Quantile Function of the Modified Half-Normal Distribution

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.

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.

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.

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)
#> [1] 0.4717434 1.0906276 1.8480472

# 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)
#> [1] TRUE

# Tail / log forms
qmhn(0.95, alpha = 2, beta = 1, gamma = 1, lower.tail = FALSE)
#> [1] 0.3394014
qmhn(log(0.05), alpha = 2, beta = 1, gamma = 1, log.p = TRUE)
#> [1] 0.3394014