Package {mhn}

mhn: The Modified Half-Normal Distribution

Description

Provides density (dmhn), distribution function (pmhn), quantile function (qmhn), random generation (rmhn), and moments / mode of the Modified Half-Normal (MHN) distribution.

MHN Distribution

The MHN distribution has probability density function

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}]}

for x > 0, where \Psi[a, z] is the Fox-Wright Psi function.

Parameters

alpha

Shape parameter (\alpha > 0). Controls the x^{\alpha-1} term.

beta

Scale (rate) parameter (\beta > 0). Controls the \exp(-\beta x^2) term.

gamma

Location (skewness) parameter (\gamma \in R). Controls the \exp(\gamma x) term.

Special Cases

\gamma = 0

Square-root Gamma distribution (X^2 \sim \textrm{Gamma}(\alpha/2, \beta)) (Sun et al., 2023, Lemma 6a).

\alpha = 1

Truncated normal distribution \textrm{TN}(\gamma/(2\beta), 1/\sqrt{2\beta}, 0, \infty) (Sun et al., 2023, Lemma 6b).

\alpha = 1, \gamma = 0

Half-normal distribution \textrm{HN}(1/\sqrt{2\beta}) (Sun et al., 2023, Lemma 6c).

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. doi: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. doi:10.1080/03610918.2025.2524551

Author(s)

Maintainer: Tomotaka Momozaki momozaki.stat@gmail.com

See Also

Useful links:

• https://github.com/t-momozaki/mhn

• https://t-momozaki.github.io/mhn/

• Report bugs at https://github.com/t-momozaki/mhn/issues

Density of the Modified Half-Normal Distribution

Description

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

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.

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).

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.

See Also

mhn_mean, mhn_var, mhn_mode

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)

Excess Kurtosis of the Modified Half-Normal Distribution

Description

Computes the excess kurtosis \gamma_2 = E[(X - \mu)^4] / \sigma^4 - 3 for X \sim \mathrm{MHN}(\alpha, \beta, \gamma).

Usage

mhn_kurtosis(alpha, beta, gamma)

Arguments

Details

Uses the moment recurrence (Sun et al., 2023, Lemma 2b) to compute raw moments up to fourth order, then converts to central moments.

Value

A numeric scalar.

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. (Lemma 2b)

See Also

mhn_skewness, mhn_mean

Examples

mhn_kurtosis(alpha = 2, beta = 1, gamma = 0)

Mean of the Modified Half-Normal Distribution

Description

Computes E(X) for X \sim \mathrm{MHN}(\alpha, \beta, \gamma).

Usage

mhn_mean(alpha, beta, gamma)

Arguments

Details

The mean is computed as a ratio of Fox-Wright Psi functions:

E(X) = \frac{\Psi[(\alpha+1)/2,\, \gamma/\sqrt{\beta}]}{ \sqrt{\beta}\, \Psi[\alpha/2,\, \gamma/\sqrt{\beta}]}

Value

A numeric scalar.

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. (Lemma 2a)

See Also

mhn_var, dmhn

Examples

mhn_mean(alpha = 2, beta = 1, gamma = 0)

Mode of the Modified Half-Normal Distribution

Description

Computes the mode (most probable value) of the MHN distribution.

Usage

mhn_mode(alpha, beta, gamma)

Arguments

Details

The mode depends on \alpha:

\alpha > 1

(\gamma + \sqrt{\gamma^2 + 8\beta(\alpha - 1)}) / (4\beta) (Sun et al., 2023, Lemma 3b).

\alpha = 1

\max(0, \gamma / (2\beta)), obtained as the mode of the truncated normal \mathrm{TN}(\gamma/(2\beta), 1/\sqrt{2\beta}, 0, \infty) that the MHN reduces to in this case (Sun et al., 2023, Lemma 6b).

0 < \alpha < 1

An interior mode exists only when \gamma > 0 and \alpha \geq 1 - \gamma^2 / (8\beta) (Sun et al., 2023, Lemma 3c); otherwise the density is monotonically decreasing (Sun et al., 2023, Lemma 3d) and NA is returned.

Value

A numeric scalar. Returns NA when no interior mode exists (density is monotonically decreasing on (0, \infty)).

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. (Lemma 3b–d, Lemma 6b)

See Also

dmhn, mhn_mean

Examples

mhn_mode(alpha = 2, beta = 1, gamma = 1)
mhn_mode(alpha = 1, beta = 1, gamma = 2)
mhn_mode(alpha = 0.5, beta = 1, gamma = -1)  # NA

Skewness of the Modified Half-Normal Distribution

Description

Computes the skewness \gamma_1 = E[(X - \mu)^3] / \sigma^3 for X \sim \mathrm{MHN}(\alpha, \beta, \gamma).

Usage

mhn_skewness(alpha, beta, gamma)

Arguments

Details

Uses the moment recurrence (Sun et al., 2023, Lemma 2b) to compute raw moments up to third order, then converts to central moments.

Value

A numeric scalar.

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. (Lemma 2b)

See Also

mhn_kurtosis, mhn_mean

Examples

mhn_skewness(alpha = 2, beta = 1, gamma = 0)

Variance of the Modified Half-Normal Distribution

Description

Computes \mathrm{Var}(X) for X \sim \mathrm{MHN}(\alpha, \beta, \gamma).

Usage

mhn_var(alpha, beta, gamma)

Arguments

Details

Uses the formula (Sun et al., 2023, Lemma 2c):

\mathrm{Var}(X) = \frac{\alpha}{2\beta} + E(X)\left(\frac{\gamma}{2\beta} - E(X)\right)

For \alpha \geq 1, the variance satisfies \mathrm{Var}(X) \leq 1/(2\beta) (Sun et al., 2023, Lemma 4c).

