کارایی براوردگر پیشآزمون انقباضی پارامتر عرض از مبدأ در مدل رگرسیونی خطی ساده.

Efficiency of shrinkage pretest estimator for the intercept parameter in simple linear regression model.

Introduction Traditionally the classical estimators of unknown parameters of the linear regression model are based exclusively on the sample information, like the maximum likelihood method. Sometimes, in practice the researcher has some prior information about the unknown intercept parameter as a guess that is called non-sample prior information. In this case, the shrinkage and shrinkage preliminary test estimators are introduced by a linear combination of sample and non-sample prior information. Now, if the non-sample prior information is available according to the experimenter's experiences or the results of previous experiments, it can be expressed in the form of a preliminary hypothesis test and the desired parameter can be estimated using the shrinkage pretest estimator. The results show that the closer the guessed value to the real parameter, the better shrinkage pretest estimator performs compared to the maximum likelihood estimator. In the past years, in the regression estimation problem, the behavior of the shrinkage and shrinkage pretest estimators under the squared error loss (SEL) and LINEX loss functions was investigated. The squared error loss (SEL) function is popularly used for estimating the unknown parameter in decision theory because of its mathematical and interpretational convenience. Due to the symmetric nature, the use of SEL function may not be appropriate, when positive and negative errors have different consequences. The SEL and LINEX loss functions are symmetric and asymmetric loss functions, respectively, but both are unbounded. Also, The SEL and LINEX loss functions have an infinite maximum value which isn’t always appropriate. Sometimes in practice it is necessary to use bounded loss functions to estimate parameters. So we used the reflected normal loss function to estimate parameters, which is the bounded loss function. Material and methods In this article, the shrinkage and shrinkage pretest estimators are introduced for the intercept parameter of the simple linear regression according to the non-sample prior information and their risk functions and relative efficiencies are derived under the reflected normal loss function. The behavior of shrinkage pretest estimator is compared with respect to the shrinkage and maximum likelihood estimator using graphical method. The intervals where the shrinkage pretest estimator has less risk compared to the maximum likelihood estimator are presented. The optimum value of the significant level of test is determined using the max-min method. Besides, several methods of finding shrinkage coefficient of the shrinkage pretest estimators are proposed. Then, the application of the proposed estimators is shown using a real data set. Results and discussion In comparison between the desired estimators, our findings show that the shrinkage and shrinkage pretest estimator are better than the maximum likelihood estimator when non-sample prior information is close to the real value. Also, the shrinkage pretest estimator with smaller level of significance has higher efficiency. An important issue for the shrinkage pretest estimator is the proper selection of the shrinkage coefficient. So, we applied several methods for finding the shrinkage coefficient to obtain shrinkage pretest estimator. Conclusion According to the reported results, the shrinkage pretest estimator has smaller risk than the maximum likelihood estimator in neighborhood of null hypothesis. Therefore, in practice, if the researcher can have prior information of the unknown intercept parameter from previous knowledge or experience that is not far from the real value, the shrinkage pretest estimator would be the best choice as intercept estimator. The effective intervals where the shrinkage pretest estimator has less risk compared to the maximum likelihood estimator was presented. By increasing the significant level of the test and the sample size, the length of the effective interval decreases. The optimal value of the significant level of the test was obtained using the max-min method. Then, the application of the proposed estimators is shown using a real data set. [ABSTRACT FROM AUTHOR]

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