Statistical Sciences 2141A/B Lecture Notes - Lecture 30: Squared Deviations From The Mean, Likelihood Function, Independent And Identically Distributed Random Variables
23 Nov 2018
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The regression line is fitted to the data points (x1, y1), (x2, y2), , (xn, yn) by finding the line that is closest to the data points. This is done by looking at the vertical distances (deviations) between the line and the data points yi ( 0 + 1 xi) and minimizing the sum of the squares of the vertical deviations. This is referred to as the least squares fit. Since the values y1, y2, , yn are observations of iid rv"s where minimizing q is equivalent to maximizing the likelihood function of the yi s. Therefore, the point estimates of 0 and 1 are maximum likelihood estimates. The parameter estimates are found by taking partial derivatives of q with respect to 0 and 1 and setting the equations equal to 0. Therefore, the parameter estimates are the solutions to the normal equations. The normal equations can be solved to give.
