from the hypothetical data using multiple combinations of survival data availability: 10 IPD with known response, 5 IPD with known response + 5 IPD with aggregate response, and 1 IPD with known response + 5 IPD with aggregate response + 4 median survival with aggregate response. Parameter estimates for each were compared for accuracy. The MLE methodology was also applied to a real-world collection of chronic lymphocytic leukemia (CLL) data. MLE of studies including IPD response data were similar (shape=0.40, scale=0.19; HR=0.89) to the actual values. The scenario with 5 studies including IPD with aggregate response also resulted in similar estimates. The final scenario including 4 studies with median survival data resulted in estimations with largest standard errors and HR=0.80. MLE-estimated HR of treatment response from published CLL data was 0.27. MLE methodology can meta-analyze multiple types of published data to estimate the overall effect of treatment response (or other covariates) on survival outcomes.

Rasch measurement model: a review of Bayesian estimation for estimating the person and item parameters

This paper focuses on the methods used for estimating the parameters in Rasch Measurement Model (RMM). These include the MLE and Bayesian Estimation (BE) techniques. The accuracy and precision of the parameter estimates based on these two MLE and BE were discussed and compared. A questionnaire is a well-known measurement instrument used by most of the researchers. It is a powerful tool for collecting data in survey research. It should be noted that the quality of a measurement instrument used plays a key role in ensuring the quality of data obtained in the survey. Therefore, it has become essential for the researchers to carefully design their questionnaire so that the quality of the data obtained can be preserved. Then, it is also vital for the researchers to assess the quality of the data obtained before it can be successfully used for further analysis. Review of the literature shows that RMM is a psychometric approach widely used as an assessment tool of many measurement instruments developed in various fields of study. At present, the Maximum Likelihood Estimation (MLE) techniques were used to estimate the parameters in the RMM. In order to obtain more precise and accurate parameter estimates, a certain number of sample size and normality assumption are usually required. However, in a small sample, MLE could produce bias, imprecise and less accurate estimates with bigger standard error. A proper selection of the parameter estimation techniques to deal with small sample and non-normality of data is required to obtain more precise and accurate parameter estimates. From the review, it reveals that BE has successfully dealt with the issues of small sample and non-normality of the data. It produced a more accurate parameter estimate with smallest Mean Squared Error (MSE), particularly in a small sample compared to MLE.

Performance of ridge estimator in inverse Gaussian regression model

The presence of multicollinearity among the explanatory variables has undesirable effects on the maximum likelihood estimator (MLE). Ridge estimator (RE) is a widely used estimator in overcoming this issue. The RE enjoys the advantage that its mean squared error (MSE) is less than that of MLE. The inverse Gaussian regression (IGR) model is a well-known model in the application when the response variable positively skewed. The purpose of this paper is to derive the RE of the IGR under multicollinearity problem. In addition, the performance of this estimator is investigated under numerous methods for estimating the ridge parameter. Monte Carlo simulation results indicate that the suggested estimator performs better than the MLE estimator in terms of MSE. Furthermore, a real chemometrics dataset application is utilized and the results demonstrate the excellent performance of the suggested estimator when the multicollinearity is present in IGR model.

Efficient Estimator of Parameters of a Multivariate Geometric Distribution

The maximum likelihood estimator (MLE) and uniformly minimum variance unbiased estimator (UMVUE) for the parameters of a multivariate geometric distribution (MGD) have been derived. A modification of the MLE estimator (modified MLE) has been derived in which case the bias is reduced. The mean square error (MSE) of the modified MLE is less than the MSE of the MLE. Variances of the parameters and the corresponding generalized variance (GV) has been obtained. It has been shown that the MLE and modified MLE are consistent estimators. A comparison of the GVs of modified MLE and UMVUE has shown that the modified MLE is more efficient than the UMVUE. In the final section its application has been discussed with an example of actual data.