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Consider the following code and output:

  par(mfrow=c(3,2))
  # generate random data from weibull distribution
  x = rweibull(20, 8, 2)
  # Quantile-Quantile Plot for different distributions
  qqPlot(x, "log-normal")
  qqPlot(x, "normal")
  qqPlot(x, "exponential", DB = TRUE)
  qqPlot(x, "cauchy")
  qqPlot(x, "weibull")
  qqPlot(x, "logistic")

enter image description here

It seems that that Q-Q plot for log-normal is almost the same as the Q-Q plot for weibull. How can we distinguish them? Also if the points are within the region defined by the two outer black lines, does that indicate that they follow the specified distribution?

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  • $\begingroup$ I believe you are using the car package, aren't you? If so, you should include the statement library(car) in your code to make it easier for people to follow. In general, you may also want to set the seed (eg, set.seed(1)) to make the example reproducible, so that anyone can get exactly the same data points you have gotten, although it's probably not as important here. $\endgroup$ – gung Mar 11 '13 at 15:41
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    $\begingroup$ This will not run on my computer as written. For example, qqPlot from the car package wants norm for normal and lnorm for log-normal. What am I missing? $\endgroup$ – Tom Mar 11 '13 at 17:45
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    $\begingroup$ @Tom, I was mistaken about the package. Evidently, it's the qualityTools package. Moreover, the example seems to be taken from here. $\endgroup$ – gung Mar 11 '13 at 19:10
  • $\begingroup$ An interesting alternative is the Cullen and Frey graph, see stats.stackexchange.com/questions/243973/… for an example $\endgroup$ – kjetil b halvorsen Nov 3 '17 at 11:23
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There are a couple of things to be said here:

  1. the shape of the CDF for the log-normal is similar enough to the shape of the CDF of the Weibull to make them harder to distinguish than the level of similarity between the Weibull and the others.
  2. the outer black lines form a confidence band. The use of the confidence band in inference is the same as any other standard form of Frequentist statistical inference. That is, when values fall within the band, we cannot reject the null hypothesis that the posited distribution is the correct one. This is not the same as saying that we know the posited distribution is the correct one. (Note that this is a great example of what I discussed in another answer here of a situation where the Fisherian perspective on hypothesis testing would be preferable to the Neyman-Pearson.)
  3. you need more data; your $N$ is only 20 here.
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  • $\begingroup$ Is there ways for examining distributions for small sample sizes? $\endgroup$ – proton Mar 11 '13 at 15:40
  • $\begingroup$ in fact it seems that the points lie in the confidence bands for all distributions. So we can't distinguish the distributions? $\endgroup$ – proton Mar 11 '13 at 15:43
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    $\begingroup$ There are tests for the goodness of fit of a dataset to a theoretical distribution, but I tend to think that they are inferior to qq-plots. Basically, you are not going to be able to distinguish between those distributions with $n=20$. If you think of this in terms of statistical power, your ability to reject each of the false nulls here is $\approx 5\%$. It may help you to read the answer I linked in point #2. $\endgroup$ – gung Mar 11 '13 at 15:48
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    $\begingroup$ +1 on the small sample size. Using 300 samples would help to distinguish things a lot. Proton: No, you can't really distinguish distributions with a small sample. How could you? It's like trying to identify a face with 20 pixels. $\endgroup$ – Wayne Mar 12 '13 at 0:16
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It seems that that Q-Q plot for log-normal is almost the same as the Q-Q plot for weibull.

Yes.

How can we distinguish them?

At that sample size, you likely can't.

Also if the points are within the region defined by the two outer black lines, does that indicate that they follow the specified distribution?

No. It only indicates that you can't tell the distribution of the data as being different from that distribution. It's lack of evidence of a difference, not evidence of a lack of difference.

You can be almost certain that the data are from a distribution that is not any of the ones you have considered (why would it be exactly from any of those?).

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  • $\begingroup$ Like the phrasing: "It's lack of evidence of a difference, not evidence of a lack of difference." $\endgroup$ – jlandercy Nov 3 '17 at 9:10

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