Research On Biased Estimation Method Considering Constraints

The estimation of unknown parameters in linear models is always a hot issue in the field of linear statistics and an important tool for data measurement in industrial research.Least square estimation is one of the most widely used unbiased estimation methods,which has the characteristic of minimum mean square error under the mean square error criterion.However,when the observation matrix has complex collinearity,the mean square error of least square estimation will increase greatly,and linear biased estimation is the most direct method to solve this problem.In biased estimation method,a smaller mean square error is obtained by introducing partial parameters and balancing the variance with the estimated deviation.With the complexity of parameter estimation in industrial research,the research of biased estimation considering constraint conditions is gradually deepened.In this paper,the method of biased estimation and the effect of data processing in industrial research with constraints are studied.Firstly,we study that the minimum values of mean square errors of typically biased estimators are equal.Under the mean square error criterion,the function extreme value method is used to study the existing biased estimate of the minimum problem,proved that although each estimation methods have different forms,but in their own slant range of parameter selection,there is always an optimal parameter value,makes the biased estimate of the value of the minimum mean square error,and the minimum of different estimation method is the same.Finally,the conclusion is verified by using typical case data and simulation data generated by Monte Carlo method.Secondly,a new biased estimation method,modified generalized ridge estimation,is proposed.The biased property of this estimation method is analyzed,its mean square error value is expressed,and the consistency of the minimum mean square error value of the biased estimator is verified.Under the mean square error criterion,the value range of partial parameter is calculated when its estimation effect is better than least square estimation and generalized ridge estimation.The conclusion is also verified by using typical case data and simulation data generated by Monte Carlo method.Finally,generalized ridge type estimators with interval constraints are studied.The properties of generalized ridge estimation with interval constraint are discussed,and it is analyzed that this estimation method is better than constrained least square estimation in data processing in industrial research under mean square error matrix criterion.And the value range of partial parameters is calculated when this estimation method is better than constrained least square estimation.The conclusion is verified by using the typical example data and simulation data generated by Monte Carlo method.