# 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? Thanks.

-

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.