Leptokurtosis

Leptokurtosis is a statistical property of a probability distribution characterized by a sharper peak around the mean and fatter tails than the normal distribution. It is indicated by a positive excess kurtosis, a measure that compares the distribution’s peakedness and tail weight to that of a normal distribution.

Mathematically, the population kurtosis is κ = E[(X − μ)4] / σ4, where μ is the mean and σ its standard deviation. Excess kurtosis is g2 = κ − 3. A distribution with g2 > 0 is described as leptokurtic, one with g2 < 0 as platykurtic, and g2 ≈ 0 as mesokurtic (similar to the normal distribution). Leptokurtic distributions have more mass in the center and in the tails, implying a higher probability of extreme values than the normal model.

Implications and examples: In applied settings, leptokurtosis suggests greater risk of extreme outcomes than a normal assumption would indicate. This matters in finance for modeling asset returns, in quality control, and in various natural phenomena. Examples of leptokurtic distributions include the Laplace distribution and, for certain degrees of freedom, the t-distribution with finite fourth moment (t with ν > 4 has positive excess kurtosis; for ν ≤ 4 the kurtosis is undefined). The normal distribution is the benchmark mesokurtic case.

Estimation and caveats: In samples, the sample excess kurtosis can be biased and sensitive to outliers. Finite-sample and heavy-tailed data can complicate interpretation, and certain distributions with infinite fourth moments render kurtosis undefined. Robust methods and alternative tail diagnostics are commonly used to supplement kurtosis.