SICHEL {gamlss.dist} | R Documentation |
The SICHEL()
function defines the Sichel distribution, a three parameter discrete distribution, for a gamlss.family
object to be used
in GAMLSS fitting using the function gamlss()
.
The functions dSICHEL
, pSICHEL
, qSICHEL
and rSICHEL
define the density, distribution function, quantile function and random
generation for the Sichel SICHEL()
, distribution. The function VSICHEL
gives the variance of a fitted Sichel model.
SICHEL(mu.link = "log", sigma.link = "log", nu.link = "identity") dSICHEL(y, mu=1, sigma=1, nu=-0.5, log=FALSE) pSICHEL(q, mu=1, sigma=1, nu=-0.5, lower.tail = TRUE, log.p = FALSE) qSICHEL(p, mu=1, sigma=1, nu=-0.5, lower.tail = TRUE, log.p = FALSE, max.value = 10000) rSICHEL(n, mu=1, sigma=1, nu=-0.5, max.value = 10000) VSICHEL(obj)
mu.link |
Defines the mu.link , with "log" link as the default for the mu parameter |
sigma.link |
Defines the sigma.link , with "log" link as the default for the sigma parameter |
nu.link |
Defines the nu.link , with "identity" link as the default for the nu parameter |
y |
vector of (non-negative integer) quantiles |
mu |
vector of positive mu |
sigma |
vector of positive despersion parameter |
nu |
vector of nu |
p |
vector of probabilities |
q |
vector of quantiles |
n |
number of random values to return |
log, log.p |
logical; if TRUE, probabilities p are given as log(p) |
lower.tail |
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x] |
max.value |
a constant, set to the default value of 10000 for how far the algorithm should look for q |
obj |
a fitted Sichel gamlss model |
The probability function of the Sichel distribution is given by
f(y|mu,sigma,nu)=mu^y Ky+n(alpha)/(alpha sigma)^(y+v) y! Knu(1/sigma)
where alpha^2=1/sigma^2 +2*mu/c*sigma, and c=Rv(1/sigma)=Kv+1(1/sigma)/Kv(1/sigma) for y=0,1,2,... where mu>0 , σ>0 and -Inf<nu<Inf and K_{λ}(t)=frac{1}{2}int_0^{infty} x^{λ-1} exp{-frac{1}{2}t(x+x^{-1})}dx is the modified Bessel function of the third kind. Note that the above parameterization is different from Stein, Zucchini and Juritz (1988) who use the above probability function but treat $μ/c$, $α$ and $nu$ as the parameters.
Returns a gamlss.family
object which can be used to fit a Sichel distribution in the gamlss()
function.
The mean of the above Sichel distribution is mu and the variance is mu^2 *( 2*sigma*(nu+1)/c + (1/c^2)-1 )
Rigby, R. A., Stasinopoulos D. M. and Akantziliotou C.
Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.
Rigby, R. A., Stasinopoulos D. M. and Akantziliotou, C. (2006) Modelling the parameters of a family of mixed Poisson distribtions including the Sichel and Delaptorte. Submitted for publication.
Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2003) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also http://www.gamlss.com/).
Stein, G. Z., Zucchini, W. and Juritz, J. M. (1987). Parameter Estimation of the Sichel Distribution and its Multivariate Extension. Journal of American Statistical Association, 82, 938-944.
Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, http://www.jstatsoft.org/v23/i07.
gamlss
, gamlss.family
, PIG
, SI
SICHEL()# gives information about the default links for the Sichel distribution #plot the pdf using plot plot(function(y) dSICHEL(y, mu=10, sigma=1, nu=1), from=0, to=100, n=100+1, type="h") # pdf # plot the cdf plot(seq(from=0,to=100),pSICHEL(seq(from=0,to=100), mu=10, sigma=1, nu=1), type="h") # cdf # generate random sample tN <- table(Ni <- rSICHEL(100, mu=5, sigma=1, nu=1)) r <- barplot(tN, col='lightblue') # fit a model to the data gamlss(Ni~1,family=SICHEL, control=gamlss.control(n.cyc=50))