Weighted Mixed Regression Estimation Under Biased Stochastic Restrictions

The use of prior information in linear regression analysis is well known to provide more efficient estimators of regression coefficients. Such prior information can be available in different forms from various sources like as past experience of the experimenter, similar kind of experiments conducted in the past, etc. The available prior information sometimes can be expressed in the form of exact, stochastic or inequality restrictions. The methods of restricted regression estimation, mixed estimation (Theil and Goldberger (1961)) and minimax estimation are preferred when prior information is available in the form of exact, stochastic and inequality restrictions, respectively. More details about these estimation procedures can be found in Rao, Toutenburg, Shalabh and Heumann (2008). When the prior information is available in the form of stochastic restrictions, then in many applications a systematic bias is also present. Such systematic bias can arise from different sources and due to various reasons like personal judgements of the persons involved in the experiment, in testing of general linear hypothesis in linear models when null hypothesis is rejected, in imputation of missing values through regression approach etc. Teräsvirta (1980) and Hill and Ziemer (1983) have given some interesting examples for this type of information. How to incorporate such systematic bias in the estimation procedure is an issue which is addressed in this article. The method of weighted mixed regression estimation is utilized for the purpose. How to choose the weights in this estimation procedure so as to have gain in efficiency under the criterion of mean dispersion error matrix is also addressed. The plan of the paper is as follows. The model description and the estimation of parameters are discussed in Section 2. The properties of the estimators are derived and analyzed in Section 3. Some conclusions are placed in Section 4.