Operational Variants of the Minimum Mean Squared Error Estimator in Linear Regression Models with Non-Spherical Disturbances

Background: Typically, modeling of health-related quality of life data is often troublesome since its distribution is positively or negatively skewed, spikes at zero or one, bounded and heteroscedasticity. Objectives: In the present paper, we aim to investigate whether Bayesian beta regression is appropriate for analyzing the SF-6D health state utility scores and respondent characteristics. Methods: A sample of 126 Lebanese members from the American University of Beirut valued 49 health states defined by the SF-6D using the standard gamble technique. Three different models were fitted for SF-6D via Bayesian Markov chain Monte Carlo (MCMC) simulation methods. These comprised a beta regression, random effects and random effects with covariates. Results from applying the three Bayesian beta regression models were reported and compared based on their predictive ability to previously used linear regression models, using mean prediction error (MPE), root mean squared error (RMSE) and deviance information criterion (DIC). Results: For the three different approaches, the beta regression model was found to perform better than the normal regression model under all criteria used. The beta regression with random effects model performs best, with MPE (0.084), RMSE (0.058) and DIC (−1621). Compared to the traditionally linear regression model, the beta regression provided better predictions of observed values in the entire learning sample and in an out-of-sample validation. Conclusions: Beta regression provides a flexible approach to modeling health state values. It also accounted for the boundedness and heteroscedasticity of the SF-6D index scores. Further research is encouraged.

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Risk Comparison of Improved Estimators in a Linear Regression Model with MultivariatetErrors under Balanced Loss Function

Journal of Applied Mathematics ◽

10.1155/2014/129205 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7

Author(s):

Guikai Hu ◽

Qingguo Li ◽

Shenghua Yu

Keyword(s):

Linear Regression ◽

Regression Model ◽

Loss Function ◽

Linear Regression Model ◽

Mean Squared Error ◽

Ordinary Least Squares ◽

Balanced Loss Function ◽

Minimum Mean Squared Error ◽

Squared Error ◽

Improved Estimators

Under a balanced loss function, we derive the explicit formulae of the risk of the Stein-rule (SR) estimator, the positive-part Stein-rule (PSR) estimator, the feasible minimum mean squared error (FMMSE) estimator, and the adjusted feasible minimum mean squared error (AFMMSE) estimator in a linear regression model with multivariateterrors. The results show that the PSR estimator dominates the SR estimator under the balanced loss and multivariateterrors. Also, our numerical results show that these estimators dominate the ordinary least squares (OLS) estimator when the weight of precision of estimation is larger than about half, and vice versa. Furthermore, the AFMMSE estimator dominates the PSR estimator in certain occasions.

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A Comparative Analysis on Some Estimators of Parameters of Linear Regression Models in Presence of Multicollinearity

Asian Journal of Probability and Statistics ◽

10.9734/ajpas/2018/v2i228773 ◽

2018 ◽

pp. 1-8

Author(s):

Warha, Abdulhamid Audu ◽

Yusuf Abbakar Muhammad ◽

Akeyede, Imam

Keyword(s):

Linear Regression ◽

Least Squares ◽

Regression Models ◽

Mean Squared Error ◽

Least Squares Method ◽

Ordinary Least Squares ◽

Least Square ◽

Linear Regression Models ◽

Independent Variables ◽

Different Levels

Linear regression is the measure of relationship between two or more variables known as dependent and independent variables. Classical least squares method for estimating regression models consist of minimising the sum of the squared residuals. Among the assumptions of Ordinary least squares method (OLS) is that there is no correlations (multicollinearity) between the independent variables. Violation of this assumptions arises most often in regression analysis and can lead to inefficiency of the least square method. This study, therefore, determined the efficient estimator between Least Absolute Deviation (LAD) and Weighted Least Square (WLS) in multiple linear regression models at different levels of multicollinearity in the explanatory variables. Simulation techniques were conducted using R Statistical software, to investigate the performance of the two estimators under violation of assumptions of lack of multicollinearity. Their performances were compared at different sample sizes. Finite properties of estimators’ criteria namely, mean absolute error, absolute bias and mean squared error were used for comparing the methods. The best estimator was selected based on minimum value of these criteria at a specified level of multicollinearity and sample size. The results showed that, LAD was the best at different levels of multicollinearity and was recommended as alternative to OLS under this condition. The performances of the two estimators decreased when the levels of multicollinearity was increased.

