The Modified Half-Normal Distribution.
mhn: The Modified Half-Normal Distribution
An R package providing density, distribution, quantile, and random generation functions for the Modified Half-Normal (MHN) distribution, together with closed-form / recurrence-based helpers for its moments and mode.
Overview
The MHN($\alpha$, $\beta$, $\gamma$) distribution has support on $(0, \infty)$ and density
$$f(x \mid \alpha, \beta, \gamma) ;\propto; x^{\alpha - 1} \exp(-\beta x^2 + \gamma x), \qquad x > 0,$$
where $\alpha, \beta > 0$ and $\gamma \in \mathbb{R}$. It arises as a conditional posterior in Bayesian MCMC for several common models (skew-elliptical regression, $t$-shrinkage priors, log-concave likelihoods with a quadratic Bayesian penalty) and generalises a number of familiar one-sided distributions:
| Constraint | Reduction |
|---|---|
| $\gamma = 0$ | $\sqrt{\mathrm{Gamma}}$: $X^2 \sim \mathrm{Gamma}(\alpha/2, \beta)$ |
| $\alpha = 1$ | Truncated normal on $(0, \infty)$, mean $\gamma / (2\beta)$ |
| $\alpha = 1,; \gamma = 0$ | Half-normal with scale $1/\sqrt{2\beta}$ |
| $\beta \to 0^+,; \gamma < 0$ | $\mathrm{Gamma}(\alpha, -\gamma)$ (limit) |
The package implements the efficient samplers of Sun, Kong & Pal (2023) (Algorithms 1 / 3) and the Gao & Wang (2025) Relaxed Transformed Density Rejection (RTDR) method with a uniform 1/e acceptance bound, and dispatches between them automatically.
Installation
# Development version from GitHub:
# install.packages("remotes")
remotes::install_github("t-momozaki/mhn")
Once the package is on CRAN it will also be installable with the usual
install.packages("mhn")
Quick start
library(mhn)
# Density, CDF, quantile, random generation
dmhn(c(0.5, 1, 2), alpha = 2, beta = 1, gamma = 1)
pmhn(1.5, alpha = 2, beta = 1, gamma = 1)
qmhn(0.95, alpha = 2, beta = 1, gamma = 1)
rmhn(10, alpha = 2, beta = 1, gamma = 1)
# Summary statistics
mhn_mean(2, 1, 1); mhn_var(2, 1, 1); mhn_mode(2, 1, 1)
Function reference
Distribution functions
| Function | Description |
|---|---|
dmhn() | Density (vectorised over x and parameters; supports log = TRUE) |
pmhn() | Cumulative distribution function (lower.tail and log.p à la pgamma) |
qmhn() | Quantile function via TOMS 748 root-finder |
rmhn() | Random generation; method dispatch via method = c("auto", "rtdr", "sun") |
dmhn(1.5, alpha = 2, beta = 1, gamma = 1, log = TRUE)
pmhn(1.5, alpha = 2, beta = 1, gamma = 1, lower.tail = FALSE)
qmhn(log(0.05), alpha = 2, beta = 1, gamma = 1, log.p = TRUE)
rmhn(5, alpha = c(1, 2, 3), beta = 1, gamma = c(0, 1, -1)) # recycled
Summary statistics
| Function | Description |
|---|---|
mhn_mean() | $E(X) = \Psi[(\alpha+1)/2,, z] ,/, (\sqrt{\beta}, \Psi[\alpha/2,, z])$, with $z = \gamma/\sqrt{\beta}$ |
mhn_var() | Variance from Sun et al. (2023, Lemma 2c) |
mhn_skewness() | Skewness $\gamma_1$ |
mhn_kurtosis() | Excess kurtosis $\gamma_2$ |
mhn_mode() | Mode (returns NA when no interior mode exists) |
mhn_mean(2, 1, 1) # 1.16...
mhn_skewness(2, 1, 1) # positive (density right-skewed)
mhn_mode(0.5, 1, -1) # NA: monotone-decreasing density
Algorithm dispatch for rmhn(method = "auto")
The default method routes each parameter triple to the cheapest sampler that is provably correct in that region. The decision rules are benchmarked in inst/benchmarks/auto_dispatch.R:
Closed-form shortcuts:
α = 1, γ = 0 -> half-normal via |rnorm|
γ = 0 -> sqrt(rgamma(...))
α = 1 -> truncated normal
Region α < 1, γ > 0:
Gao & Wang (2025) RTDR (Sun et al. (2023) Algorithm 2 is
intentionally not implemented; RTDR is
uniformly faster here)
Region γ > 0, α > 1:
Sun et al. (2023, Algorithm 1) (Normal or sqrt-Gamma proposal,
closed-form optimal parameters)
Region γ ≤ 0:
n-dependent (the crossover at 25 is benchmarked, not theoretical;
see vignette("theory") §7 for the cost-decomposition derivation):
samples per setup ≥ 25 -> RTDR (lighter per-proposal cost)
samples per setup < 25 -> Sun Algorithm 3 (lighter setup cost)
Forcing a specific sampler:
rmhn(1000, 2, 1, 1, method = "rtdr") # always RTDR
rmhn(1000, 2, 1, 1, method = "sun") # always Sun (errors for α < 1, γ > 0)
Documentation
Full documentation, with a searchable function reference and rendered vignettes, is published at https://t-momozaki.github.io/mhn/.
vignette("introduction", package = "mhn")— a 5-minute tour of every exported function with worked examples.?dmhn,?pmhn,?qmhn,?rmhn— full argument documentation including the recycling rules and themethodargument.citation("mhn")— package + underlying papers.
Citation
If you use this package in academic work, please cite both the package and the methodology papers (citation("mhn") prints all three):
- Momozaki, T. (2026). mhn: The Modified Half-Normal Distribution. R package version 0.1.0.
- 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. https://doi.org/10.1080/03610926.2021.1934700
- 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. https://doi.org/10.1080/03610918.2025.2524551
References
- Robert, C. P. (1995). Simulation of truncated normal variables. Statistics and Computing, 5(2), 121–125. (Used by the α = 1 truncated-normal special case of
rmhn.)
License
MIT © 2026 Tomotaka Momozaki. See LICENSE.