| borel.tanner {VGAM} | R Documentation |
Borel-Tanner Distribution Family Function
Description
Estimates the parameter of a Borel-Tanner distribution by maximum likelihood estimation.
Usage
borel.tanner(Qsize = 1, link = "logitlink", imethod = 1)
Arguments
Qsize |
A positive integer.
It is called |
link |
Link function for the parameter;
see |
imethod |
See |
Details
The Borel-Tanner distribution (Tanner, 1953) describes the
distribution of the total number of customers served before a
queue vanishes given a single queue with random arrival times
of customers (at a constant rate r per unit time, and
each customer taking a constant time b to be served).
Initially the queue has Q people and the first one starts
to be served.
The two parameters appear in the density only in the form of the
product rb, therefore we use a=rb, say, to denote
the single parameter to be estimated. The density function is
f(y;a) =
\frac{ Q }{(y-Q)!} y^{y-Q-1} a^{y-Q} \exp(-ay)
where y=Q,Q+1,Q+2,\ldots.
The case Q=1 corresponds to the Borel distribution
(Borel, 1942).
For the Q=1 case it is necessary for 0 < a < 1 for the
distribution to be proper.
The Borel distribution is a basic Lagrangian distribution of the
first kind.
The Borel-Tanner distribution is an Q-fold convolution of the
Borel distribution.
The mean is Q/(1-a) (returned as the fitted values) and the
variance is Q a / (1-a)^3.
The distribution has a very long tail unless a is small.
Fisher scoring is implemented.
Value
An object of class "vglmff"
(see vglmff-class).
The object is used by modelling functions
such as vglm
and vgam.
Author(s)
T. W. Yee
References
Tanner, J. C. (1953). A problem of interference between two queues. Biometrika, 40, 58–69.
Borel, E. (1942). Sur l'emploi du theoreme de Bernoulli pour faciliter le calcul d'une infinite de coefficients. Application au probleme de l'attente a un guichet. Comptes Rendus, Academie des Sciences, Paris, Series A, 214, 452–456.
Johnson N. L., Kemp, A. W. and Kotz S. (2005). Univariate Discrete Distributions, 3rd edition, p.328. Hoboken, New Jersey: Wiley.
Consul, P. C. and Famoye, F. (2006). Lagrangian Probability Distributions, Boston, MA, USA: Birkhauser.
See Also
Examples
bdata <- data.frame(y = rbort(n <- 200))
fit <- vglm(y ~ 1, borel.tanner, bdata, trace = TRUE, crit = "c")
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)