Confidence and prediction bands for custom nonlineal for CDF from ProbabilityDistribution

Question

I have a CDF obtained with the code:

dist = ProbabilityDistribution[{"CDF", 
   Exp[-a Exp[-x b] - c Exp[-x d]]}, {x, 0, Infinity}, 
  Assumptions -> {{0 < a < 10}, {0 < b < 1}, {0 < c < 10}, {0 < d < 
      1}}]
res = FindDistributionParameters[data, 
  dist, {{a, 3.8}, {b, 0.006}, {c, 0.08}, {d, 0.0002}}]

I can plot the confidence and prediction intervals following this demo for nonlinear pairs of data adjustment: http://demonstrations.wolfram.com/MeanAndSinglePredictionBandsForANonlinearModel/

But for the custom CDF obtained with ProbabilityDistritution, I tryed this:

Plot[{Func[x], bands90[x]}, {x, 0, 3000}, Filling -> {2 -> {1}}]

Without results.

Someone knows how to get the confidence and prediction bands in the case of CDF obtained with that function, as well as the ANOVA or the table that one can shows with ParamterTable (when fitting with NonlinearModelFit)?

Thanks you very much.

Answer 1

Somebody could verify my code regarding the theory?

To start getting the variance-covariance matrix I followed the code found here: Standard errors for maximum likelihood estimates in FindDistributionParameters And I elimitated the last part (Sqrt[Diagonal[cov]/len]]) as I also want to abtain the covariance matrix, so my code is:

covariance[data_, dist_, paramlist_, mleRule_] := 
 Block[{len, infmat, cov}, len = Length[data];
  (*compute negative of expected Fisher information*)
  infmat = -D[LogLikelihood[dist, data], {paramlist, 2}]/len /. 
    mleRule;
  (*invert to get asymptotic covariance*)
  cov = Inverse[infmat]]

In this way I got the central part of the Hat matrix, (X^T X)^-1.