Fixed Design (Chosen )

Sunday, 18 August 2019

8:59 PM

10. Regression Analysis Properties of the Least Squares Estimators We noted that Yil(X/ Xi) JV(/30 + (31>0, a) Then, because Ei(Xi — R) 0, we can write sxy 31 sxx which is a linear combination of the normal random variables Yi, 31 is normally distributed! Its expectation is Ei(Xi - Ei(Xi - + ßIXi) -R) E(31) Sxx + unbiased estimator of [31 sxx Similarly, its variance is Var(fil ) — Hence, the sampling distribution of 31 is sxx —R)2 Term 2019 [31 2 sxx MATH2099/2859 (Statistics) Dr Jia Deng sxx - Lecture g 19/50

 

Now, we can write — 31k, i=l which is again a linear combination of normal r.v.'s (the Yi's and 31) the estimator is also normally distributed! Its expectation is E(30) E i=l — [31 R n i=l unbiased estimator of [30 Similarly, we find Var(30) 02 1 + n sxx Hence, the sampling distribution of o is n sxx

 

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