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Math Help - Linear Regression: parameter estimation of alpha and beta?

  1. #1
    LHS
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    Exclamation Linear Regression: parameter estimation of alpha and beta?

    Could anyone help me with the part of this question highlighted in the link? I can derive the first part, but i'm having problems canceling alpha, any help would be greatly appreciated!

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  2. #2
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    Quote Originally Posted by LHS View Post
    Could anyone help me with the part of this question highlighted in the link? I can derive the first part, but i'm having problems canceling alpha, any help would be greatly appreciated!

    Let Q be the function you want to minimize.

    \displaystyle<br />
\frac {dQ}{d\alpha} = 0 \iff \underbrace{\sum \frac {y_i}{\rho_i^2}}_{A_1} - \alpha \underbrace{\sum \frac 1 {\rho_i^2}}_{B_1} - \beta \underbrace{\sum \frac{x_i}{\rho_i^2}}_{C_1} = 0<br />
    \displaystyle<br />
\frac {dQ}{d\beta} = 0 \iff \underbrace{\sum \frac {y_i x_i}{\rho_i^2}}_{A_2} - \alpha \underbrace{\sum \frac {x_i} {\rho_i^2}}_{B_2} - \beta \underbrace{\sum \frac{x_i^2}{\rho_i^2}}_{C_2} = 0<br />

    Just solve the simultaneous equation A_1 - \alpha B_1 - \beta C_1 = 0 and A_2 - \alpha B_2 - \beta C_2 = 0 in the usual way.

    Alternatively, we might notice that in the usual matrix notation
    <br />
Q(\vec \beta) = (Y - \vec \beta X)^T W (Y - \vec \beta X)<br />
    where <br />
W = \mbox{diag}(\rho_i^{-2}, i = 1, 2, ..., n)<br />
and solve with matrix calculus.

    An obvious application is where we think that the variance grows proportionally with x_i, so we could take \rho_i = \sqrt {x_i}
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  3. #3
    LHS
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    Thank you very much! that is all sorted now!
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