Error Matrix Criterion Research Articles

A new general biased estimator in linear measurement error model

Numerous biased estimators are known to circumvent the multicollinearity problem in linear measurement error models. This article proposes a general biased estimator with the ridge regression and the Liu estimators as special cases. The efficiency of the suggested estimator is compared with ridge regression and Liu estimators under the mean squared error matrix criterion. In addition, a Monte Carlo simulation study and a numerical evaluation have been conducted to elucidate the superiority of the new general biased estimator over other estimators.

A New Modified Generalized Two Parameter Estimator for linear regression model

The Ordinary Least Squares estimator estimates the parameter vectors in a linear regression model. However, it gives misleading results when the input variables are highly correlated, emanating the issue of multicollinearity. In light of multicollinearity, we wish to obtain more accurate estimators of the regression coefficients than the least square estimators. The main problem of least square estimation is to tackle multicollinearity so as to get more accurate estimates. In this paper, we introduce a New Modified Generalized Two Parameter Estimator by merging the Generalized Two Parameter Estimator and the Modified Two Parameter Estimator and compare it with other known estimators like Ordinary Least Squares Estimator, Ridge Regression Estimator, Liu estimator, Modified Ridge Estimator, Modified Liu Estimator and Modified Two Parameter Estimator. Mean Squared Error Matrix criterion was used to compare the new estimator over existing estimators. The estimation of the biased parameters is discussed. Necessary and sufficient conditions are derived to compare the proposed estimator with the existing estimators. The excellence of the new estimator over existing estimators is illustrated with the help of real data set and a Monte Carlo simulation study. The results indicate that the newly developed estimator is more efficient as it has lower mean square error.

Bayesian instrumental variable estimation in linear measurement error models

AbstractIn this article, we study the problem of parameter estimation for measurement error models by combining the Bayes method with the instrumental variable approach, deriving the posterior distribution of parameters under different priors with known and unknown variance parameters, respectively, and calculating the Bayes estimator (BE) of the parameters under quadratic loss. However, it is difficult to obtain an explicit expression for BE because of the complex multiple integrals involved. Therefore, we adopt the linear Bayes method, which does not specify the form of the prior and avoids these complicated integral calculations, to obtain an expression for the linear Bayes estimator (LBE) for different priors. We prove that this LBE is superior to the two‐stage least squares estimator under the mean squared error matrix criterion. Numerical simulations show that our LBE is very close to the real parameter whether the variance parameters are known or unknown, and it gradually approaches BE as the sample size increases. Our results indicate that this instrumental variable approach is valid for measurement error models.

A generalized Liu-type estimator for logistic partial linear regression model with multicollinearity

<abstract><p>This paper is concerned with proposing a generalized Liu-type estimator (GLTE) to address the multicollinearity problem of explanatory variable of the linear part in the logistic partially linear regression model. Using the profile likelihood method, we propose the GLTE as a general class of Liu-type estimator, which includes the profile likelihood estimator, the ridge estimator, the Liu estimator and the Liu-type estimator as special cases. The conditional superiority of the proposed GLTE over the other estimators is derived under the asymptotic mean square error matrix (MSEM) criterion. Moreover, the optimal choices of biasing parameters and function of biasing parameter are given. Numerical simulations demonstrate that the proposed GLTE performs better than the existing estimators. An application on a set of real data arising from the study of Indian Liver Patient is shown for illustrating our theoretical results.</p></abstract>

Statistical inference of a stochastically restricted linear mixed model

<abstract><p>This article compares a predictor with the best linear unbiased predictor (BLUP) for a unified form of all unknown parameters under a stochastically restricted linear mixed model (SRLMM) in terms of the mean squared error matrix (MSEM) criterion. The methodology of block matrix inertias and ranks is employed to compare the MSEMs of these predictors. The comparison results are also demonstrated for a linear mixed model with and without an exact restriction, as well as special cases of the unified form of all unknown parameters in the SRLMM.</p></abstract>

Estimation of structural parameters in balanced Bühlmann credibility model with correlation risk

In this paper, the longitudinal data analysis is used to interpret the balanced Bühlmann credibility model with correlation risk, and the homogeneous credibility estimator is derived. We obtain the restricted maximum likelihood estimators (RMLE) for the structural parameters involved in the credibility factor and show that they are unbiased. In addition, the linear Bayes method is employed to estimate the structural parameters, and the proposed linear Bayes estimators (LBE) appear to outperform RMLE in terms of the mean squared error matrix (MSEM) criterion. Simulation studies show that the proposed LBE performs well.

