I have two variables as below: inventory turnover (multiple to 100 already) and inventory day. And I have the kurtosis as below: enter image description here

I do not know how to judge if there is any "fat-tail kurtosis" among these two variables. And what should we do to address it? I could not see any benchmark to compare to. And I am quite surprise that I could not find many papers representing Kurtosis in their summary statistic but standard error, min, max. And what is the standard solution when facing "fat-tail" kurtosis apart from winsorizing or getting the log of this variable?


1 Answer 1


It probably does not have a mean or a variance. Ratio variables often don't. As these are accounting ratios, there are several candidate distributions and their ratios wouldn't have a first moment.

There are a couple of things to remember. If none of the variables in a ratio can be negative, then you likely have a truncated distribution.

In practice, this absolutely excludes least squares related formulas in most cases. The exception is if a dependent variable has a restricted range such as in probit or logit.

While the log transformation would permit least-squares style algorithms, there are two issues that cannot be ignored. The first is the interpretation. You cannot just transform something because it is woefully inconvenient. The second is that the likely log-transformed distributions will lack a covariance matrix. That creates an obvious problem. After all, if $$\beta=\frac{\text{cov}(x,y)}{\text{var}(x)}$$ and $\text{cov}(x,y)$ is an undefined quantity, what does it mean?

There isn't a standard solution because the field likes to pretend things are normally distributed. The general field of statistics has solutions, however.

First, if you are going to use it in a linear or polynomial regression, your best choice is to use Theil's regression.


Theil, H. (1950), "A rank-invariant method of linear and polynomial regression analysis. I, II, III", Nederl. Akad. Wetensch., Proc., 53: 386–392, 521–525, 1397–1412

It is intimately related to bootstrapping and resampling methods.

The next best choice would be quantile regression around the median. See

Koenker, Roger (2005). Quantile Regression. Cambridge: Cambridge University Press.

If you were doing scientific work, then either of those would be the better choice.

If you were doing applied work, neither of those would be a good choice. Instead, you should derive the distribution and solve it with a Bayesian method. Again, it depends on the intent. If the purpose were to be used in finance, for example, you should use a proper subjective prior distribution or you could be Dutch Booked.


Vineberg, Susan, "Dutch Book Arguments", The Stanford Encyclopedia of Philosophy (Spring 2016 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/spr2016/entries/dutch-book/.

You cannot use Frequentist statistics in any form of pricing or risk based purpose because Frequentist statistics give rise to a Dutch Book automatically. There is a clash between Kolmogorov's third axiom and de Finetti's Dutch Book Theorem. Savage and de Finetti argued, and I agree, that probability methods or statistics built on countable additivity will give rise to Dutch Books, forcing the market maker to take sure losses. Frequentist methods are built on countable additivity. De Finetti's axioms give rise to Bayesian probability and statistics, but they cannot be countably additive, only finitely additive.

To understand the difference, a Frequentist statistician, if they wanted to, could cut the standard normal distribution into disjoint sets at every integer, such as from 0 to 1, 1 to 2, 2 to 3 and so on for an infinite number of sets. A Bayesian statistician cannot do that. You have to pick how many sets you want and stick to it. You can cut it three ways or six million, two hundred and fifty-one thousand, six hundred and two ways and you are fine.

I wrote an example so that you can see what I mean at How To Cut Cakes.

If it were for an applied purpose, but for something such as taxation policy, where the idea of a Dutch Book doesn't mean much, then you could get away with proper but diffuse prior distributions. You could use things such as maximum entropy priors, assuming you were using prior research or information as an element of your prior construction.


Jaynes, E.T. (2003) Probability Theory The Language of Science. Bretthorst, Larry (ed). Cambridge University Press.

I have a guess or two as to which distributions are involved, but it would change, for example, if there were a strong budget constraint.

To derive the distribution, look at

Weisstein, Eric W. "Ratio Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RatioDistribution.html

You should look up a basic undergraduate book in statistics and read up on the addition, subtraction, multiplication and division of random variables. You can get in trouble with truncated distributions if you ignore the truncation, so also pick up basic knowledge of the consequences of truncation.

The good news is that the distributions, since they lack a mean or a variance, also lack a skew or kurtosis that is defined. There are two possible sources of perceived skew. First, if the variables are truncated then the entire left tail is gone. Even if it were symmetric, but for truncation, it would appear skewed. Second, probabilistic constraints will dampen the distribution as it goes to the right.

As for kurtosis, well, ignoring the fact that it doesn't exist, the 99.99% interval for the Cauchy distribution is $\pm{636}\sigma$ not $\pm{3}\sigma$. However, $\sigma$ is not a standard deviation, it is the half-width at half-maximum. It is roughly half the interquartile range if there is no truncation. If there is truncation, then it is the location from the center to the point where the density is exactly half the height of the mode.

EDIT Okay, let me take your comments in order. Not all probability distributions have an expected value, a population mean. Some that have a population mean don’t have a variance.

