# The Sichel dustribution for fitting a GAMLSS model

### 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 |

`sigma.link` |
Defines the |

`nu.link` |
Defines the |

`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 |

`what` |
take values 1 or 2, for function |

### 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.

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.

### See Also

`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))
``` |