Erlang Distribution
Estimates the scale parameter of the Erlang distribution by maximum likelihood estimation.
erlang(shape.arg, lscale = "loglink", imethod = 1, zero = NULL)
shape.arg |
The shape parameters.
The user must specify a positive integer, or integers for multiple responses.
They are recycled |
lscale |
Link function applied to the (positive) scale parameter.
See |
imethod, zero |
See |
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*scale and
its variance is shape*scale^2.
The linear/additive predictor, by default, is
eta=log(scale).
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm
and vgam.
T. W. Yee
Most standard texts on statistical distributions describe this distribution, e.g.,
Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011). Statistical Distributions, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.
rate <- exp(2); myshape <- 3 edata <- data.frame(y = rep(0, nn <- 1000)) for (ii in 1:myshape) edata <- transform(edata, y = y + rexp(nn, rate = rate)) fit <- vglm(y ~ 1, erlang(shape = myshape), data = edata, trace = TRUE) coef(fit, matrix = TRUE) Coef(fit) # Answer = 1/rate 1/rate summary(fit)
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