# YULE: Yule distribution for fitting a GAMLSS model In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

## 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

 ```1 2 3 4 5 6``` ```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) ```

## 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≤q 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.

## Details

The Yule distribution has density

f(y|μ)=(1/μ+1) B(y+1, 1/μ +2)

for y=0,1,2,… and mu>0.

## Value

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

## Author(s)

Fiona McElduff, Bob Rigby and Mikis Stasinopoulos.

## References

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

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.

`gamlss.family`
 ``` 1 2 3 4 5 6 7 8 9 10``` ```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)) ```