A New Two-Parameter Estimator for Beta Regression Model: Method, Simulation, and Application

The Beta Regression Model (BRM) is commonly used while analyzing data where the dependent variable is restricted to the interval $[0,1]$ for example proportion or probability. The Maximum Likelihood Estimator (MLE) is used to estimate the regression coefficients of BRMs. But in the presence of multicollinearity, MLE is very sensitive to high correlation among the explanatory variables. For this reason, we introduce a new biased estimator called the Beta Liu-Type Estimator (BLTE) to overcome the multicollinearity problem in the case that dependent variable follows a Beta distribution. The proposed estimator is a general estimator which includes other biased estimators, such as the Ridge Estimator, Liu Estimator, and the estimators with two biasing parameters as special cases in BRM. The performance of the proposed new estimator is compared to the MLE and other biased estimators in terms of the Estimated Mean Squared Error (EMSE) criterion by conducting a simulation study. Finally, a numerical example is given to show the benefit of the proposed estimator over existing estimators.

Mixed Poisson regression models (MPRMs) are widely used for analyzing overdispersed count data. However, the presence of multicollinearity among explanatory variables poses challenges when estimating regression coefficients using the maximum likelihood estimator (MLE), leading to inflated variances. The Poisson Modification of the Quasi-Lindley regression model (PMQLRM), a recently introduced alternative within MPRMs, faces similar issues. To address this, we propose a Liu-type estimator for the PMQLRM as an effective remedy for multicollinearity. Several existing methods are utilized to estimate the Liu-type parameter, and the theoretical superiority conditions of the proposed estimator over the MLE, ridge regression estimator, and Liu estimator are established using the scalar mean squared error (MSE) criterion. A Monte Carlo simulation study is conducted to compare the performance of different estimators based on the MSE. Additionally, a real-world dataset is analyzed to demonstrate the practical advantages of the proposed method. The findings indicate that the Poisson-modification of the Quasi-Lindley Liu-type estimator outperforms the MLE and other biased estimators when multicollinearity is present, offering a more stable and reliable alternative for parameter estimation in mixed Poisson regression models.

The beta regression model is commonly used when analyzing data that come in the form of rates or percentages. However, a problem that may encounter when analyzing these kinds of data that has not been investigated for this model is the multicollinearity problem. It is well known that the maximum likelihood (ML) method is very sensitive to high inter-correlation among the explanatory variables. Therefore, this paper proposes some ridge estimators for the beta regression model to remedy the problem of instability of the traditional ML method and increase the efficiency of estimation. The performance of ridge estimators is compared to the ML estimator through the mean squared error (MSE) and the mean absolute error (MAE) criteria by conducting a Monte-Carlo simulation study and through an empirical application. According to the simulation and application results, the proposed estimators outperform the ML estimator in terms of MSE and MAE.

Beta regression model is used for modeling proportions measured on a continuous scale; its parameters are estimated with the maximum likelihood method. Classical regression models, such as linear regression model and nonlinear regression models like logistic regression are not suitable for such situations. As in linear regression model, the independent variables are assumed to be uncorrelated if this assumption is not met, then the multicollinearity appears. Multicollinearity problem means that there is a near dependency between the independent variables. Biased estimators are commonly used for correcting the multicollinearity problem. In this study, we propose a generalized biased estimator for correcting multicollinearity in beta regression that is generalize beta ridge regression estimator (GBRRE). The performance of the proposed generalized biased estimator is evaluated theoretically via the matrix mean squared errors and the scalar mean squared errors; and practically using a Monte Carlo simulation study. The simulation results show that the optimal shrinkage estimator is K1 and the worst one is K2. Also, the proposed generalized estimator is applied to a real data set of pre-university education students in Egypt during the academic year (2018/2019) and we found the application results agree with the simulation results. Finally based on the results of the simulation study and the application the performance of the suggested generalized biased estimator is better than maximum likelihood estimators.

