gamlss.dist (version 6.1-1)

YULE: Yule distribution for fitting a GAMLSS model

Description

The function YULE defines the Yule distribution, a one parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(), with mean equal to the parameter mu. The functions dYULE, pYULE, qYULE and rYULE define the density, distribution function, quantile function and random generation for the YULE parameterization of the Yule distribution.

Usage

YULE(mu.link = "log")
dYULE(x, mu = 2, log = FALSE)
pYULE(q, mu = 2, lower.tail = TRUE, log.p = FALSE) 
qYULE(p, mu = 2, lower.tail = TRUE, log.p = FALSE, 
    max.value = 10000)
rYULE(n, mu = 2)

Value

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

Arguments

mu.link

Defines the mu.link, with "log" link as the default for the mu parameter

x

vector of (non-negative integer) quantiles.

q

vector of quantiles.

p

vector of probabilities.

n

number of random values to return.

mu

vector of positive mu values.

lower.tail

logical; if TRUE (default) probabilities are \(P[Y\leq y]\), otherwise, \(P[Y>y]\).

log, log.p

logical; if TRUE probabilities p are given as log(p).

max.value

constant; generates a sequence of values for the cdf function.

Author

Fiona McElduff, Bob Rigby and Mikis Stasinopoulos.

Details

The Yule distribution has density $$P(Y=y|\mu) = (\mu^{-1}+1) B(y+1,\mu^{-1}+2)$$ for \(y=0,1,2,\ldots\) and \(\mu>0\), see pp 477-478 of Rigby et al. (2019).

References

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, tools:::Rd_expr_doi("10.1201/9780429298547"). An older version can be found in https://www.gamlss.com/.

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. tools:::Rd_expr_doi("10.1201/b21973")

Wimmer, G. and Altmann, G. (1999) Thesaurus of univariate discrete probability distributions. Stamm.

(see also https://www.gamlss.com/).

See Also

gamlss.family

Examples

par(mfrow=c(2,2))
y<-seq(0,20,1)
plot(y, dYULE(y), type="h")
q <- seq(0, 20, 1)
plot(q, pYULE(q), type="h")
p<-seq(0.0001,0.999,0.05)
plot(p , qYULE(p), type="s")
dat <- rYULE(100)
hist(dat)
#summary(gamlss(dat~1, family=YULE))