Linear Regression Analysis: Theory and Computing by Xin Yan

By Xin Yan

This booklet is sort of concise, yet there are various mistakes or oversight. for example, phrases brought in proofs weren't given or mentioned prior, making it tricky to keep on with the facts. one other instance is simply sheer sloppiness, similar to on p.45, the place the final line of the 1st paragraph is obviously fallacious. at the similar web page, one other user-friendly errors seems within the definition of orthonormal foundation (orthognality is missing). Readers may have little self assurance within the proofs or formulation.

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Nqq. /noline mse caxis=red ctext=blue cframe=ligr; run; *Compute confidence band of regression mean; plot density*stiffness/conf caxis=red ctext=blue cframe=ligr legend=legend1; run; *Compute confidence band of regression prediction; plot density*stiffness/pred caxis=red ctext=blue cframe=ligr legend=legend1; run; ods graphics off; ods rtf close; quit; The regression scatterplot, residual plot, 95% confidence bands for regression mean and prediction are presented in Fig. 1. 11:50 World Scientific Book - 9in x 6in Regression˙master Simple Linear Regression 33 (b) Residual Plot of Stiffness Data 0 Residual −6 5 −4 10 −2 15 density 20 2 25 4 30 (a) density = a + b(stiffness) 2e+04 4e+04 6e+04 8e+04 1e+05 10 15 20 25 (c) 95% Regression Band (d) 95% Prediction Band 30 25 20 15 5 5 10 15 Density 20 25 30 Fitted Value 30 stiffness 10 Density April 29, 2009 2e+04 4e+04 6e+04 Stiffness 8e+04 1e+05 2e+04 4e+04 6e+04 8e+04 1e+05 Stiffness Fig.

Likewise, when we test H0 : β1 = β10 we only test H0 regardless of the values of β0 . If we would like to test whether or not a regression line falls into certain region we need to test the multiple hypothesis: H0 : β0 = β00 , β1 = β10 simultaneously. This falls into the scope of multiple inference. For the multiple inference on β0 and β1 we notice that b0 − β0 , b1 − β1 n n i=1 xi n i=1 xi n 2 i=1 xi b0 − β0 b1 − β1 ∼ 2s2 F2,n−2 . Thus, the (1 − α)100% confidence region of the β0 and β1 is given by b0 − β0 , b1 − β1 n n i=1 xi n i=1 xi n 2 i=1 xi b0 − β0 b1 − β1 ≤ 2s2 Fα,2,n−2 , where Fα,2,n−2 is the upper tail of the αth percentage point of the Fdistribution.

5. Var(b0 ) = 1 x ¯2 + σ2 . n nSxx Proof. Var(b0 ) = Var(¯ y − b1 x ¯) = Var(¯ y ) + (¯ x)2 Var(b1 ) 2 σ σ2 = +x ¯2 n nSxx 1 x ¯2 = + σ2 n nSxx The properties 1 − 5, especially the variances of b0 and b1 , are important when we would like to draw statistical inference on the intercept and slope of the simple linear regression. Since the variances of least squares estimators b0 and b1 involve the variance of the error term in the simple regression model. This error variance is unknown to us. Therefore, we need to estimate it.

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