The Maximum Likelihood (ML) and Best Linear Unbiased (BLU) estimators of the location and scale parameters of an extreme value distribution (Lawless [1982]) are compared under conditions of small sample sizes and Type I censorship. The comparisons were made in terms of the mean square error criterion. According to this criterion, the ML estimator of σ in the case of very small sample sizes (n < 10) and heavy censorship (low censoring time) proved to be more efficient than the corresponding BLU estimator. However, the BLU estimator for σ attains parity with the corresponding ML estimator when the censoring time increases even for sample sizes as low as 10. The BLU estimator of σ attains equivalence with the ML estimator when the sample size increases above 10, particularly when the censoring time is also increased. The situation is reversed when it came to estimating the location parameter μ, as the BLU estimator was found to be consistently more efficient than the ML estimator despite the improved performance of the ML estimator when the sample size increases. However, computational ease and convenience favor the ML estimators. Keywords: maximum likelihood estimatorsbest linear unbiased estimatorslife-testingreliability

The beta regression model (BRM) is used when the dependent variable may take continuous values and be bounded in the interval (0, 1), such as rates, proportions, percentages and fractions. Generally, the parameters of the BRM are estimated by the method of maximum likelihood estimation (MLE). However, the MLE does not offer accurate and reliable estimates when the explanatory variables in the BRM are correlated. To solve this problem, the ridge and Liu estimators for the BRM were proposed by different authors. In the current study, the James Stein Estimator (JSE) for the BRM is proposed. The matrix mean squared error (MSE) and the scalar MSE properties are derived and then compared to the available ridge estimator, Liu estimator and MLE. The performance of the proposed estimator is evaluated by conducting a simulation experiment and analyzing two real-life applications. The MSE of the estimators is considered as a performance evaluation criterion. The findings of the simulation experiment and applications indicate the superiority of the suggested estimator over the competitive estimators for estimating the parameters of the BRM.

Logistic regression is a widely used method to model categorical response data, and maximum likelihood (ML) estimation has widespread use in logistic regression. Although ML method is the most used method to estimate the regression coefficients in logistic regression model, multicollinearity seriously affects the ML estimator. To remedy the undesirable effects of multicollinearity, estimators alternative to ML are proposed. Drawing on the similarities between the multiple linear and logistic regressions, ridge, Liu and two parameter estimators are proposed which are based on the ML estimator. On the other hand, first-order approximated ridge estimator is proposed for use in logistic regression. This study will present further solutions to the problem in the form of alternative estimators which reduce the effect of collinearity. Owing to this, first-order approximated Liu, iterative Liu and iterative two parameter estimators are proposed. A simulation study as well as real life application are carried out to ascertain the effect of sample size and degree of multicollinearity, in which the ML based, first-order approximated and iterative biased estimators are compared. Graphical representations are presented which support the effect of the shrinkage parameter on the mean square error and prediction mean square error of the biased estimators.

Decomposition of Kullback–Leibler risk and unbiasedness for parameter-free estimators

The Multiple Comparison Procedures with Modeling Techniques (MCP-Mod) framework has been recently approved by the U.S. Food, Administration, and European Medicines Agency as fit-for-purpose for phase II studies. Nonetheless, this approach relies on the asymptotic properties of Maximum Likelihood (ML) estimators, which might not be reasonable for small sample sizes. In this paper, we derived improved ML estimators and correction for their covariance matrices in the censored Weibull regression model based on the corrective and preventive approaches. We performed two simulation studies to evaluate ML and improved ML estimators with their covariance matrices in (i) a regression framework (ii) the Multiple Comparison Procedures with Modeling Techniques framework. We have shown that improved ML estimators are less biased than ML estimators yielding Wald-type statistics that controls type I error without loss of power in both frameworks. Therefore, we recommend the use of improved ML estimators in the MCP-Mod approach to control type I error at nominal value for sample sizes ranging from 5 to 25 subjects per dose.