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Some Estimators of the Dispersion Parameter of a Chi-distributed Radial Error with Applications to Target Analysis

Austrian Journal of Statistics ◽

10.17713/ajs.v34i1.398 ◽

2016 ◽

Vol 34 (1) ◽

Cited By ~ 1

Author(s):

Sharad Saxena ◽

Housila P. Singh

Keyword(s):

Mean Squared Error ◽

Practical Importance ◽

Dispersion Parameter ◽

Point Estimate ◽

Error Estimator ◽

Shrinkage Estimators ◽

Target Analysis ◽

Radial Error ◽

Minimum Mean Squared Error ◽

Squared Error

The dispersion parameter of a chi-distributed radial error is of interest in numerous target analysis problems as a measure of weapon-system accuracy, and it is often of practical importance to estimate it. This paper presents a few classical estimators including the maximum likelihood estimator, an unbiased estimator and a minimum mean squared error estimator of this dispersion for both when the origin or “center of impact” is knownor can be assumed as known and when it is unknown. Some families of shrinkage estimators have also been suggested when a prior point estimate of the dispersion parameter is available in addition to sample information. The estimators of circular error probable and spherical error probable have been obtained as well. A simulation study has been carried out to demonstrate the performance of the proposed estimators.

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Modified Liu estimators in the linear regression model: An application to Tobacco data

PLoS ONE ◽

10.1371/journal.pone.0259991 ◽

2021 ◽

Vol 16 (11) ◽

pp. e0259991

Author(s):

Iqra Babar ◽

Hamdi Ayed ◽

Sohail Chand ◽

Muhammad Suhail ◽

Yousaf Ali Khan ◽

...

Keyword(s):

Linear Regression ◽

Mean Squared Error ◽

Prediction Interval ◽

Evaluation Criteria ◽

Absolute Error ◽

Vital Role ◽

Predictor Variables ◽

Linear Regression Models ◽

Squared Error ◽

Optimal Value

Background The problem of multicollinearity in multiple linear regression models arises when the predictor variables are correlated among each other. The variance of the ordinary least squared estimator become unstable in such situation. In order to mitigate the problem of multicollinearity, Liu regression is widely used as a biased method of estimation with shrinkage parameter ‘d’. The optimal value of shrinkage parameter plays a vital role in bias-variance trade-off. Limitation Several estimators are available in literature for the estimation of shrinkage parameter. But the existing estimators do not perform well in terms of smaller mean squared error when the problem of multicollinearity is high or severe. Methodology In this paper, some new estimators for the shrinkage parameter are proposed. The proposed estimators are the class of estimators that are based on quantile of the regression coefficients. The performance of the new estimators is compared with the existing estimators through Monte Carlo simulation. Mean squared error and mean absolute error is considered as evaluation criteria of the estimators. Tobacco dataset is used as an application to illustrate the benefits of the new estimators and support the simulation results. Findings The new estimators outperform the existing estimators in most of the considered scenarios including high and severe cases of multicollinearity. 95% mean prediction interval of all the estimators is also computed for the Tobacco data. The new estimators give the best mean prediction interval among all other estimators. The implications of the findings We recommend the use of new estimators to practitioners when the problem of high to severe multicollinearity exists among the predictor variables.

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Preliminary-Test Estimation of the Error Variance in Linear Regression

Econometric Theory ◽

10.1017/s0266466600010355 ◽

1987 ◽

Vol 3 (2) ◽

pp. 299-304 ◽

Cited By ~ 22

Author(s):

Judith A. Clarke ◽

David E. A. Giles ◽

T. Dudley Wallace

Keyword(s):

Linear Regression ◽

Mean Squared Error ◽

Preliminary Test ◽

Error Variance ◽

Error Component ◽

Quadratic Loss ◽

Finite Sample ◽

Minimum Mean Squared Error ◽

Squared Error ◽

Preliminary Test Estimation

We derive exact finite-sample expressions for the biases and risks of several common pretest estimators of the scale parameter in the linear regression model. These estimators are associated with least squares, maximum likelihood and minimum mean squared error component estimators. Of these three criteria, the last is found to be superior (in terms of risk under quadratic loss) when pretesting in typical situations.

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An Improved Estimation of Parameter of Morgenstern-Type Bivariate Exponential Distribution Using Ranked Set Sampling

10.4018/978-1-7998-7556-7.ch001 ◽

2022 ◽

pp. 1-25

Author(s):

Vishal Mehta

Keyword(s):

Exponential Distribution ◽

Mean Squared Error ◽

Ranked Set Sampling ◽

Error Estimator ◽

Bivariate Exponential Distribution ◽

Ranked Set Sample ◽

Minimum Mean Squared Error ◽

Squared Error ◽

Bivariate Exponential ◽

Special Cases

In this chapter, the authors suggest some improved versions of estimators of Morgenstern type bivariate exponential distribution (MTBED) based on the observations made on the units of ranked set sampling (RSS) regarding the study variable Y, which is correlated with the auxiliary variable X, where (X,Y) follows a MTBED. In this chapter, they firstly suggested minimum mean squared error estimator for estimation of 𝜃2 based on censored ranked set sample and their special case; further, they have suggested minimum mean squared error estimator for best linear unbiased estimator of 𝜃2 based on censored ranked set sample and their special cases; they also suggested minimum mean squared error estimator for estimation of 𝜃2 based on unbalanced multistage ranked set sampling and their special cases. Efficiency comparisons are also made in this work.

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