Observational Evidence Favors a Static Universe - Journal of ...

where σi is the uncertainty in yi, εi is the uncertainty in xi and η is described
below. It is apparent that this equation has the basic symmetry in that if η is
zero then interchanging x and y will give the reciprocal of the slope b. If all the
ε are zero it has an analytic solution otherwise the minimization can be done
numerically.
A problem arises when the χ 2 value is significantly different from the number
of degrees of freedom. The case where the value of χ 2 is much less than the
number of degrees of freedom is handled by multiplying the uncertainties by
the square root of the ratio of χ 2 to the degrees of freedom. If the χ 2 is larger
than the number of degrees of freedom the approach used here is to assume
that there is an additional, unknown contribution which is uncorrelated with
the existing uncertainties. This is a common occurrence when the uncertainties
are measurement errors and may be precise but some other factor makes a
significant contribution the scatter of values. the solution is to add a quadratic
term, η 2 , to the denominator and estimate its value by making the χ 2 equal to
the degrees of freedom. The prime advantage of this technique is that it provides
a smooth transition from the situation where the given uncertainties dominate
to the alternative where they are negligible. The main effect is a change in the
relative weights.
References
Andreon, A., Lobo, C. & Iovino, A., (2004). Extending the Butcher-Oemler
effect up to z 0.7. MNRAS, 349, 889.
Astier, A., Guy, J., Regnault, N., Pain, R., Aubourg, E., Balam, D., Basa, S.,
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