SICHEL: The Sichel distribution for fitting a GAMLSS model

In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

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.

The functions ZASICHEL() and ZISICHEL() are the zero adjusted (hurdle) and zero inflated versions of the Sichel distribution, respectively. That is four parameter distributions.

The functions dZASICHEL, dZISICHEL, pZASICHEL,pZISICHEL, qZASICHEL qZISICHEL rZASICHEL and rZISICHEL define the probability, cumulative, quantile and random generation functions for the zero adjusted and zero inflated Sichel distributions, ZASICHEL(), ZISICHEL(), respectively.

Usage

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)

ZASICHEL(mu.link = "log", sigma.link = "log", nu.link = "identity", 
         tau.link = "logit")
dZASICHEL(x, mu = 1, sigma = 1, nu = -0.5, tau = 0.1, log = FALSE)
pZASICHEL(q, mu = 1, sigma = 1, nu = -0.5, tau = 0.1, 
          lower.tail = TRUE, log.p = FALSE)
qZASICHEL(p, mu = 1, sigma = 1, nu = -0.5, tau = 0.1, 
          lower.tail = TRUE, log.p = FALSE, max.value = 10000)
rZASICHEL(n, mu = 1, sigma = 1, nu = -0.5, tau = 0.1, 
          max.value = 10000)

ZISICHEL(mu.link = "log", sigma.link = "log", nu.link = "identity", 
          tau.link = "logit")
dZISICHEL(x, mu = 1, sigma = 1, nu = -0.5, tau = 0.1, log = FALSE)
pZISICHEL(q, mu = 1, sigma = 1, nu = -0.5, tau = 0.1, 
         lower.tail = TRUE, log.p = FALSE)
qZISICHEL(p, mu = 1, sigma = 1, nu = -0.5, tau = 0.1, 
         lower.tail = TRUE, log.p = FALSE, max.value = 10000)
rZISICHEL(n, mu = 1, sigma = 1, nu = -0.5, tau = 0.1, 
         max.value = 10000)

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
tau.link	Defines the tau.link, with "logit" link as the default for the tau parameter
x	vector of (non-negative integer) quantiles
mu	vector of positive mu
sigma	vector of positive dispersion parameter sigma
nu	vector of nu
tau	vector of probabilities tau
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 SICHEL is given by

f(y|\mu,\sigma,\nu)= \frac{(\mu/b)^y K_{y+\nu}(\alpha)}{y!(\alpha \sigma)^{y+\nu} K_\nu(\frac{1}{\sigma})}

for y=0,1,2,...,\infty, \mu>0 , \sigma>0 and -\infty <\nu<\infty where

\alpha^2=\frac{1}{\sigma^2}+\frac{2\mu}{\sigma}

c=K_{\nu+1}(1/\sigma) / K_{\nu}(1/\sigma)

and K_{\lambda}(t) is the modified Bessel function of the third kind see pp 508-510 of Rigby et al. (2019). Note that the above parametrization is different from Stein, Zucchini and Juritz (1988) who use the above probability function but treat \mu, \alpha and \nu as the parameters.

The definition of the zero adjusted Sichel distribution, ZASICHEL and the the zero inflated Sichel distribution, ZISICHEL, are given in pp. 517-518 and pp. 519-520 of of Rigby et al. (2019), respectively.

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 \left[\frac{2\sigma (\nu+1)}{c} + \frac{1}{c^2}-1\right]

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., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/9780429298547")}. An older version can be found in https://www.gamlss.com/.

Rigby, R. A., Stasinopoulos, D. M., & Akantziliotou, C. (2008). A framework for modelling overdispersed count data, including the Poisson-shifted generalized inverse Gaussian distribution. Computational Statistics & Data Analysis, 53(2), 381-393.

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, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v023.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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/b21973")}

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 https://www.gamlss.com/).

See Also

gamlss.family, PIG , SI

Examples

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))

gamlss.dist documentation built on Aug. 24, 2023, 1:06 a.m.