Tail ratios that separate light from power-law count tails

The ratio \(S(2k)/S(k)\) is \(q^k \to 0\) for a geometric tail, the constant \(2^{-\alpha}\) for a power tail (Research.P7.TailShape.geometric_ratio_tendsto_zero, power_ratio_const), and the negative-binomial pmf ratio \(p(2k)/p(k)\) is at most \((2k+r)^{r-1}q^k \to 0\) (nb_pmf_ratio_le, nb_pmf_ratio_tendsto_zero); the empirical survival ratio of counts is returned beside them.

Usage

morie_tail_ratio(k, q = 0.5, alpha = 1, r = 2L, counts = NULL)

Arguments

k

Lags at which to evaluate.

q

Geometric / negative-binomial success parameter in (0, 1).

alpha

Power-law exponent.

r

Negative-binomial integer shape.

counts

Optional observed counts for the empirical ratio.

Value

A data frame with one row per k.

References

Coles S (2001). An Introduction to Statistical Modeling of Extreme Values, Theorem 4.1. Springer.

Examples

morie_tail_ratio(k = c(5, 10, 20), q = 0.7, alpha = 1.5, r = 3)
#>    k    geometric     power nb_pmf_ratio  nb_bound
#> 1  5 0.1680700000 0.3535534  0.528220000 28.403830
#> 2 10 0.0282475249 0.3535534  0.098866337 14.942941
#> 3 20 0.0007979227 0.3535534  0.002974075  1.475359