For a balanced two‐way mixed model, the maximum likelihood (ML) and restricted ML (REML) estimators of the variance components were obtained and compared under the non‐negativity requirements of the variance components by Lee and Kapadia (1984). In this note, for a mixed (random blocks) incomplete block model, explicit forms for the REML estimators of variance components are obtained. They are always non‐negative and have smaller mean squared error (MSE) than the analysis of variance (AOV) estimators. The asymptotic sampling variances of the maximum likelihood (ML) estimators and the REML estimators are compared and the balanced incomplete block design (BIBD) is considered as a special case. The ML estimators are shown to have smaller asymptotic variances than the REML estimators, but a numerical result in the randomized complete block design (RCBD) demonstrated that the performances of the REML and ML estimators are not much different in the MSE sense.

Over-the-air computation (OAC) is a promising technique to realize fast model aggregation in the uplink of federated edge learning (FEEL). OAC, however, hinges on accurate channel-gain precoding and strict synchronization among edge devices, which are challenging in practice. As such, how to design the maximum likelihood (ML) estimator in the presence of residual channel-gain mismatch and asynchronies is an open problem. To fill this gap, this paper formulates the problem of misaligned OAC for FEEL and puts forth a whitened matched filtering and sampling scheme to obtain oversampled, but independent samples from the misaligned and overlapped signals. Given the whitened samples, a sum-product ML (SP-ML) estimator and an aligned-sample estimator are devised to estimate the arithmetic sum of the transmitted symbols. In particular, the computational complexity of our SP-ML estimator is linear in the packet length, and hence is significantly lower than the conventional ML estimator. Extensive simulations on the test accuracy versus the average received energy per symbol to noise power spectral density ratio (EsN0) yield two main results: 1) In the low EsN0 regime, the aligned-sample estimator can achieve superior test accuracy provided that the phase misalignment is not severe. In contrast, the ML estimator does not work well due to the error propagation and noise enhancement in the estimation process. 2) In the high EsN0 regime, the ML estimator attains the optimal learning performance regardless of the severity of phase misalignment. On the other hand, the aligned-sample estimator suffers from a test-accuracy loss caused by phase misalignment.

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%.

Latent growth models (LGMs) are an application of structural equation modeling and frequently used in developmental and clinical research to analyze change over time in longitudinal outcomes. Maximum likelihood (ML), the most common approach for estimating LGMs, can fail to converge or may produce biased estimates in complex LGMs especially in studies with modest samples. Bayesian estimation is a logical alternative to ML for LGMs, but there is a lack of research providing guidance on when Bayesian estimation may be preferable to ML or vice versa. This study compared the performance of Bayesian versus ML estimators for LGMs by evaluating their accuracy via Monte Carlo (MC) simulations. For the MC study, longitudinal data sets were generated and estimated using LGM via both ML and Bayesian estimation with three different priors, and parameter recovery across the two estimators was evaluated to determine their relative performance. The findings suggest that ML estimation is a reasonable choice for most LGMs, unless it fails to converge, which can occur with limiting data situations (i.e., just a few time points, no covariate or outcome, modest sample sizes). When models do not converge using ML, we recommend Bayesian estimation with one caveat that the influence of the priors on estimation may have to be carefully examined, per recent recommendations on Bayesian modeling for applied researchers.

A restricted maximum likelihood estimator for truncated height samples

A maximum likelihood (ML) estimator is developed for determining time delay between signals received at two spatially separated sensors in the presence of uncorrelated noise. This ML estimator can be realized as a pair of receiver prefilters followed by a cross correlator. The time argument at which the correlator achieves a maximum is the delay estimate. The ML estimator is compared with several other proposed processors of similar form. Under certain conditions the ML estimator is shown to be identical to one proposed by Hannan and Thomson [10] and MacDonald and Schultheiss [21]. Qualitatively, the role of the prefilters is to accentuate the signal passed to the correlator at frequencies for which the signal-to-noise (S/N) ratio is highest and, simultaneously, to suppress the noise power. The same type of prefiltering is provided by the generalized Eckart filter, which maximizes the S/N ratio of the correlator output. For low S/N ratio, the ML estimator is shown to be equivalent to Eckart prefiltering.