The assessment of kurtosis in the case of serially dependent data

Though kurtosis is often identified with Pearson's standardized fourth moment (Formula presented.), a long debate has been carried over about its actual meaning and the aspects of the shape which (Formula presented.) can really capture. This paper wants to add some further lines of criticism about that identification, by looking at a practical need: the assessment of kurtosis when the sampled population is unknown and data are serially correlated. Under suitable mixing conditions on the underlying stochastic process, new theorems are first proved to get knowledge of the sample distribution of some kurtosis estimators, so that their standard errors and the related confidence intervals can be computed. The problems in the implementation of such theorems are then discussed, to gain some guidance in the choice of the most suitable tools to be used under various circumstances. To this purpose, a wide simulation study is provided with the conclusion that, beyond the level of serial dependence, the reliability of the assessed level of kurtosis mostly depends on the regularity of the marginal distribution of the process. Thus indexes based on low-order moments are to be preferred in many cases. An application to real data is finally reported to support this claim.