Estimates the scale parameter of the Erlang distribution
by maximum likelihood estimation.
Usage
erlang(shape.arg, link = "loge", earg=list(), method.init = 1)
Arguments
shape.arg
The shape parameter.
The user must specify a positive integer.
link
Link function applied to the (positive) $scale$ parameter.
See Links for more choices.
earg
List. Extra argument for the link.
See earg in Links for general information.
method.init
An integer with value 1 or 2 which
specifies the initialization method. If failure to converge occurs
try the other value.
Value
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm
and vgam.
Details
The Erlang distribution is a special case of the gamma distribution
with shape that is a positive integer. If shape.arg=1
then it simplifies to the exponential distribution. As illustrated
in the example below, the Erlang distribution is the distribution of
the sum of shape.arg independent and identically distributed
exponential random variates.
The probability density function of the Erlang
distribution is given by
$$f(y) = \exp(-y/scale) y^{shape-1} scale^{-shape} / \Gamma(shape)$$
for known positive integer $shape$,
unknown $scale > 0$ and $y > 0$.
Here,
$\Gamma(shape)$ is the gamma
function, as in gamma.
The mean of Y
is $\mu=shape \times scale$ and
its variance is $shape \times scale^2$.
The linear/additive predictor, by default, is
$\eta=\log(scale)$.
References
Most standard texts on statistical distributions describe
this distribution, e.g.,
Evans, M., Hastings, N. and Peacock, B. (2000)
Statistical Distributions,
New York: Wiley-Interscience, Third edition.