Tail Index of the Beta-Danish Distribution

Usage

bd_tail_index(a, b, c, k)

Arguments

a: Shape parameter (beta generator). Accepted for interface consistency; it does not affect the index.
b: Shape parameter. This is the tail index.
c: Shape parameter (baseline). Accepted for interface consistency.
k: Scale parameter (baseline). Accepted for interface consistency.

Value

A list with the tail index, the highest finite moment order, and a note on the moment generating function.

Details

The survival function is regularly varying at infinity with index \(-b\): \(S(t) \sim (c/k)^b t^{-b} / \{b B(a,b)\}\). Three consequences follow.

The \(r\)-th moment is finite if and only if \(b > r\), so the mean needs \(b > 1\), the variance \(b > 2\), skewness \(b > 3\) and kurtosis \(b > 4\).

The moment generating function does not exist for any \(t > 0\): a regularly varying tail decays polynomially, so \(E(e^{tZ})\) diverges. Characteristic-function or Laplace-transform arguments must be used instead of MGF ones anywhere in this family.

The distribution lies in the Frechet domain of attraction, so sample maxima normalise to a Frechet limit with shape \(b\), not to a Gumbel limit.

Examples

bd_tail_index(a = 1.5, b = 3, c = 2, k = 1)
#> $tail_index
#> [1] 3
#> 
#> $survival_exponent
#> [1] -3
#> 
#> $highest_finite_moment
#> [1] 2
#> 
#> $moment_condition
#> [1] "E(Z^r) finite if and only if b > r"
#> 
#> $mgf_exists
#> [1] FALSE
#> 
#> $domain_of_attraction
#> [1] "Frechet"
#>