Particle Physics Booklet - Particle Data Group - Lawrence Berkeley ...

33. Statistics 273
The minimum of Equation (33.13) defines the least-squares estimators
�θ for the more general case where the yi are not Gaussian distributed
as long as they are independent. If they are not independent but rather
have a covariance matrix Vij =cov[yi,yj], then the LS estimators are
determined by the minimum of
χ 2 (θ) =(y− F (θ)) T V −1 (y − F (θ)) , (33.14)
where y =(y1,...,yN) is the vector of measurements, F (θ) is the
corresponding vector of predicted values (understood as a column vector
in (33.14)), and the superscript T denotes transposed (i.e., row) vector.
In many practical cases, one further restricts the problem to the
situation where F (xi; θ) is a linear function of the parameters, i.e.,
m�
F (xi; θ) = θjhj(xi) . (33.15)
j=1
Here the hj(x)aremlinearly independent functions, e.g., 1,x,x2 ,...,xm−1 ,
or Legendre polynomials. We require m