Approximate Bayesian estimator for the parameter vector in linear models with multivariate t distribution errors

This article constructs an approximate Bayes estimator for the parameter vector consisted of regression coefficients and variance parameter in the linear model in which the error terms follow multivariate t distribution. Its superiorities over the classical estimators are strictly proved in terms of the mean squared error matrix (MSEM) criterion. Compared with the Bayes estimator computed via the MCMC method, the proposed Bayes estimator is simple and easy to interpret and compute, which only requires relatively little prior designation. The numerical computations further verify that the approximate Bayes estimator performs well. Also, the proposed procedure can be easily extended to other multivariate distribution cases.

A new stochastic restricted two-parameter estimator in multiplelinear regression model

In this paper, we proposed a biased estimator, a new stochastic restricted two-parameter estimator (NSRTPE), for the multiple linear regression model to tackle the multicollinearity problem when the stochastic restrictions are available. Necessary and sufficient conditions for the superiority of the proposed estimator over the ordinary least square estimator (OLSE), ridge estimator (RE), Liu estimator (LE), almost unbiased Liu estimator (AULE), modified new two-parameter estimator (MNTPE), mixed estimator (ME), stochastic restricted Liu estimator (SRLE) were derived in the mean square error matrix (MSEM) criterion. Finally, we showed the superiority of the estimator proposed using a simulation study and a real-world example in the scalar mean square error (SMSE) criterion.

An almost unbiased Liu-type estimator in the linear regression model

A biased estimator, compared to least squares estimators, is one of the most used statistical procedures to overcome the problem of multicollinearity. Liu-type estimators, which are biased estimators, are preferred in a wide range of fields. In this article, we propose an almost unbiased Liu-type (AUNL) estimator and discuss its performance under the mean square error matrix criterion among existing estimators. The proposed AUNL estimator is a general estimator and is based on the function of a single biasing parameter. It includes an ordinary least squares estimator, an almost unbiased ridge estimator, an almost unbiased Liu estimator, and an almost unbiased two-parameter estimator. Finally, real data examples and a Monte Carlo simulation are provided to illustrate the theoretical results.

Linear Bayes estimator of the extreme value distribution based on type II censored samples

In this paper, a linear Bayes method is employed to simultaneously estimate the location parameter and the scale parameter of the extreme value distribution. Based on type II censored samples, we construct the linear Bayes estimator (LBE) of the parameter vector and establish its superiority over the classical unbiased estimator in terms of the mean square error matrix criterion. The proposed LBE is easy to calculate, and numerical results are presented to verify that the LBE works well. The estimator is further applied to a real data case to demonstrate its feasibility.

Stochastic restricted Liu predictors in linear mixed models

In this article, we propose the stochastic restricted Liu predictors by augmenting the stochastic restrictions to the linear mixed models. The Liu biasing parameter is selected via generalized cross validation (GCV) criterion. Comparisons between the stochastic restricted Liu estimators and several other estimators, namely the BLUE, the mixed and Liu estimators are made through the mean square error matrix criterion. Finally, a numerical example and a simulation study are done to show the performance of the estimators.

On the Restricted Poisson Ridge Regression Estimator

For modeling count data, the Poisson regression model is widely used in which the response variable takes non-negative integer values. However, the presence of strong correlation between the explanatory variables causes the problem of multicollinearity. Due to multicollinearity, the variance of the maximum likelihood estimator (MLE) will be inflated causing the parameters estimation to become unstable. Multicollinearity can be tackled by using biased estimators such as the ridge estimator in order to minimize the estimated variance of the regression coefficients. An alternative approach is to specify exact linear restrictions on the parameters in addition to regression model. In this paper, the restricted Poisson ridge regression estimator (RPRRE) is suggested to handle multicollinearity in Poisson regression model with exact linear restrictions on the parameters. In addition, the conditions of superiority of the suggested estimator in comparison to some existing estimators are discussed based on the mean squared error (MSE) matrix criterion. Moreover, a simulation study and a real data application are provided to illustrate the theoretical results. The results indicate that the suggested estimator, RPRRE, outperforms the other existing estimators in terms of scalar mean squared error (SMSE). Therefore, it is recommended to use the RPRRE for the Poisson regression model when the problem of multicollinearity is present.

Linear Bayesian estimators for linear models with constraints

The paper employs a linear Bayesian procedure to simultaneously estimate regression parameters and variance parameter in a linear model with equality constraints. We obtain the expression of the linear Bayesian estimator (LBE) for the parameter vector, which consists of the regression parameters and the variance parameter, without specifying the specific form of the prior. The superiorities of the proposed LBE over some classical estimators are established in terms of mean squared error matrix criterion. Monte Carlo simulations and a numerical example are presented to compare its performances with those of the usual Bayesian estimator and the Lindley approximation as well.

