Weighted Least Squares¶

Suppose we know that the variance of \(Y\) is dependent on \(Y\) itself in the form \(Var(Y_{i}) \propto \sigma^{2}/w_{i}\), i.e., the weights are known only upto a constant. In this case, we minimize the weighted least squares to obtain the coefficients

\[ \begin{aligned} \minimize_{\theta_{0}, \theta_{1}} \sum_{i=1}^{n} w_{i}(Y_{i} - \theta_{0} - \theta_{1}x_{i})^{2} \end{aligned} \]