VGAM (version 1.1-8)

inv.gaussianff: Inverse Gaussian Distribution Family Function

Description

Estimates the two parameters of the inverse Gaussian distribution by maximum likelihood estimation.

Usage

inv.gaussianff(lmu = "loglink", llambda = "loglink",
      imethod = 1, ilambda = NULL,
      parallel = FALSE, ishrinkage = 0.99, zero = NULL)

Value

An object of class "vglmff"

(see vglmff-class). The object is used by modelling functions such as vglm,

rrvglm

and vgam.

Arguments

lmu, llambda

Parameter link functions for the \(\mu\) and \(\lambda\) parameters. See Links for more choices.

ilambda, parallel

See CommonVGAMffArguments for more information. If parallel = TRUE then the constraint is not applied to the intercept.

imethod, ishrinkage, zero

See CommonVGAMffArguments for information.

Author

T. W. Yee

Details

The standard (``canonical'') form of the inverse Gaussian distribution has a density that can be written as $$f(y;\mu,\lambda) = \sqrt{\lambda/(2\pi y^3)} \exp\left(-\lambda (y-\mu)^2/(2 y \mu^2)\right)$$ where \(y>0\), \(\mu>0\), and \(\lambda>0\). The mean of \(Y\) is \(\mu\) and its variance is \(\mu^3/\lambda\). By default, \(\eta_1=\log(\mu)\) and \(\eta_2=\log(\lambda)\). The mean is returned as the fitted values. This VGAM family function can handle multiple responses (inputted as a matrix).

References

Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd edition, Volume 1, New York: Wiley.

Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011). Statistical Distributions, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.

See Also

Inv.gaussian, waldff, bisa.

The R package SuppDists has several functions for evaluating the density, distribution function, quantile function and generating random numbers from the inverse Gaussian distribution.

Examples

idata <- data.frame(x2 = runif(nn <- 1000))
idata <- transform(idata, mymu   = exp(2 + 1 * x2),
                          Lambda = exp(2 + 1 * x2))
idata <- transform(idata, y = rinv.gaussian(nn, mu = mymu, Lambda))
fit1 <-   vglm(y ~ x2, inv.gaussianff, data = idata, trace = TRUE)
rrig <- rrvglm(y ~ x2, inv.gaussianff, data = idata, trace = TRUE)
coef(fit1, matrix = TRUE)
coef(rrig, matrix = TRUE)
Coef(rrig)
summary(fit1)