Biased Estimation Of Parameters And Its Relative Efficiency In Linear Model With Aggregated Data

For the linear model,when the collected data is a linear combination of observation vectors,it is called the linear model of aggregated data.This paper focuses on the parameter estimation and relative efficiency of linear models and stochastic constrained linear models in the case of aggregated data.Firstly,based on the generalized aggregated two-parameter(GATP)estimation,an improved generalized aggregated two-parameter(IGATP)estimation and a generalized aggregated compression two-parameter(GACTP)estimation are proposed.Under the mean square error matrix criterion,the superiority of GACTP estimation is mainly studied.The superiority of GACTP estimation is further verified by Monte Carlo simulation method.The two relative efficiency of GACTP estimation and Peter-Karsten(PK)estimation and LS estimation are defined respectively,and the upper and lower bounds are obtained.Secondly,combined with aggregated(AM)estimation and aggregated Liu(AL)estimation,aggregated mixed Liu(AML)estimation is defined.The selection method of Liu parameters in AML estimation is given.Under the mean square error matrix criterion,the necessary and sufficient conditions for AML estimation are better than AL estimation,PK estimation and AM estimation.The Monte Carlo simulation method is used to further verify the excellent properties of AML estimation.The relative efficiency of AML estimation and PK estimation is defined,and the upper and lower bounds are obtained.Thirdly,considering the imbalance between prior information and sample information.Based on AM estimation,an aggregate weighted mixture(AWM)estimation is proposed.Under the mean square error matrix criterion,the necessary and sufficient conditions for AWM estimation are better than AM estimation and PK estimation.The Monte Carlo simulation method is used to further verify the superiority of AWM estimation.The relative efficiency of AWM estimation and PK estimation is defined,and the upper and lower bounds are obtained.Finally,the general form of PK estimation,AML estimation,AM estimation,AWM estimation,and AL estimation—aggregate weighted mixed Liu(AWML)estimation is proposed.The selection method of Liu parameters in AWML estimation is given.Under the mean square error matrix criterion,the necessary and sufficient conditions for AWML estimation to be better than other estimates are obtained.The superiority of new estimation isfurther verified by Monte Carlo simulation method.Two relative effects of AWML estimation and PK estimation are defined,and the upper and lower bounds are obtained.