Excess Kurtosis of the Modified Half-Normal Distribution

Computes the excess kurtosis \(\gamma_2 = E[(X - \mu)^4] / \sigma^4 - 3\) for \(X \sim \mathrm{MHN}(\alpha, \beta, \gamma)\).

Usage

mhn_kurtosis(alpha, beta, gamma)

Arguments

alpha: Shape parameter (\(\alpha > 0\)).
beta: Scale parameter (\(\beta > 0\)).
gamma: Location parameter (\(\gamma \in R\)).

Value

A numeric scalar.

Details

Integrates the fourth central moment against the unnormalised kernel, in the variable centred on its peak. Building it from raw moments by the Lemma 2b recurrence is exact algebra but cancels once the standard deviation is small next to the mean, which is what a large tilt produces; that expansion is kept only for the parameter values where the integration window cannot be established.

References

Sun, J., Kong, M., & Pal, S. (2023). The Modified-Half-Normal distribution: Properties and an efficient sampling scheme. Communications in Statistics - Theory and Methods, 52(5), 1591–1613. (Lemma 2b)

Examples

mhn_kurtosis(alpha = 2, beta = 1, gamma = 0)
#> [1] 0.2450893