Fundamentals of Probability and Statistics for Engineers

Linear Models and Linear Regression 359Let us note in this example that, since x 2 ˆ x 2 1 , matrix C is constrained in thatits elements in the third column are the squared values of their correspondingelements in the second column. It needs to be cautioned that, for high-orderpolynomial regression models, constraints of this type may render matrix C T Cill-conditioned and lead to matrix-inversion difficulties.REFERENCERao, C.R., 1965, Linear Statistical Inference and Its Applications, John Wiley & SonsInc., New York.FURTHER READINGSome additional useful references on regression analysis are given below.Anderson, R.L., and Bancroft, T.A., 1952, Statistical Theory in Research, McGraw-Hill,New York.Bendat, J.S., and Piersol, A.G., 1966, Measurement and Analysis of Random Data, JohnWiley & Sons Inc., New York.Draper, N., and Smith, H., 1966, Applied Regression Analysis, John Wiley & Sons Inc.,New York.Graybill, F.A., 1961, An Introduction to Linear Statistical Models, Volume 1 . McGraw-Hill, New York.PROBLEMS11.1 A special case of simple linear regression is given byY ˆ x ‡ E:Determine:(a) The least-square estimator ^B for ;(b) The mean and variance of ^B ;(c) An unbiased estimator for 2 , the variance of Y .11.2 In simple linear regression, show that the maximum likelihood estimators for and are identical to their least-square estimators when Y is normally distributed.11.3 Determine the maximum likelihood estimator for variance 2 of Y in simple linearregression assuming that Y is normally distributed. Is it a biased estimator?11.4 Since data quality is generally not uniform among data points, it is sometimesdesirable to estimate the regression coefficients by minimizing the sum of weightedsquared residuals; that is, ^ and ^ in simple linear regression are found by minimizingX niˆ1w i e 2 i ;TLFeBOOK