Why does the calculation of AIC, the Akaike Criterion, use (k+1) versus k?

Questions
1

  • The formula for AIC = - 2 max loglikelihood + 2 k, where k is the number of estimated parameters.

  • But Mathematica strangely uses (k+1) rather than just k in computing AIC

  • Example:

      data = {{51,55},{100,68},{63,60},{52,40},{67,45},{42,49},{81,62},{70,56},{108,93},{90,76}};
  lm=LinearModelFit[data,x,x];
  k = 2;
  (* 1 for the slope parameter + 1 for the intercept parameter  *)
  n = Length[data];
  ser=Sqrt[Total[lm["FitResiduals"]^2]/(n-k)];
  (* this is the standard error of the regression *)
  lm["AIC"]
  -2 LogLikelihood[NormalDistribution[0,ser],lm["FitResiduals"]]+ 2 (k+1)
  75.5076
  75.5076

I had to add 1 to k to make the results agree.
So, why does Mathematica use k + 1 rather than just k?

2

This is not an answer to your question (as I don’t know the answer) but as I’m sure you know, it won’t make any difference when comparing AIC values as the increase by 2 just subtracts out when taking the difference in two AIC values.

It’s also not clear that Mathematica uses the standard formula for AIC_c, too.