Tailindex

Tailindex is a parameter that quantifies how heavy the tails of a probability distribution are. In the language of regular variation, a distribution has tailindex alpha > 0 if its survival function satisfies P(X > x) ~ x^(-alpha) L(x) as x grows large, where L is a slowly varying function. A larger tailindex corresponds to a lighter tail; smaller values indicate heavier tails. If alpha > k, moments up to order k exist; if alpha ≤ k, the k-th moment is infinite. The Pareto distribution is a canonical example whose tailindex equals its shape parameter alpha.

In extreme value theory, the tailindex links to the behavior of extremes. Distributions with regularly varying tails have significant probability of very large observations, a feature captured by alpha. The tailindex also relates to the Generalized Pareto distribution and the maximum domain of attraction, with positive values indicating heavy tails and, in some formulations, infinite variance or mean for sufficiently small alpha.

Estimation methods focus on the tail observations. The Hill estimator uses the largest order statistics to estimate 1/alpha (or alpha directly in alternative parameterizations). Other approaches include the Pickands estimator and various moment-based or kernel-based techniques. Practical estimation requires careful threshold or tail sample size selection to balance bias and variance, and results can be sensitive to non-stationarity, censoring, or data quality.

Applications of tailindex span finance, insurance, environmental science, and risk assessment, where understanding tail behavior informs measures such as Value-at-Risk, expected shortfall, and premium pricing for extreme events.

See also: heavy-tailed distributions, regularly varying functions, Pareto distribution, Hill estimator, extreme value theory.