Linear Bayesian Estimation Under Constraint Conditions

The linear model can be used to describe the phenomena of biology,medicine,economy,etc.,which is one of the most widely used models in modern statistics.The parameter estimation method of the linear model under the unconstrained condition is very mature,but in many cases,the parameters under the constraint conditions need to be estimated.At present,there are least square estimator,biased estimator and numer-ical solutions for the parameter estimation with imposed constraints.The least square estimator satisfies the unbiasedness,but it does not perform well on the multicollinearity problem.However,the biased estimator sacrifice unbiasedness to reduce the variance of the estimator.This paper proposes a new unbiased estimation-constrained linear Bayesian esti-mation to deal with the above situation.First,the expression of the constrained least square estimator is given.Then the constrained linear Bayes estimator is derived for the regression parameters in a linear model with equality constraints,which is based on the constrained least square estimator.Finally,the superiority of the constrained linear Bayes estimator over the constrained least square estimator is studied under the mean square error matrix criterion.Monte Carlo simulation is used to obtain the numerical solution of Bayes estima-tor,because it is difficult to acquire the integral and obtain its display expression.Given the same distribution and the same sample size,we can find that the distance between the constrained linear Bayes estimation and the Bayesian estimation is not greater than the constrained least square estimator.As the sample size increases,the distance gets smaller between the constrained linear Bayesian estimation and the Bayesian estima-tion.The degree of dispersion under different prior distributions has different effects on the degree of convergence of the constrained linear Bayes estimation.Finally,we em-ploy Portland cement data to further illustrate the superiority of the constrained linear Bayesian estimation over the constrained least square estimation.