Density of the Modified Half-Normal Distribution

Computes the probability density function (or log-density) of the Modified Half-Normal (MHN) distribution with parameters alpha, beta, and gamma.

Usage

dmhn(x, alpha = 1, beta = 1, gamma = 0, log = FALSE)

Arguments

x: Numeric vector of evaluation points.
alpha: Shape parameter ($\alpha > 0$). Scalar or numeric vector. Default: 1.
beta: Scale parameter ($\beta > 0$). Scalar or numeric vector. Default: 1.
gamma: Location parameter ($\gamma \in R$). Scalar or numeric vector. Default: 0.
log: Logical; if TRUE, log-density is returned. Default: FALSE.

Value

A numeric vector. The output length equals max(length(x), length(alpha), length(beta), length(gamma)); each input is recycled to that length following standard R recycling rules. For x < 0, the density is 0 (-Inf if log = TRUE).

Details

The MHN density is $$f(x \mid \alpha, \beta, \gamma) = \frac{2 \beta^{\alpha/2} x^{\alpha-1} \exp(-\beta x^2 + \gamma x)}{\Psi[\alpha/2, \gamma/\sqrt{\beta}]} \quad (x > 0)$$ where $\Psi[a, z]$ is the Fox-Wright Psi function (Sun et al., 2023, Lemma 1a).

The default parameters alpha = 1, beta = 1, gamma = 0 correspond to the half-normal distribution $\mathrm{HN}(1/\sqrt{2})$.

Special cases are detected and dispatched to closed-form solutions:

$\gamma = 0$: sqrt-Gamma distribution
$\alpha = 1$: truncated normal distribution

Computation is performed in log-space to avoid numerical underflow/overflow.

When any of alpha, beta, gamma is a vector, the density is evaluated element-wise. The Fox-Wright $\Psi$ normalizing constant is recomputed only when consecutive elements present a different $(\alpha, \beta, \gamma)$ triple, so passing grouped parameters is significantly faster than calling dmhn inside an R loop.

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.

Examples

x <- seq(0, 5, length.out = 100)
plot(x, dmhn(x, alpha = 2, beta = 1, gamma = 1), type = "l")


# Log-density
dmhn(1, alpha = 2, beta = 1, gamma = 1, log = TRUE)
#> [1] -0.3112403