# Robust estimation of kurtosis?

I am using the usual estimator for kurtosis, $$\hat{K}=\frac{\hat{\mu}_4}{\hat{\sigma}^4}$$, but I notice that even small 'outliers' in my empirical distribution, i.e. small peaks far from the center, affect it tremendously. Is there a kurtosis estimator which is more robust?

There are several. You will find an exhaustive comparison in this link to an ungated version of the paper (proper reference at the bottom of this answer).

Because of the constraints of the problem, the breakdown of the most robust of these algorithms (the L/RMC) is at most 12.5%. An advantage to the the L/RMC is that it is based on quantiles and remain interpretable even when the underlying distribution has no moments. Another advantage is that it does not assume symmetry of the distribution of the uncontaminated part of the data to measure tail weight: in fact, the algorithm returns two numbers: the RMC for the right tail weight and the LMC for the left tail weight.

The robustness of an estimator can be measured by its breakdown point. However, the notion of breakdown point is a complicated one in this context. Intuitively, it means that an adversary would need to control at least 12.5% of your sample to make this estimator take on arbitrary values (that is to be understood as an arbitrary value within the range of values that the estimator can return, since the measure of tail weight is always in $[0,1]$ by construction: no amount of contamination can for example cause the algorithm to return -1!). In practice, one finds that one can replace about 5% of the sample with even very pathological outliers without causing the most affected of the estimates (there are always two) to depart too much from the value it had on the uncontaminated sample.

The L/RMC is also widely implemented. For example you can find an R implementation here. As explained in the article linked above, to compute the L/RMC, you need to compute the MC (the estimator implemented in the link) separately on the left and right half of you data. Here, (left) right half are the sub-samples formed of the observation (smaller) larger than the median of your original sample.

• Brys, Hubert, Struyf. (2006). Robust Measures of Tail Weight.
• Aren't these alternative measures of tail weight rather than robust estimators of kurtosis per say? This may be what he really wants. but it is not exactly what he asked for. Do any/all of these estimators converge to kurtosis for large samples? – andrewH Mar 26 '15 at 19:16
• Summary from the paper: At uncontaminated data satifying the conditions on convex ordering of Van Zwet (under which the measure of kurtosis is meaningful) they converge to a monotone function of kurtosis. – user603 Mar 27 '15 at 10:33
• Pearson's kurtosis measures outliers (rare extreme observations), plain and simple. So what are you looking for instead? A measure of "peakedness"? First, that is not at all what Pearson's kurtosis measures. Second, if you want a measure of "peakedness", you first have to define what that means. If you can define it, you can estimate it. One possibility is the second derivative of the pdf of the standardized data, evaluated at the peak. (You're welcome). I am sure there are others. – Peter Westfall Nov 21 '17 at 1:33
• Actually, I give three mathematical theorems that relate kurtosis to the tails of the distribution, so these cannot be falsified: (i) For all distributions with finite fourth moment, kurtosis is between E(Z^4 * I(|Z| >1)) and E(Z^4 * I(|Z| >1)) +1. (ii) In the sub-class for which the density of Z^2 is continuous and decreasing on (0,1), the "+1" can be replaced by "+.5". (iii) For any sequence of distributions having kurtosis -> infinity, E(Z^4 * I(|Z| >b))/kurtosis -> 1, for any real b. It's all here: ncbi.nlm.nih.gov/pmc/articles/PMC4321753 – Peter Westfall Nov 21 '17 at 16:23