# DistributionFitTest with WeightedData?

My problem is related to statistical testing with weighted data. I have a data sample and I assume that a part of the data follows a specific distribution (Pareto). So I fit my data to a distribution by using EstimatedDistribution and in turn I obtain a p-value related to this fit according to different statistical tests (Cramer-Von Mises, Kolmogorov-Smirnov and possibly others).

Everything goes well as long as I do not include weights (I really know it works, since I have replicated my results in another software package...).

When including weights by using the function WeightedData I run into problems. EstimatedDistribution still works well on WeightedData (again results are replicated in another package), but the DistributionFitTest does not work any longer.

When trying to apply DistributionFitTest on WeightedData I get an error message of the following form:

"rctnln: The argument [my weighted data object] at position 1 should be a rectangular array of real numbers with length greater than the dimension of the array."

I got the intuition behind these error message and intensely searched for an appropriate work-around or alternative Function, but could not find any suitable tools.

Could anybody help me out on this?

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A little bit of code to reproduce this: data = {8, 3, 5, 4, 9, 0, 4, 2, 2, 3}; w = {0.15, 0.09, 0.12, 0.10, 0.16, 0., 0.11, 0.08, 0.08, 0.09}; wd = WeightedData[data, w]; DistributionFitTest[wd]. Seems like DistributionFitTest wasn't updated when WeightedData was introduced. However, DistributionFitTest[EmpiricalDistribution[w -> data]] doesn't work either and EmpiricalDistribution was introduced a version ago. –  Sjoerd C. de Vries Jul 11 '13 at 20:51
I also tried the variant you suggest (i.e. using DistributionFitTest[EmpiricalDistribution[w -> data]]) when I was still using version 8. Basically it was the same issue as far as I remember. Another variant I tried was to integrate a RandomVariate to draw form the empirical distribution and test this artificial data against the distribution. This worked out well, but I was unsure whether this really represents what I am trying to do... –  Joko Jul 11 '13 at 22:40
You may consider reporting this at support@wolfram.com. –  Sjoerd C. de Vries Jul 11 '13 at 23:03
I got the following comment from Wolfram: "Unfortunately, DistributionFitTest does not currently support WeightedData objects. Our developers are aware of user interest in this feature, and are working to include it in a future version of Mathematica. On your behalf, I've filed your comments with our development team for their review as they move forward. We are always interested in improving Mathematica, and I want to thankyou ...." So there will be no solution to my problem on this front - has anyone a suggestion for an appropriate work-around? –  Joko Jul 20 '13 at 10:50

The problem is that EmpiricalDistribution[] does not generate a PDF function, only a CDF. A work around could be to create an un-weighted data set out of your data. It's sub-optimal to say the least, and you may run out of memory on some data sets before getting your answer, but it works.
Flatten[ConstantArray @@ # & /@ Transpose[{data, Ceiling[100*w]}]]