Package 'mhn'

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

x	Numeric vector of evaluation points.
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.
log	Logical; if TRUE, log-density is returned. Default: FALSE.

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

alpha	Shape parameter ($\alpha > 0$).
beta	Scale parameter ($\beta > 0$).
gamma	Location parameter ($\gamma \in R$).

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

alpha	Shape parameter ($\alpha > 0$).
beta	Scale parameter ($\beta > 0$).
gamma	Location parameter ($\gamma \in R$).

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

alpha	Shape parameter ($\alpha > 0$).
beta	Scale parameter ($\beta > 0$).
gamma	Location parameter ($\gamma \in R$).

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

alpha	Shape parameter ($\alpha > 0$).
beta	Scale parameter ($\beta > 0$).
gamma	Location parameter ($\gamma \in R$).

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

alpha	Shape parameter ($\alpha > 0$).
beta	Scale parameter ($\beta > 0$).
gamma	Location parameter ($\gamma \in R$).

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

q	Numeric vector of quantiles.
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.
lower.tail	Logical; if TRUE (default), probabilities are $P(X \le q)$, otherwise $P(X > q)$.
log.p	Logical; if TRUE, probabilities are returned on the log scale. Default: FALSE.

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

p	Numeric vector of probabilities.
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.
lower.tail	Logical; if TRUE (default), probabilities are $P(X \le q)$, otherwise $P(X > q)$.
log.p	Logical; if TRUE, probabilities are provided on the log scale. Default: FALSE.

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

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.

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))

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