# SI: The Sichel dustribution for fitting a GAMLSS model In mstasinopoulos/GAMLSS-Distibutions: Distributions for Generalized Additive Models for Location Scale and Shape

## Description

The SI() 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 dSI, pSI, qSI and rSI define the density, distribution function, quantile function and random generation for the Sichel SI(), distribution.

## Usage

 1 2 3 4 5 6 7 8 SI(mu.link = "log", sigma.link = "log", nu.link = "identity") dSI(x, mu = 0.5, sigma = 0.02, nu = -0.5, log = FALSE) pSI(q, mu = 0.5, sigma = 0.02, nu = -0.5, lower.tail = TRUE, log.p = FALSE) qSI(p, mu = 0.5, sigma = 0.02, nu = -0.5, lower.tail = TRUE, log.p = FALSE, max.value = 10000) rSI(n, mu = 0.5, sigma = 0.02, nu = -0.5) tofyS(y, mu, sigma, nu, what = 1) 

## 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 y the y variable. The function tofyS() should be not used on its own. what take values 1 or 2, for function tofyS().

## 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/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. Note that σ=[(μ^2+α^2)^{\frac{1}{2}} -μ ]^{-1}.

## Value

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

## Author(s)

Akantziliotou C., Rigby, R. A., Stasinopoulos D. M. 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.

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/).

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.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC.

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.

gamlss.family, PIG, NBI, NBII

## Examples

  1 2 3 4 5 6 7 8 9 10 11 SI()# gives information about the default links for the Sichel distribution #plot the pdf using plot plot(function(y) dSI(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),pSI(seq(from=0,to=100), mu=10, sigma=1, nu=1), type="h") # cdf # generate random sample tN <- table(Ni <- rSI(100, mu=5, sigma=1, nu=1)) r <- barplot(tN, col='lightblue') # fit a model to the data # library(gamlss) # gamlss(Ni~1,family=SI, control=gamlss.control(n.cyc=50)) 

mstasinopoulos/GAMLSS-Distibutions documentation built on Sept. 23, 2017, 10:31 p.m.