Linear Regression Analysis: Theory and Computing by Xin Yan

By Xin Yan

This e-book is kind of concise, yet there are numerous blunders or oversight. for example, phrases brought in proofs weren't given or mentioned past, making it tricky to stick with the evidence. one other instance is simply sheer sloppiness, akin to on p.45, the place the final line of the 1st paragraph is obviously flawed. at the related web page, one other hassle-free mistakes seems to be within the definition of orthonormal foundation (orthognality is missing). Readers could have little self assurance within the proofs or formulation.

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In particular, in a multiple linear regression model with p parameters the denominator should be n − p in order to construct an unbiased estimator of the error variance σ 2 . Detailed discussion can be found in later chapters. The unbiasness of estimator s2 for the simple linear regression can be shown in the following derivations. yi − yˆi = yi − b0 − b1 xi = yi − (¯ y − b1 x ¯) − b1 xi = (yi − y¯) − b1 (xi − x ¯) It follows that n n (yi − yˆi ) = n (yi − y¯) − b1 i=1 i=1 (xi − x ¯) = 0. i=1 Note that (yi − yˆi )xi = [(yi − y¯) − b1 (xi − x ¯)]xi , hence we have n n (yi − yˆi )xi = i=1 n = [(yi − y¯) − b1 (xi − x ¯)]xi i=1 [(yi − y¯) − b1 (xi − x ¯)](xi − x ¯) i=1 n = n (xi − x ¯)2 (yi − y¯)(xi − x ¯) − b1 i=1 i=1 Sxy Sxx = 0 Sxx To show that s2 is an unbiased estimate of the error variance, first we note that = n(Sxy − b1 Sxx ) = n Sxy − (yi − yˆi )2 = [(yi − y¯) − b1 (xi − x ¯)]2 , therefore, n n (yi − yˆi )2 = i=1 n [(yi − y¯) − b1 (xi − x ¯)]2 i=1 n 2 = (yi − y¯) − 2b1 i=1 n n (xi − x ¯)(yi − y¯i ) + i=1 (yi − y¯)2 − 2nb1 Sxy + nb21 Sxx = i=1 n (yi − y¯)2 − 2n = i=1 n (yi − y¯)2 − n = i=1 2 Sxy Sxy Sxy + n 2 Sxx Sxx Sxx 2 Sxy Sxx b21 (xi − x ¯)2 i=1 April 29, 2009 11:50 World Scientific Book - 9in x 6in Regression˙master Simple Linear Regression 17 Since (yi − y¯)2 = [β1 (xi − x ¯) + (εi − ε¯)]2 and (yi − y¯)2 = β12 (xi − x ¯)2 + (εi − ε¯)2 + 2β1 (xi − x ¯)(εi − ε¯), therefore, E(yi − y¯)2 = β12 (xi − x ¯)2 + E(εi − ε¯)2 = β12 (xi − x ¯ )2 + n−1 2 σ , n and n n E(yi − y¯)2 = nβ12 Sxx + i=1 i=1 n−1 2 σ = nβ12 Sxx + (n − 1)σ 2 .

Myers, Classical and Modern Regression Analysis With Applications, P76. April 29, 2009 11:50 World Scientific Book - 9in x 6in This page intentionally left blank Regression˙master April 29, 2009 11:50 World Scientific Book - 9in x 6in Regression˙master Chapter 3 Multiple Linear Regression The general purpose of multiple linear regression is to seek for the linear relationship between a dependent variable and several independent variables. Multiple regression allows researchers to examine the effect of more than one independent variables on response at the same time.

A confidence interval of interest can be used to evaluate the accuracy of a single future value of y at a chosen value of regressor x. Confidence interval estimator for a future value of y provides confidence interval for an estimated value y at x with a desirable confidence level 1 − α. It is of interest to compare the above two different kinds of confidence interval. The second kind has larger confidence interval which reflects the less accuracy resulting from the estimation of a single future value of y rather than the mean value computed for the first kind confidence interval.

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