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I have two examples of a Cullen and Frey graph (obtained using the fitdistrplus package in R). The first is from this question, and the second is from some data I have. I don't understand what the use of the bootstrapped values are. In the first graph, do they simply confirm that the empirical bootstrap sample have observed kurtosis and skew that match that of a beta distribution, thus giving support to the hypothesis that the original data is beta distributed? What about in my data where the bootstrapped values are all over the graph. What am I meant to conclude?

enter image description here

enter image description here

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  • $\begingroup$ Can you graph either qqnorm or ecdf? Can you list numeric summary statistics for your data? What was the text output of the descdist function? $\endgroup$ Commented May 22, 2017 at 23:52
  • $\begingroup$ Why would doing these things help me understand what the bootstrapped values do? $\endgroup$
    – Alex
    Commented May 23, 2017 at 1:37
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    $\begingroup$ They are alternative pictures of the data that can help show the distribution. Think of each tool a a foggy window, but with multiple windows you can get a better picture. Ecdf, qqnorm, this, and others are tools, hammers, but not every problem is a nail. They will show not only the tails, but the center. They can show modality. They are also frameworks for determining the nature of the distribution. $\endgroup$ Commented May 23, 2017 at 12:31
  • $\begingroup$ I see. But using your analogy, I am really asking about how to use a hammer. I ended up choosing a distribution for my data but I am still curious to know how I am meant to interpret the bootstrapped kurtosis and skew values. $\endgroup$
    – Alex
    Commented May 23, 2017 at 23:22
  • $\begingroup$ I am still figuring out this graph, for my own uses too. My best guess is to look at the blue point, my distro, and what it is near. If it is on one of the lines, such as gamma or lognormal, then it can be gamma, lognormal, or weibull. I think (my guess) that the bootstrap gives you a heuristic sense of where the point could be, and helps pick nearer vs. farther points. That is why I asked my related question about how to plot a 2d nonparametric density on the bootstrap values. $\endgroup$ Commented May 24, 2017 at 0:37

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What is the point of the bootstrapped skewness and kurtosis in the plot? Well, they represent some reference distribution to judge the plot. Referring to your first plot, which shows both skewness and kurtosis way too large for the named distributions shown with only one point in the plot.

Look at the dotted line representing the lognormal family. The bootstrapped cloud lies clearly above this curve, so the skewness/kurtosis relationship for the lognormal cannot represent your data.

Then look at the dashed line representing the gamma family. The bootstrapped point cloud at least intersects with this curve, so at least the gamma family could be contemplated.

Then finally look at the blue-grey area representing the (four-parameter) beta family. Most of the bootstrapped points lie within that area, so the (4-parameter) beta family could also be contemplated.

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