# Does this look like a Cauchy distribution?

This is my data

there is more actually... but I just want to know if this looks like a Cauchy distribution, such as these:

(source: mathworks.com)

I'm asking about just that bump... here is the whole distribution , it's clearly not Cauchy , here is the entire histogram. .. the tail off to the right is exponential.

UPDATE: here it is with an exponential for comparison

• In general I do not think it is a good idea to "chop off" your (kernel?) density estimate to smaller chunks and try to associate each chunk with particular distribution. To go to the main question: Nope! Not by a long shot. Finally check your original process that generates this data. Such peaks (as the one you are observe roughly at the 1.2-1.3 of your $x$ axis) might resonate from some binning or threshold that occurs in your sample generating process around these values. – usεr11852 Mar 1 '15 at 5:10

No -- except in the most superficial of senses (like, "they're unimodal").

For starters, those curves don't seem to even be density functions:

... since if that increase going toward the left continues beyond the left edge of the plot, it won't even have a finite integral.

(If it doesn't continue increasing as you move left, it implies bimodality, so it would still certainly not be Cauchy.)

Even ignoring that, they're not even close to symmetric. If we flip over and overlay the plots so the peaks nearly coincide (see the pink one), then we can clearly see the asymmetry:

I illustrated my points on the pink one because it's easier to see - some of the others are somewhat obscured by other series overlaid -- but some of the other colours are actually worse.

So no, they don't look like densities to me, and they're not symmetric, let alone Cauchy.

In response to the now-changed question:

A Cauchy isn't "just that bump". A Cauchy is a distribution over $(-\infty,\infty)$. If you're willing to take a truncated distribution and compare it to "just that bump", it arguably looks similar to an infinity of distributions on just a part of the range. I see no particular reason to say it's Cauchy rather than any of a very long list of distributions

How would you choose between the Cauchy and the infinity of other candidates with very similar-looking bumps in the middle?

Here's an example -- two scaled densities that look more or less like your curves near the peak.

Which do you chooose? (Now imagine instead there was another hundred similar looking curves there. Now how do you begin choose?)

You can't identify the mathematical form "by eye". Theory, where possible, is a better way to arrive at a suitable description of a peak.

If you're saying "is it reasonable to model a short region around those peaks with a function of the form $y=\alpha+\frac{\beta}{1+(\frac{x-\theta}{\tau})^2}$?" ... then quite possibly, since models aren't necessarily intended to be exact descriptions (they're models, after all), and near the peak it does look at least something like a constant plus a scaled truncated Cauchy. But any number of other choices may do as well or better.

• good point, thanks Glen_b, my stats books are not handy right now.. I was perusing Wikipedia, hoping there was some sort of page with just a list of all the PDF plots and their characteristic shapes so one could go down the list eyeballing it as it were... the -inf..+inf range is the real deal-breaker, I need to find a similar distribution supported on just the positive half-line. – crow Mar 1 '15 at 22:27
• No website or book can list "all the pdf plots". They're always restricted to choosing some subset, usually some idiosyncratic selection of ones in popular use. For any you could find, there will be many others - even among only the tiny subset that is listed in wikipedia - that are almost indistinguishable from some other symmetric unimodal distribution. I'll add a display to my answer to help illustrate the dilemma. – Glen_b Mar 1 '15 at 22:50
• Oh come on now, wikipedia has a huge selection of data on this topic in a standard format.. it could be done, and yes I know theory is a better way but in that instance I would be either indifferent to the selection of the distribution, or use some other aspect of the model to match it up with some other aspect (e.g. theory). I don't wish to go into the data-generating process at this moment because it's too much of a headache, and off-topic.. but the resonances interpretation of the Cauchy distribution does make sense in the context I am finding it – crow Mar 1 '15 at 23:58
• 1) I think you've missed the point on wikipedia. They only have pages for about 90 continuous distributions -- only 90! ... one of the two I drew above isn't even in the list! You merely have an arbitrary selection from only the tiny subset of distributions in common use. (Most of those 90 aren't symmetric.) 2) The fact that the interpretation of the Cauchy makes some sense in your application is a better basis for a choice than looking through any arbitrary list of named distributions. – Glen_b Mar 2 '15 at 0:28
• Outside the 'makes sense for your application', I don't see a particular a priori reason to think that "distributions listed in wikipedia" and "an ideal model for the way the curves behave near the peak" ought to necessarily contain elements in common. – Glen_b Mar 2 '15 at 0:36

yes, the Cauchy distribution appears in the power-spectral distribution of a self-exciting process which is what I conjectured the data source to come from.. or rather an ergodic mixture of a Cauchy process and a self-exciting process perhaps. See Remark 6 on page 11 of https://arxiv.org/abs/1507.02822 "Remark 6 The power spectral density appearing in Theorem 2 is a shifted scaled Cauchy p.d.f."