Fundamentals of Probability and Statistics for Engineers

Linear Models and Linear Regression 343then E is a zero-mean random vector with covariance matrix L ˆ 2 I, , I beingthe n n identity matrix.The mean and variance of estimator ^Q are now easily determined. In view ofEquations (11.17) and (11.19), we haveEf^Qg ˆ…C T C† 1 C T EfYgˆ…C T C† 1 C T ‰Cq ‡ EfEgŠˆ…C T C† 1 …C T C†q ˆ q:…11:20†Hence, estimators ^A and ^B for and , respectively, are unbiased.The covariance matrix associated with ^Q is given by, as seen from Equation(11.17),covf^Qg ˆEf…^Q q†…^Q q† T gˆ…C T C† 1 C T covfYgC…C T C† 1 :But covfYgˆ2 I; we thus havecovf^Qg ˆ 2 …C T C† 1 C T C…C T C† 1 ˆ 2 …C T C† 1 :…11:21†The diagonal elements of the matrix in Equation (11.21) give the variances of^A and ^B . In terms of the elements of C, we can write" #" # 1varf ^Ag ˆ…x i x† 2 ; …11:22†varf ^Bg ˆ 2 2 Xnx 2 i n Xniˆ1 iˆ1" # 1 Xn…x i x† 2 : …11:23†iˆ1It isseen that thesevariancesdecreaseassamplesizenincreases, accordingto 1/n.Thus, it followsfrom our discussion in Chapter 9that theseestimatorsareconsistent –a desirable property. We further note that, for a fixed n, the variance of ^B can bereduced byselectingthex i in such a waythat thedenominator ofEquation (11.23)ismaximized; this can be accomplished by spreading the x i as far apart as possible. InExample 11.1, for example, assuming that we are free to choose the values of x i ,thequality of ^ is improved if one-half of the x readings are taken at one extreme of thetemperature range and the other half at the other extreme. However, the samplingstrategy for minimizing var( ^A ) for a fixed n is to make x as close to zero as possible.Are the variances given by Equations (11.22) and (11.23) minimum variancesassociated with any unbiased estimators for and ? An answer to this importantquestion can be found by comparing the results given by Equations (11.22)TLFeBOOK