# Confidence and prediction bands for custom nonlineal for CDF from ProbabilityDistribution

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.

• Related link and link. – user9660 Jul 25 '14 at 3:08
• I found this post that I'll use to try starting with the variance-covariance matrix: mathematica.stackexchange.com/questions/6498/…. Then I want apply this to find the pointwise intervals. – JosGranada Jul 26 '14 at 11:08
• I don't think you're distribution is properly specified. Your CDF is not zero at x = 0. As a consequence, the PDF that Mma computes does not integrate to 1. – mef Feb 22 '15 at 12:48

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.

• To cross-check your theory and or results the guys over at Cross Validated might be happy to anwser your questions. – user9660 Jul 26 '14 at 15:54