Transcription of Chapter 7 Least Squares Estimation
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Chapter 7 Least Squares Estimation
7-1 Least Squares EstimationVersion 7 Least Squares IntroductionLeast Squares is a time-honored Estimation procedure, that was developed independently by Gauss(1795), Legendre (1805) and Adrain (1808) and published in the first decade of the nineteenthcentury. It is perhaps the most widely used technique in geophysical data analysis. Unlikemaximum likelihood, which can be applied to any problem for which we know the general formof the joint pdf, in Least Squares the parameters to be estimated must arise in expressions for themeans of the observations. When the parameters appear linearly in these expressions then theleast Squares Estimation problem can be solved in closed form, and it is relatively straightforwardto derive the statistical properties for the resulting parameter very simple example which we will treat in some detail in order to illustrate the more generalproblem is that of fitting a straight line to a collection of pairs of observations (xi, yi) wherei= 1,2.
7-2 Least Squares Estimation Version 1.3 Solving for the βˆ i yields the least squares parameter estimates: βˆ 0 = P x2 i P y i− P x P x y n P x2 i − ( P x i)2 βˆ 1 = n P x iy − x y n P x 2 i − ( P x i) (5) where the P ’s are implicitly taken to be from i = 1 to n in each case.
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