Bayesian Linear Restricted Least Squares Estimators

Abstract

Bayesian Statistics had been outshined classical Statistics over the years due to its probabilistic approach to modelling and data analytics. In recent times, much of model complexity and data ambiguity emerged of which the existing methods have their drawbacks. Model complexity and data uncertainty most times exhibit multicollinearity which render the inferences of least squares invalid. This then call for improvement in the existing methods so as to capture the trend in model complexity and data uncertainty. The Monte Carlo simulations was adopted to compare the restricted least squares family of estimators (RLSFE) with the presence of multicollinearity and two sets of linear restrictions were imposed on the model. The risk values and relative efficiency of the RLSFE under quadratic loss function were estimated to compare the estimators. Bayesian restricted least squares estimator was derived, the posterior mean and Bayes estimate were deprioritised with prior mean. The study found out that posterior mean and Bayes estimates outperformed classical restricted least squares estimators due to their lower risk values. The relative efficiency pointed to the fact that posterior mean and Bayes estimates are most preferred and efficient. The study further observed monotonic increase in the risk values as the increases which proved the negative impact of multicollinearity of the inferences of regression parameters.