Can we infer other distributions from the distribution of the 99th percentile? This is a fairly weird question, but I'm analysing a paper that reports its findings in a curious way; for every subject, they passively took measurements every 10 seconds for 24 hours (8640 measurements per person). Then, for unclear reasons, they took the 99th percentile of that data and defined it as a person's index measurement. All they give is their quartile data for this distribution. I believe that from this data, I can construct a CDF (please correct me if I'm wrong!), and from that the underlying PDF. If so, it looks like an exponential distribution, which is what you'd expect for this kind of data:
 
Firstly, is this a valid inference? And if it is, here's a trickier question - can I infer anything about the underlying data distribution? I am dubious why the 99th percentile was considered the metric, and even if I have a CDF/PDF, I have this only for the 99th percentile - but can I then presume that, say, the median / 75th percentile etc are also exponential distributions? Or that each individual's range of measurements also gave rise to an exponential distribution? 
Appreciate this is a weird question, but figured it was the right place to ask! 
 A: Yes, it looks strange---but something can be done. If we are assuming exponential distribution, as you have, with density function $f(x;\theta)=\frac1\theta e^{-x/\theta}\quad (x>0)$, cdf $F(x;\theta)=1-e^{-x/\theta}$ and quantile function $Q(p;\theta)=-\theta\log(1-p)$ where $\theta>0$ is the expectation. So the quantile function is explicit, and for $p=0.99$ we get $q=4.60517\cdot \theta$. So, yes, under this strong assumption the distribution can be recovered. But that is a strong assumption---and if we weaken the assumption to gamma distribution (which generalizes the exponential), this is no longer possible. More generally, observed one quantile as in your case can identify any one-parameter family of distributions (loosely speaking, assuming that quantile in in one-one relation to the parameter.)  
But from only your quantile data, the exponential assumption cannot be tested or criticized. In this answer I did not enter into formal inference, that should be possible, but since you do not have the full sample, you cannot get the full likelihood function, so efficient inference will not be possible. 
