The Modified Half-Normal Distribution • mhn

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 the method argument.
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.)

mhn: The Modified Half-Normal Distribution