The generalized new two-type parameter estimator in linear regression model

In this paper, a new two-type parameter estimator is proposed. This estimator is a generalization of the new two parameter (NTP) estimator introduced by Yang and Chang, which includes the ordinary least squares (OLS), the generalized ridge (GR) and the generalized Liu (GL) estimators, as special cases. Here, the performance of this new estimator is, theoretically, investigated over the OLS, the GR, the GL and the NTP estimators in terms of mean squared error matrix criterion. Furthermore, the estimation of the biasing parameters is obtained to minimize the scalar mean squared error. In addition, a complementary algorithm is proposed for the estimator presented by Yang and Chang. As well, a numerical example is given and a simulation study is done.

Improvement of mixed predictors in linear mixed models

In this paper, we introduce stochastic-restricted Liu predictors which will be defined by combining in a special way the two approaches followed in obtaining the mixed predictors and the Liu predictors in the linear mixed models. Superiorities of the linear combination of the new predictor to the Liu and mixed predictors are done in the sense of mean square error matrix criterion. Finally, numerical examples and a simulation study are done to illustrate the findings. In numerical examples, we took some arbitrary observations from the data as the prior information since we did not have historical data or additional information about the data sets. The results show that this case does the new estimator gain efficiency over the constituent estimators and provide accurate estimation and prediction of the data.

The modified Liu-ridge-type estimator: a new class of biased estimators to address multicollinearity

This article proposes another general class of biased estimators which includes some popular estimators as special cases and discusses its properties for multiple linear regression models with the issue of multicollinearity. We proposed the estimator by modifying the Liu estimator (LE) with the use of the existing modified ridge-type estimator and provide its properties. Performance of the proposed estimator is compared with many of the leading estimators, using the mean squared error (MSE) matrix criterion. We conducted an extensive simulation study to illustrate the superiority of the proposed estimator.

The 𝑟-𝑑 class predictions in linear mixed models

Abstract In this paper, we propose the r-d class predictors which are general predictors of the best linear unbiased predictor (BLUP), the principal components regression (PCR) and the Liu predictors in the linear mixed models. Superiorities of the linear combination of the new predictors to each of these predictors are done in the sense of the mean square error matrix criterion. Finally, numerical examples and a simulation study are done to illustrate the findings.

Multisensor temporal approach for transplanted paddy fields mapping using fuzzy-based classifiers

Paddy transplantation rate mapping can provide important information regarding water demand for its transplantation. It is a time-tagged activity that cannot be accurately mapped using a single date image. Further, many times, single sensor data may not have enough temporal resolution required to map it. Furthermore, due to cloud cover, a single optical sensor is not enough to provide required temporal scenes for mapping the transplanted paddy fields. Therefore, to have a finer temporal resolution, a multisensor approach was applied in our research work by integrating an optical sensor data with a synthetic aperture radar data to map transplanted paddy fields at a frequency of 5- to 10-day intervals. In our research, the transplanted paddy fields were mapped as a specific class of interest using suitable fuzzy-based classifiers. The specific fuzzy-based classifiers explored here were modified possibilistic c-mean (MPCM) and noise clustering (NC). For evaluating the results, mean membership difference (MMD) and entropy values were computed within transplanted paddy fields and for noninterest classes. Moreover, an error matrix criterion was also used to validate the results. The computed MMD and entropy values were found to be lower within transplanted paddy fields in comparison to the other classes of noninterest, indicating accurate classification. NC algorithm outperformed the MPCM algorithm while observing MMD and entropy values for assessment. However, using the fuzzy error matrix criterion, the performance of both algorithms was identical with an average overall accuracy and kappa coefficient of 90.88% and 0.82 for MPCM and 90.66% and 0.81 for NC, respectively.

Another proposal about the new two-parameter estimator for linear regression model with correlated regressors

In this article, we present a new general class of biased estimators which includes some popular estimators as special cases and discuss its properties for multiple linear regression models when regressors are correlated. This proposal is based on some modification in the existing new two-parameter estimator. Performance of the proposed estimator is compared with many of the leading estimators, using the mean squared error matrix criterion, mitigating the adverse effects of multicollinearity. An extensive simulation study has been provided with a numerical example to illustrate the superiority of the proposed estimator.

Modifying Two-Parameter Ridge Liu Estimator Based on Ridge Estimation

In this paper, we introduce the new biased estimator to deal with the problem of multicollinearity. This estimator is considered a modification of Two-Parameter Ridge-Liu estimator based on ridge estimation. Furthermore, the superiority of the new estimator than Ridge, Liu and Two-Parameter Ridge-Liu estimator were discussed. We used the mean squared error matrix (MSEM) criterion to verify the superiority of the new estimate. In addition to, we illustrated the performance of the new estimator at several factors through the simulation study.