Hill estimator of the tail index

Hill (1975): the mean log-excess of the top k order statistics over the (k+1)-st. Consistent ONLY for xi > 0 (Frechet-type, regularly varying tails); non-positive data at the threshold are an error, not a silent log of a negative number. The choice of k is THE problem – variance falls and bias grows with k – so when k is omitted the whole Hill plot is returned alongside the default, because a single number hides exactly the instability a user needs to see. The asymptotic SE xi/sqrt(k) is honest only in the bias-free regime and says so.

Usage

morie_evt_hill(x, k = NULL)

Arguments

x

sample; the top order statistics must be positive.

k

number of top order statistics; sqrt(n) when NULL, with the full plot returned.

Value

list: xi, tail_alpha, se, k, threshold, and when k was omitted hill_plot_k / hill_plot_xi; n, method.

References

Hill (1975), Annals of Statistics 3:1163-1174.

Examples

set.seed(1)
morie_evt_hill((1 - stats::runif(500))^(-1 / 4), k = 50)$xi
#> [1] 0.234371