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For an exponential QQ plot, we fix the theoretical distribution to have $λ=1$. In order to compare the plotted points with the line $y=x$, what kind of rescaling should be applied to the data and why?

“Hint“ given with this problem: If $X$ follows an exponential distribution with parameter $λ$, then $λX$ follows an exponential distribution with parameter 1.

I don’t know how to use this hint in answering this question. This is a homework problem. So far, I have considered the graph of $\ln(x)$ but I am not sure what to do with that and the hint.

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    $\begingroup$ Is this homework? Please add the self-study tag and read our policy on that meta.stackexchange.com/questions/10811/… $\endgroup$ – Momo Jul 17 '14 at 23:32
  • $\begingroup$ Maybe it's just me, but I feel like there is some information missing. Is the exponential qq plot a qq plot with an exponential distribution with $\lambda=1$ as the distribution from which the theoretical quantiles are derived? And you have some data $x$ that are supposedly from this distribution and you want to check that? Why would you need to rescale $x$ anyway, what is the overall goal? $\endgroup$ – Momo Jul 17 '14 at 23:54
  • $\begingroup$ I'm still confused. What are the plotted points, $x$ and $y$? $\endgroup$ – Momo Jul 17 '14 at 23:57
  • $\begingroup$ Yes, exactly as you say. The theoretical distribution is exponential with lamda 1. The rescaling idea is similar to the idea of rescaling data in order to "normalize" it. For example, if a normal qq plot shows positively skewed data you would rescale the data by taking the log or square root. I hope this makes it more clear. I am a bit confused as well, as I am just learning this stuff, but I would appreciate the input. $\endgroup$ – user162381 Jul 18 '14 at 0:24
  • $\begingroup$ here, you would not want to "normalize" the data, but instead to make it fit to the exponential distribution, so that the qq plot would lie on the y=x line. $\endgroup$ – user162381 Jul 18 '14 at 0:25
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Time to put this one to bed, I think

For an exponential qq plot, we fix the theoretical distribution to have λ=1.

Sure, because in a QQ plot we care about how far it is from exponential, not what the parameter value is. But in any case, you can see approximately what it is from the slope of the line the points should lie along in the QQ plot.

In order to compare the plotted points with the line y=x, what kind of rescaling should be applied to the data and why?

While you could rescale the data by dividing by their mean (yielding data with a mean of 1), I think that you should not do so.

If you do this with exponential data, the result will be data that no longer has an exponential distribution. In small samples the effect may be strong. Further, if the data aren't exponential, that adjustment may be badly impacted by large outliers.

It may do okay with very large samples, but there's really no need.

“Hint“ given with this problem: If X follows an exponential distribution with parameter λ, then λX follows an exponential distribution with parameter 1.

Yeah, as I thought -- my first guess is they probably want you to estimate $\lambda$ by $\hat{\lambda}=1/\bar{x}$, and then scale to unit mean -- but as I already explained, I think that's actually not a great idea, because it alters the very distribution you're checking for.

Better I think to plot the the data, and if you want a line to compare to, put some moderately robust estimate of the trend as a line on the graph ... without putting too much stock in the position of the line (it's more a guide for the eye, to help you spot where the thing curves).

So far, I have considered the graph of ln (x) but not sure what to do with that and the hint.

Okay, that's another way to go. If you do it that way, consider $Z\sim \text{Exp}(1)$, and $X=Z/\lambda$. Then $\ln(X)=\ln(Z)-\ln\lambda$, implying that a plot of quantiles of $\ln(X)$ should be parallel to expected quantiles of $\ln(Z)$, and the distance to the $y=x$ line tells you about $\lambda$. But to be honest I don't think this is the intent of the question.

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