Random Generation from the Modified Half-Normal Distribution

Draws random variates from the Modified Half-Normal (MHN) distribution with parameters alpha, beta, and gamma.

Usage

rmhn(n, alpha = 1, beta = 1, gamma = 0, method = c("auto", "rtdr", "sun"))

Arguments

n: Non-negative integer giving the number of variates to draw. n = 0 returns numeric(0).
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.
method: Sampling algorithm. One of "auto" (default), "rtdr", or "sun". See Details.

Value

A numeric vector of length n. If any of alpha, beta, gamma (after recycling to length n) is NA or non-finite (Inf, -Inf, NaN), the corresponding output element is NA.

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. rmhn does not evaluate $\Psi$; the rejection-sampling kernels cancel it out.

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

The method argument selects the rejection sampler:

"auto": Special-case shortcuts when applicable ($\gamma \approx 0$ -> sqrt-Gamma, $\alpha \approx 1$ -> truncated normal). Otherwise dispatches to RTDR (Gao & Wang, 2025).
"rtdr": Force the Relaxed Transformed Density Rejection method of Gao & Wang (2025). The acceptance probability is bounded below by $1/e \approx 0.368$ uniformly over the parameter space. Note: Gao & Wang (2025) use the parameterization $(\lambda, \alpha, \beta)$ with density proportional to $x^{\lambda - 1} \exp(-\alpha x^2 - \beta x)$; the mapping to the Sun et al. parameterization used here is $\lambda \leftrightarrow \alpha$, $\alpha \leftrightarrow \beta$, $\beta \leftrightarrow -\gamma$ (sign flip on the linear term).
"sun": Force the Sun et al. (2023) algorithms. Algorithm 1 is used when $\gamma > 0$ and $\alpha > 1$; Algorithm 3 is used when $\gamma \le 0$. The combination $\alpha < 1$ with $\gamma > 0$ is unsupported and triggers an error.

Vector parameters are recycled to length n following standard R rules. Trailing parameter elements beyond index n - 1 are silently ignored, matching the convention of rnorm.

Internally the setup state of the chosen sampler is reused as long as consecutive $(\alpha, \beta, \gamma)$ triples are equal, so passing parameters grouped by triple is faster than calling rmhn 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.

Gao, F. & Wang, H.-B. (2025). Generating modified-half-normal random variates by a relaxed transformed density rejection method. Communications in Statistics - Simulation and Computation.

Robert, C. P. (1995). Simulation of truncated normal variables. Statistics and Computing, 5(2), 121–125.

Examples

set.seed(1)
rmhn(10, alpha = 2, beta = 1, gamma = 0.5)
#>  [1] 0.3277054 0.9876283 0.2750659 0.7977315 1.2334905 1.0218017 0.6370495
#>  [8] 0.5679219 0.2958218 1.9631443

# Vector parameters are recycled to length n.
set.seed(1)
rmhn(5, alpha = c(1, 2, 3, 4, 5))
#> [1] 0.4429697 0.7989284 0.5821857 1.4320656 2.2998471