Linear Approximate Bayesian Estimators

Perhaps one of the most important distributions is the uniform distribution for con-tinuous random variables and it has been applied in many fields including natural and social sciences.Therefore,it is still of great significance to study the estimation of parameters for the uniform distribution.we first illustrate some properties of the uni-form distribution and its sufficient and complete statistic.Then,we employ the linear Bayesian procedure to construct linear approximate B ayesian estimator for the param-eter.We also investigate its superiorities over some classical estimators under the mean squared error criterion.Numerical simulations are carried out to make comparisons be-tween the linear approximate B ayesian estimator and the Bayesian estimator,the latter is obtained via numerical integration.Also,we introduce the Lindley approximation as a comparison.Linear model can be used to describe the phenomena in the fields of biology,engi-neering,economics and management,etc.,among which parameter estimator problem is always one of the important research directions for domestic and foreign scholars.The paper employs a linear Bayesian procedure to simultaneously estimate regression parameters and variance parameter in a normal linear model with equality constraints.The superiorities of the proposed linear approximate Bayesian estimator over some classical estimators are established in terms of mean squared error matrix criterion.Furthermore,Monte Carlo simulations and a numerical example show that the linear approximate Bayesian estimator has a better estimation effect to the true value than those of the usual Bayesian estimator and the Lindley approximation.Finally,we extend the research on normal linear model with constraints to the constrained linear model with uniform distributed errors.Similarly,we obtain the ex-pression of the linear approximate Bayesian estimator 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 linear B ayesian method are illustrated from both theoretical and simulation perspectives as well.