LINEAR-MODEL ESTIMATION ERRORS WITH ESTIMATED DESIGN PARAMETERS

The problem of error estimation of parameters b in a linear model, Y = Xb + e, is considered when the elements of the design matrix X are functions of an unknown 'design' parameter vector c. An estimated value c is substituted in X to obtain a derived design matrix X. Even though the usual linear model conditions are not satisfied with X, there are situations in physical applications where the least squares solution to the parameters is used without concern for the magnitude of the resulting error. Such a solution can suffer from serious errors. This paper examines bias and covariance errors of such estimators. Using a first-order Taylor series expansion, we derive approximations to the bias and covariance matrix of the estimated parameters. The bias approximation is a sum of two terms: One is due to the dependence between c and Y; the other is due to the estimation errors of c and is proportional to b, the parameter being estimated. The covariance matrix approximation, on the other hand, is composed of three components: One component is due to the dependence between c and Y; the second is the covariance matrix SIGMA(b) corresponding to the minimum variance unbiased estimator b, as if the design parameters were known without error; and the third is an additional component due to the errors in the design parameters. It is shown that the third error component is directly proportional to bb'. Thus, estimation of large parameters with wrong design matrix X will have larger errors of estimation. The results are illustrated with a simple linear example.