As an example, note that if a population mean exists, then $$E(x)=\int_{-\infty}^\infty{x}f(x)\mathrm{d}x.$$ For the normal distribution, that is if $f$ is a normal distribution, then $E(x)=\mu$, where $\mu$ is the population mean.

However, consider the simple case where $$f(x)=\pi^{-1}\frac{1}{1+x^2.}$$ This distribution is called the Cauchy distribution. The name is probably not appropriate, but lots of things are named for Cauchy.

The distribution was discovered by Poisson as a counter-example to the central limit theorem. The central limit theorem does not hold for this distribution. Cauchy found it as an example where least squares always fail as a regression method.

So, from the above definition, $$E(x)=\int_{-\infty}^\infty\frac{1}{\pi}\frac{x}{1+x^2}\mathrm{d}x.$$

The indefinite integral is $$\int\frac{1}{\pi}\frac{x}{1+x^2}\mathrm{d}x=\frac{1}{2}\log(1+x^2)+C.$$ Obviously, as $x$ goes to either positive or negative infinity, the upper and lower terms of the integral go to infinity. This is an improper integral, so the exact way we could discuss it depends a little on how we define the integral, as infinity is not part of the real numbers, the integral diverges.

Since least-squares finds the mean slope passing through the mean value and the mean value does not exist, the estimator falls apart and is unrelated to any real population values. Since the population mean does not exist, the sample mean is meaningless. Since the mean is undefined, the variance, skew, and kurtosis are undefined.

Of course, since return is defined as the ratio of prices, the same problem exists for equity returns. That is also the source of the fat tails.

The median exists for all distributions, including distributions where all the mass sits on a single point. So parameter-free and distribution-free methods work that involve the median. The interpretation of the regression will change.

For example, the slope estimator for Theil’s is the median slope of the set of all slopes passing between every point in the sample. Because of that, a relationship between variables has to exist at least half the time, but it doesn’t have to exist at all for the rest of the time. Actors have to be rational in at least half of the transactions but can be bat-shit crazy the rest of the time. It won’t impact the regression at all if people behave in a totally insane manner 41% of the time. It reduces the requirements for human behavior.

As to whether it is truncated, it matters that the quotient can be negative. If it cannot, then likelihood-based tools such as Bayesian methods or the method of maximum likelihood will produce results that do not converge to the parameter unless you take the truncation into account in your formula. They will be sensitive to the missing element of the distribution.

As to whether or not you are doing scientific work versus applied work, the issue revolves around the importance of the Dutch Book Theorem.

If I ran a casino and you came in to play, and my calculations violated the Dutch Book Theorem, then there would exist times where you could be guaranteed a sure win. Why would I want to play at all if you could force me to take losses?

In addition to creating valid probabilities, Bayesian estimators cannot be dominated by other estimators. Consider the case where $$f(x|\mu)=\left[\frac{\pi}{2}+\tan^{-1}(\mu)\right]^{-1}\frac{1}{1+(x-\mu)^2},x\ge{0}.$$ That is the truncated case of the above distribution where $\mu\in\Re$ rather than $\mu=0$. In that case, $\mu$ is not at the median.

The median-based statistics are still valid for it, but they don’t represent the behavior of the marginal actor. Instead, they represent the behavior of the average actor.

Likelihood-based methods will be superior in locating $\mu$ over median-based methods, which would be guaranteed to miss it.

Although median-based Frequentist methods such as Theil’s regression are guaranteed to miss the population parameter, they still offer valid probability guarantees about the average actor. The p-values will give correct inferences. The confidence intervals will provide correct coverage. So, for example, if you would ask about the applicability of a Monetarist model to a particular question where a mean was not defined, it would provide valid inference.

If your goal is truth or falsehood, then you are fine to use Theil’s regression. If you need to gamble money with your model, then you cannot be.

As to moments, you can find a discussion of the moment generating function at the wiki article on it. You should also look at the characteristic function as well as a characteristic function can exist even when moments do not, see the wiki article on that.

  • $\begingroup$ Hi @Dave Harris, thank you for your dedicated explanation. There are some points I am still ambiguous in: (1) Can you clarify this sentence "It probably does not have a mean or a variance. Ratio variables often don't". (2) "If none of the variables in a ratio can be negative" do you mean the moninator and denominator of a quotient ? (3) "If you were doing scientific work, then either of those would be the better choice." So, do you mean: in my case with the data description above, I should not run normal OLS but choose Theil's regression and/or Quantile Regression? Warm regards. $\endgroup$ May 17, 2021 at 5:31
  • $\begingroup$ And could you please let me know what do "scientific work" and "applied work" imply in your statement. I am working on my thesis. $\endgroup$ May 17, 2021 at 5:33
  • $\begingroup$ And what is "the first moment of ratio" in the sentence "As these are accounting ratios, there are several candidate distributions and their ratios wouldn't have a first moment."please? I googled but cannot find the definition for such a term $\endgroup$ May 17, 2021 at 5:35

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