Skip to contents

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 \((0, \infty)\) with location \(\gamma / (2\beta)\) and scale \(1/\sqrt{2\beta}\)
\(\alpha = 1,\; \gamma = 0\) 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

The development version is on GitHub:

# install.packages("remotes")
remotes::install_github("t-momozaki/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, Sun et al. (2023, Lemma 2c) where that form is well conditioned
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)       # 1.1337...
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 is benchmarked, not theoretical; see
  vignette("theory") for the cost-decomposition derivation):
    α ≥ 10                        -> Sun Algorithm 3 (lower per-proposal
                                     cost than RTDR once the mode sharpens)
    samples per setup ≥ N*, α < 10 -> RTDR (lighter per-proposal cost)
    samples per setup <  N*        -> Sun Algorithm 3 (lighter setup cost)
  where N* = 25, raised to 100 for α < 0.1: the crossover moves to larger
  batches as the shape shrinks.

Choosing a specific sampler:

rmhn(1000, 2, 1, 1, method = "rtdr")   # RTDR for the general case
rmhn(1000, 2, 1, 1, method = "sun")    # Sun for the general case
                                       # (errors for α < 1, γ > 0)

method selects the sampler used for the general case. The closed-form special cases are taken first whatever it is set to: a negligible tilt draws from the square root of a Gamma, and α ≈ 1 from a truncated normal. Both are exact and cheaper than any rejection scheme.

Documentation

Full documentation, with a searchable function reference and rendered vignettes, is published at https://t-momozaki.github.io/mhn/.

Citation

If you use this package in academic work, please cite both the package and the methodology papers (citation("mhn") prints all four):

  • Momozaki, T. (2026). mhn: The Modified Half-Normal Distribution. R package version 0.1.1.
  • 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), 1591–1613. 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.