Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
9.6. A Regression Problem 545 As a final note, in Model 9.6.1 we have centered thex’s; i.e., subtractedxfrom xi. In practice, us ...
546 Inferences About Normal Linear Models ci; -0.0054678 -0.0032842 So with approximate confidence 95%, we estimate the drop in ...
9.6. A Regression Problem 547 Model (9.6.1) can be expressed equivalently as Y = α 1 +βxc+e =[1xc] ( α β ) +e = Xβ+e, (9.6.13) w ...
548 Inferences About Normal Linear Models EXERCISES 9.6.1.Obtain the least squares estimates for the modelyi=α∗+βxi+eiby min- im ...
9.6. A Regression Problem 549 9.6.6.Consider Model (9.6.1). Letη 0 =E(Y|x=x 0 −x). The least squares estimator ofη 0 is ˆη 0 =ˆα ...
550 Inferences About Normal Linear Models 9.6.11.Let the independent random variablesY 1 ,...,Ynhave the joint pdf L(α, β, σ^2 ) ...
9.7. A Test of Independence 551 (c)As depicted in Figure 9.6.3, show that the angle between the vectorsˆθandˆe is a right angle. ...
552 Inferences About Normal Linear Models bivariate normal distribution; that is, the joint pdf of these 2nrandom variables is g ...
9.7. A Test of Independence 553 ThusT 2 ,givenX 1 =x 1 ,...,Xn=xn, has a conditionalt-distribution withn− 2 degrees of freedom. ...
554 Inferences About Normal Linear Models withρ=0sothat^12 log[(1 +ρ)/(1−ρ)] = 0. However, usingW, we can also test a hypothesis ...
9.8. The Distributions of Certain Quadratic Forms 555 (a)Using the baseball data, determine the estimate and the confidence inte ...
556 Inferences About Normal Linear Models quadratic forminX. Due to the symmetry ofA, there are several ways we can writeQ: Q=X′ ...
9.8. The Distributions of Certain Quadratic Forms 557 proof thatS^2 is an unbiased estimate ofσ^2. Note that the mean of the ran ...
558 Inferences About Normal Linear Models BecauseW 1 ,...,Wrare independentN(0,1) random variables,W 12 ,...,Wr^2 are independen ...
9.8. The Distributions of Certain Quadratic Forms 559 variables. We would like to treat this problem generally, but limitations ...
560 Inferences About Normal Linear Models Example 9.8.3.Based on this last theorem, we can obtain quickly the distri- bution of ...
9.8. The Distributions of Certain Quadratic Forms 561 (a)Use Theorem 9.8.1 to find theE(Q). (b)Justify the assertion thatQisχ^2 ...
562 Inferences About Normal Linear Models 9.9 The Independence of Certain Quadratic Forms We have previously investigated the in ...
9.9. The Independence of Certain Quadratic Forms 563 Finally, using (9.9.1) and (9.9.3), we have the identity AB={Γ′ 11 Λ 11 }Γ ...
564 Inferences About Normal Linear Models for (t 1 ,t 2 )inanopenneighborhoodof(0,0). The coefficient of (− 2 t 1 )ron the right ...
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