Value

A numeric scalar.

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. (Lemma 2c)

See Also

mhn_mean, dmhn

Examples

mhn_var(alpha = 2, beta = 1, gamma = 0)

Distribution Function of the Modified Half-Normal Distribution

Description

Computes the cumulative distribution function (CDF) of the Modified Half-Normal (MHN) distribution with parameters alpha, beta, and gamma.

Usage

pmhn(q, alpha = 1, beta = 1, gamma = 0, lower.tail = TRUE, log.p = FALSE)

Arguments

Details

The CDF is computed via the series representation

F(x \mid \alpha, \beta, \gamma) = \frac{1}{\Psi[\alpha/2, \gamma/\sqrt{\beta}]} \sum_{i=0}^{\infty} \frac{z^i}{i!}\, \Gamma(s_i)\, P(s_i, \beta x^2)

where z = \gamma/\sqrt{\beta}, s_i = (\alpha + i)/2, and P(s, y) is the regularized lower incomplete gamma function (Sun et al., 2023, Lemma 1b; equivalent to the paper's form via the identity \Gamma(s)\, P(s, y) = \gamma(s, y), where \gamma(s, y) is the lower incomplete gamma function used in the paper). The infinite sum is truncated at the constructive bound K = \max\{K_1, K_2\} from Sun et al. (2023), Supplementary Lemma 10(d), which makes the truncation residual bounded by the user's tolerance divided by \Psi. When double-precision cancellation in the alternating-sign accumulator for \gamma < 0 would exceed that tolerance, the series is replaced by a Gauss-Kronrod (or tanh-sinh for \alpha < 1) numerical integration of the density on [0, q].

Special cases are detected and dispatched to standard R primitives:

• \gamma = 0: pgamma(q^2, alpha/2, scale = 1/beta)

• \alpha = 1: truncated-normal CDF via pnorm

When any of alpha, beta, gamma is a vector, the CDF 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 pmhn inside an R loop.

Value

A numeric vector. The output length equals max(length(q), length(alpha), length(beta), length(gamma)); each input is recycled to that length following standard R recycling rules. For q <= 0 the CDF is 0; for q = Inf it is 1.

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.

See Also

dmhn, qmhn, rmhn

Examples

# Basic evaluation
pmhn(c(0.5, 1, 1.5), alpha = 2, beta = 1, gamma = 1)

# Tail / log forms
pmhn(2, alpha = 2, beta = 1, gamma = 1, lower.tail = FALSE)
pmhn(2, alpha = 2, beta = 1, gamma = 1, log.p = TRUE)

# Special case: gamma = 0 reduces to sqrt-Gamma
all.equal(pmhn(1.5, alpha = 2, beta = 1, gamma = 0),
          pgamma(1.5^2, shape = 1, rate = 1))

Quantile Function of the Modified Half-Normal Distribution

Description

Computes the quantile (inverse cumulative) function of the Modified Half-Normal (MHN) distribution with parameters alpha, beta, and gamma.

Usage

qmhn(p, alpha = 1, beta = 1, gamma = 0, lower.tail = TRUE, log.p = FALSE)

Arguments

Details

For the general case, q = F^{-1}(p) is obtained by a TOMS 748 root-finder applied to the series CDF (Sun et al., 2023, Lemma 1b). The initial bracket is [\sqrt{\epsilon},\; E(X) + 8 \sqrt{\mathrm{Var}(X)}] and is doubled on the right (up to 30 times) until it brackets the target probability.

Special cases are detected and dispatched to standard R primitives:

• \gamma = 0: sqrt(qgamma(p, alpha/2, scale = 1/beta))

• \alpha = 1: truncated-normal inverse via qnorm

When any of alpha, beta, gamma is a vector, the quantile is evaluated element-wise. The Fox-Wright \Psi normalizing constant and moments E(X), \mathrm{Var}(X) (used to size the root-finder bracket) are recomputed only when consecutive elements present a different (\alpha, \beta, \gamma) triple.

Value

A numeric vector. The output length equals max(length(p), length(alpha), length(beta), length(gamma)); each input is recycled to that length following standard R recycling rules. qmhn(0) = 0 and qmhn(1) = Inf. Probabilities outside [0, 1] yield NaN.

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.

See Also

dmhn, pmhn, rmhn

Examples

# Basic evaluation
qmhn(c(0.1, 0.5, 0.9), alpha = 2, beta = 1, gamma = 1)

# Round-trip: F(F^-1(p)) ~ p
p <- c(0.05, 0.25, 0.5, 0.75, 0.95)
all.equal(pmhn(qmhn(p, alpha = 2, beta = 1, gamma = 1),
               alpha = 2, beta = 1, gamma = 1),
          p, tolerance = 1e-6)

# Tail / log forms
qmhn(0.95, alpha = 2, beta = 1, gamma = 1, lower.tail = FALSE)
qmhn(log(0.05), alpha = 2, beta = 1, gamma = 1, log.p = TRUE)

Random Generation from the Modified Half-Normal Distribution

Description

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

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.

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.

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.

See Also

dmhn, mhn_mean, mhn_var

Examples

set.seed(1)
rmhn(10, alpha = 2, beta = 1, gamma = 0.5)

# Vector parameters are recycled to length n.
set.seed(1)
rmhn(5, alpha = c(1, 2, 3, 4, 5))