1\name{inv.gaussianff}
2\alias{inv.gaussianff}
3%- Also NEED an '\alias' for EACH other topic documented here.
4\title{ Inverse Gaussian Distribution Family Function }
5\description{
6  Estimates the two parameters of the inverse Gaussian distribution
7  by maximum likelihood estimation.
8
9
10}
11\usage{
12inv.gaussianff(lmu = "loglink", llambda = "loglink",
13               imethod = 1, ilambda = NULL,
14               parallel = FALSE, ishrinkage = 0.99, zero = NULL)
15}
16%- maybe also 'usage' for other objects documented here.
17%apply.parint = FALSE,
18\arguments{
19  \item{lmu, llambda}{
20  Parameter link functions for the \eqn{\mu}{mu} and
21  \eqn{\lambda}{lambda} parameters.
22  See \code{\link{Links}} for more choices.
23
24
25  }
26  \item{ilambda, parallel}{
27  See \code{\link{CommonVGAMffArguments}} for more information.
28  If \code{parallel = TRUE} then the constraint is not applied to the
29  intercept.
30
31
32  }
33  \item{imethod, ishrinkage, zero}{
34  See \code{\link{CommonVGAMffArguments}} for information.
35
36
37  }
38}
39\details{
40  The standard (``canonical'') form of the
41  inverse Gaussian distribution has a density
42  that can be written as
43  \deqn{f(y;\mu,\lambda) = \sqrt{\lambda/(2\pi y^3)}
44       \exp\left(-\lambda (y-\mu)^2/(2 y \mu^2)\right)}{%
45       f(y;mu,lambda) = sqrt(lambda/(2*pi*y^3)) *
46       exp(-lambda*(y-mu)^2/(2*y*mu^2))
47  }
48  where \eqn{y>0},
49  \eqn{\mu>0}{mu>0}, and
50  \eqn{\lambda>0}{lambda>0}.
51  The mean of \eqn{Y} is \eqn{\mu}{mu} and its variance is
52  \eqn{\mu^3/\lambda}{mu^3/lambda}.
53  By default, \eqn{\eta_1=\log(\mu)}{eta1=log(mu)} and
54  \eqn{\eta_2=\log(\lambda)}{eta2=log(lambda)}.
55  The mean is returned as the fitted values.
56  This \pkg{VGAM} family function can handle multiple
57  responses (inputted as a matrix).
58
59
60
61}
62\value{
63  An object of class \code{"vglmff"} (see \code{\link{vglmff-class}}).
64  The object is used by modelling functions such as \code{\link{vglm}},
65  \code{\link{rrvglm}}
66  and \code{\link{vgam}}.
67
68
69}
70\references{
71
72Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1994).
73\emph{Continuous Univariate Distributions},
742nd edition, Volume 1, New York: Wiley.
75
76
77Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011).
78\emph{Statistical Distributions},
79Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.
80
81
82}
83\author{ T. W. Yee }
84\note{
85  The inverse Gaussian distribution can be fitted (to a certain extent)
86  using the usual GLM framework involving a scale parameter. This family
87  function is different from that approach in that it estimates both
88  parameters by full maximum likelihood estimation.
89
90
91}
92
93\seealso{
94  \code{\link{Inv.gaussian}},
95  \code{\link{waldff}},
96  \code{\link{bisa}}.
97
98
99  The \R{} package \pkg{SuppDists} has several functions for evaluating
100  the density, distribution function, quantile function and generating
101  random numbers from the inverse Gaussian distribution.
102
103
104}
105\examples{
106idata <- data.frame(x2 = runif(nn <- 1000))
107idata <- transform(idata, mymu   = exp(2 + 1 * x2),
108                          Lambda = exp(2 + 1 * x2))
109idata <- transform(idata, y = rinv.gaussian(nn, mu = mymu, lambda = Lambda))
110fit1 <-   vglm(y ~ x2, inv.gaussianff, data = idata, trace = TRUE)
111rrig <- rrvglm(y ~ x2, inv.gaussianff, data = idata, trace = TRUE)
112coef(fit1, matrix = TRUE)
113coef(rrig, matrix = TRUE)
114Coef(rrig)
115summary(fit1)
116}
117\keyword{models}
118\keyword{regression}
119
120