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--- standard_error_of_regression_coefficients [2026/06/05 11:59] – [Linear algebra] hkimscil
+++ standard_error_of_regression_coefficients [2026/06/05 23:04] (current) – hkimscil
@@ Line 1: / Line 1: @@
-====== Standard Error of Regression Coefficient ======
+====== Standard Error of Regression Coefficients ======
 see https://academicweb.nd.edu/~rwilliam/stats1/x91.pdf
 $ \hat{Y} = a + b X $ 에서 b 에 대한 standard error 값을 말한다. 이 standard error는 샘플에서 구한 b값으로 모집단의 (population의) b값을 추정하는 데 쓰인다.
@@ Line 35: / Line 35: @@
 & = & (y - X b)'(y - X b) \\
 & = & y'y - b' X' y - y' X b + 'b X' X b \\
-& & b' X y = y' X b \\
+& & \because \;\; b' X y = y' X b \\
 & = & \underbrace{y'y}_{(1)} - \underbrace{2 b' X' y}_{(2)} +  \underbrace{b' X' X b}_{(3)} \\
 \\
-\dfrac {\partial {S(b)} } {\partial{b} } & = & {(1)} - {(2)} + {(3)} \\
+\dfrac {\partial {S(b)} } {\partial{b} } & = & \dfrac { \partial{ ( (1) - (2) + (3)) }} {\partial {b}} \\
+\\
+\\
 \dfrac {\partial {(y'y)} } {\partial{b} }  & = & 0 \\
-\dfrac {\partial {(2 b' X' y)} } {\partial{b} }  & = & 2 b X' y \\
+\dfrac {\partial {(2 b' X' y)} } {\partial{b} }  & = & 2 X' y \\
 \dfrac {\partial {(b' X' X b)} } {\partial{b} }  & = & (X'X) b + (X' X)' b\\
-& = & b (X'X)(X'X)' \\
+& = & b ((X'X)+(X'X)') \;\;\;  \because {X'X = (X'X)'} \\
-& = & 2 b (X'X) \because {X'X = (X'X)'} \\
+& = & 2 (X'X) b \\
 \\
 \dfrac {\partial {S(b)} } {\partial{b} } & = & 0 - 2 X' y + 2 X' X b  \\
-& & \text{in order for the above minimal value} \\
+& & \text{in order for the above to have minimal value} \\
   & = &  -2 X' y + 2 X' X b\\
 X' X b  & = &  X' y  \\