The Sichel distribution for fitting a GAMLSS model
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.
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)mu.link |
Defines the |
sigma.link |
Defines the |
nu.link |
Defines the |
tau.link |
Defines the |
x |
vector of (non-negative integer) quantiles |
mu |
vector of positive |
sigma |
vector of positive dispersion parameter |
nu |
vector of |
tau |
vector of probabilities |
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 |
The probability function of the Sichel distribution is given by
f(y|μ,σ,ν)=[μ^y K_{y+n}(α)] / [y! (α σ)^(y+ν) K_ν(1/σ)]
for y=0,1,2,..., mu>0 , σ>0 and -Inf<ν<Inf where
α^2= 1/σ^2 +2*μ/σ
c=K_{ν+1} (1/σ)/K_{ν}(1/σ)
and K_{lambda}(t) is the modified Bessel function of the third kind. 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.
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., Akantziliotou C and Marco Enea.
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.
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. An older version can be found in https://www.gamlss.com/.
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 https://www.gamlss.com/).
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, https://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 , 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 # library(gamlss) # gamlss(Ni~1,family=SICHEL, control=gamlss.control(n.cyc=50))
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