Mode of the Modified Half-Normal Distribution

Computes the mode (most probable value) of the MHN distribution.

Usage

mhn_mode(alpha, beta, gamma)

Arguments

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

Value

A numeric scalar. Returns NA when no interior mode exists (density is monotonically decreasing on \((0, \infty)\)).

Details

The mode depends on \(\alpha\):

\(\alpha > 1\): \((\gamma + \sqrt{\gamma^2 + 8\beta(\alpha - 1)}) / (4\beta)\) (Sun et al., 2023, Lemma 3b).
\(\alpha = 1\): \(\max(0, \gamma / (2\beta))\), obtained as the mode of the truncated normal \(\mathrm{TN}(\gamma/(2\beta), 1/\sqrt{2\beta}, 0, \infty)\) that the MHN reduces to in this case (Sun et al., 2023, Lemma 6b).
\(0 < \alpha < 1\): An interior mode exists only when \(\gamma > 0\) and \(\alpha \geq 1 - \gamma^2 / (8\beta)\) (Sun et al., 2023, Lemma 3c); otherwise the density is monotonically decreasing (Sun et al., 2023, Lemma 3d) and NA is returned.

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), 1507–1536. (Lemma 3b–d, Lemma 6b)

Examples

mhn_mode(alpha = 2, beta = 1, gamma = 1)
#> [1] 1
mhn_mode(alpha = 1, beta = 1, gamma = 2)
#> [1] 1
mhn_mode(alpha = 0.5, beta = 1, gamma = -1)  # NA
#> [1] NA