Distribution Function of the Modified Half-Normal Distribution

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.

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.

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.

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.

Examples

# Basic evaluation
pmhn(c(0.5, 1, 1.5), alpha = 2, beta = 1, gamma = 1)
#> [1] 0.1129101 0.4338796 0.7614259

# Tail / log forms
pmhn(2, alpha = 2, beta = 1, gamma = 1, lower.tail = FALSE)
#> [1] 0.06369619
pmhn(2, alpha = 2, beta = 1, gamma = 1, log.p = TRUE)
#> [1] -0.06581527

# 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))
#> [1] TRUE