# invgauss

##### Inverse Gaussian Distribution

Density, cumulative probability, quantiles and random generation for the inverse Gaussian distribution.

- Keywords
- distribution

##### Usage

```
dinvgauss(x, mean=1, shape=NULL, dispersion=1, log=FALSE)
pinvgauss(q, mean=1, shape=NULL, dispersion=1, lower.tail=TRUE, log.p=FALSE)
qinvgauss(p, mean=1, shape=NULL, dispersion=1, lower.tail=TRUE, log.p=FALSE,
maxit=200L, tol=1e-14, trace=FALSE)
rinvgauss(n, mean=1, shape=NULL, dispersion=1)
```

##### Arguments

- x,q
vector of quantiles.

- p
vector of probabilities.

- n
sample size. If

`length(n)`

is larger than 1, then`length(n)`

random values are returned.- mean
vector of (positive) means.

- shape
vector of (positive) shape parameters.

- dispersion
vector of (positive) dispersion parameters. Ignored if

`shape`

is not`NULL`

, in which case`dispersion=1/shape`

.- lower.tail
logical; if

`TRUE`

, probabilities are P(X<q) otherwise P(X>q).- log
logical; if

`TRUE`

, the log-density is returned.- log.p
logical; if

`TRUE`

, probabilities are on the log-scale.- maxit
maximum number of Newton iterations used to find

`q`

.- tol
small positive numeric value giving the convergence tolerance for the quantile.

- trace
logical, if

`TRUE`

then the working estimate for`q`

from each iteration will be output.

##### Details

The inverse Gaussian distribution takes values on the positive real line.
It is somewhat more right skew than the gamma distribution, with variance given by `dispersion*mean^3`

.
The distribution has applications in reliability and survival analysis and is one of the response distributions used in generalized linear models.

Giner and Smyth (2016) show that the inverse Gaussian distribution is asympotically equal to an inverse chisquare distribution as the mean becomes large.

The functions provided here implement numeric algorithms developed by Giner and Smyth (2016) that achieve close to full machine accuracy for all possible parameter values. Giner and Smyth (2016) show that the probability calculations provided by these functions are considerably more accurate, and is most cases faster, than previous implementations of inverse Gaussian functions. The improvement in accuracy is most noticeable for extreme probability values and for large parameter values.

The shape and dispersion parameters are alternative parametrizations for the variability, with `dispersion=1/shape`

.
Only one of these two arguments needs to be specified.
If both are set, then `shape`

takes precedence.

##### Value

Output values give density (`dinvgauss`

), probability (`pinvgauss`

), quantile (`qinvgauss`

) or random sample (`rinvgauss`

) for the inverse Gaussian distribution with mean `mean`

and dispersion `dispersion`

.
Output is a vector of length equal to the maximum length of any of the arguments `x`

, `q`

, `mean`

, `shape`

or `dispersion`

.
If the first argument is the longest, then all the attributes of the input argument are preserved on output, for example, a matrix `x`

will give a matrix on output.
Elements of input vectors that are missing will cause the corresponding elements of the result to be missing, as will non-positive values for `mean`

or `dispersion`

.

##### References

Giner, G., and Smyth, G. K. (2016).
statmod: Probability calculations for the inverse Gaussian distribution.
*R Journal* 8(1), 339-351.
https://journal.r-project.org/archive/2016-1/giner-smyth.pdf

##### Examples

```
# NOT RUN {
q <- rinvgauss(10, mean=1, disp=0.5) # generate vector of 10 random numbers
p <- pinvgauss(q, mean=1, disp=0.5) # p should be uniformly distributed
# Quantile for small right tail probability:
qinvgauss(1e-20, mean=1.5, disp=0.7, lower.tail=FALSE)
# Same quantile, but represented in terms of left tail probability on log-scale
qinvgauss(-1e-20, mean=1.5, disp=0.7, lower.tail=TRUE, log.p=TRUE)
# }
```

*Documentation reproduced from package statmod, version 1.4.32, License: GPL-2 | GPL-3*