# SICHEL: The Sichel distribution for fitting a GAMLSS model

### Description

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.

### Usage

 1 2 3 4 5 6 7 8 9 SICHEL(mu.link = "log", sigma.link = "log", nu.link = "identity") dSICHEL(x, 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) tofySICHEL(y, mu, sigma, nu) 

### Arguments

 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 x 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 y the y variable, the tofySICHEL() should not be used on its own.

### Details

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^{∞} 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 mu, alpha and nu as the parameters.

### Value

Returns a gamlss.family object which can be used to fit a Sichel distribution in the gamlss() function.

### Note

The mean of the above Sichel distribution is mu and the variance is mu^2 *( 2*sigma*(nu+1)/c + (1/c^2)-1 )

### Author(s)

Rigby, R. A., Stasinopoulos D. M., Akantziliotou C and Marco Enea.

### References

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 distributions 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.org/).

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.family, PIG , SI

### Examples

  1 2 3 4 5 6 7 8 9 10 11 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 # library(gamlss) # gamlss(Ni~1,family=SICHEL, control=gamlss.control(n.cyc=50)) 

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