Inverse Gaussian Distribution Functions

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Inverse Gaussian (Inverse Normal) distribution.

With mean \(\mu\) and scale \(\lambda\), the PDF is

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

and the CDF is

$$F(x;\mu,\lambda) = \Phi\left(\sqrt{\lambda x}(x\mu-1)\right) + e^{2\mu/\lambda}\Phi\left(-\sqrt{\lambda/\mu}(1+x/\mu)\right)$$

where \(\Phi\) is the standard normal CDF.

Accuracy and Implementation Notes: Implemented using the standard normal distribution and the exponential function. logpdf is specialised for numerical accuracy. Quantiles have no closed form and are computed via Newton-Raphson refinement.

Usage

inverse_gaussian_distribution(mu = 1, lambda = 1)

inverse_gaussian_pdf(x, mu = 1, lambda = 1)

inverse_gaussian_lpdf(x, mu = 1, lambda = 1)

inverse_gaussian_cdf(x, mu = 1, lambda = 1)

inverse_gaussian_lcdf(x, mu = 1, lambda = 1)

inverse_gaussian_quantile(p, mu = 1, lambda = 1)

Arguments

mu

Mean parameter (mu > 0; default is 1).

lambda

Scale parameter (lambda > 0; default is 1).

x

Quantile value (x >= 0).

p

Probability (0 <= p <= 1).

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See also

Boost Documentation for more details on the mathematical background.

Examples

# Inverse Gaussian distribution with mu = 3, lambda = 4
dist <- inverse_gaussian_distribution(3, 4)
# Apply generic functions
cdf(dist, 0.5)
#> [1] 0.01617264
logcdf(dist, 0.5)
#> [1] -4.124435
pdf(dist, 0.5)
#> [1] 0.1403174
logpdf(dist, 0.5)
#> [1] -1.963848
hazard(dist, 0.5)
#> [1] 0.142624
chf(dist, 0.5)
#> [1] 0.01630484
mean(dist)
#> [1] 3
median(dist)
#> [1] 2.202698
mode(dist)
#> [1] 1.140598
range(dist)
#> [1]  0.000000e+00 1.797693e+308
quantile(dist, 0.2)
#> [1] 1.161488
standard_deviation(dist)
#> [1] 2.598076
support(dist)
#> [1]  0.000000e+00 1.797693e+308
variance(dist)
#> [1] 6.75
skewness(dist)
#> [1] 2.598076
kurtosis(dist)
#> [1] 8.25
kurtosis_excess(dist)
#> [1] 11.25

# Convenience functions
inverse_gaussian_pdf(2, 3, 4)
#> [1] 0.2524295
inverse_gaussian_lpdf(2, 3, 4)
#> [1] -1.376623
inverse_gaussian_cdf(2, 3, 4)
#> [1] 0.4512408
inverse_gaussian_lcdf(2, 3, 4)
#> [1] -0.7957542
inverse_gaussian_quantile(0.5, 3, 4)
#> [1] 2.202698