d_ig: Inverse Gaussian Distribution Functions

In IGPFrailty: Inverse Gaussian Process Degradation Models with Frailty

Inverse Gaussian Distribution Functions

Description

Density, distribution function, quantile function, and random generation for the Inverse Gaussian (Wald) distribution with mean mu and shape/scale parameter lambda.

Usage

d_ig(x, mu = 1, lambda = 1, log = FALSE)

p_ig(q, mu = 1, lambda = 1, lower.tail = TRUE, log.p = FALSE)

q_ig(p, mu = 1, lambda = 1, lower.tail = TRUE, log.p = FALSE)

r_ig(n, mu = 1, lambda = 1)

Arguments

x, q	Numeric vector of quantiles. Must be positive.
mu	Mean parameter. Must be strictly positive. Default is 1.
lambda	Shape parameter. Must be strictly positive. Default is 1.
log, log.p	Logical; if TRUE, probabilities/densities are returned on the log scale. Default is FALSE.
lower.tail	Logical; if TRUE (default), probabilities are P[X \le x], otherwise P[X > x].
p	Numeric vector of probabilities (between 0 and 1).
n	Number of observations to generate. Must be a single positive integer.

Details

The probability density function of the Inverse Gaussian distribution is:

f(x; \mu, \lambda) = \sqrt{\frac{\lambda}{2\pi x^3}} \exp\left(-\frac{\lambda (x - \mu)^2}{2\mu^2 x}\right), \quad x > 0

The cumulative distribution function is evaluated using:

F(x; \mu, \lambda) = \Phi\left(\sqrt{\frac{\lambda}{x}}\left(\frac{x}{\mu} - 1\right)\right) + \exp\left(\frac{2\lambda}{\mu}\right) \Phi\left(-\sqrt{\frac{\lambda}{x}}\left(\frac{x}{\mu} + 1\right)\right)

where \Phi(\cdot) is the standard normal cumulative distribution function.

Value

d_ig returns the density, p_ig returns the distribution function, q_ig returns the quantile function, and r_ig returns a vector of random variates of length n.

References

Wasan, M. T. (1968). On an inverse Gaussian process. Scandinavian Actuarial Journal, 1968(1-2), 69-96.

See Also

igp_fit, sim_igp

Examples

# Density and CDF
d_ig(1.5, mu = 2, lambda = 5)
p_ig(1.5, mu = 2, lambda = 5)

# Quantiles and random generation
q_ig(0.5, mu = 2, lambda = 5)
set.seed(123)
r_ig(5, mu = 2, lambda = 5)

IGPFrailty documentation built on Aug. 25, 2026, 9:08 a.m.