The function `WEI2`

can be used to define the Weibull distribution, a two parameter distribution, for a
`gamlss.family`

object to be used in GAMLSS fitting using the function `gamlss()`

.
This is the parameterization of the Weibull distribution usually used in proportional hazard models and is defined in details below.
[Note that the GAMLSS function `WEI`

uses a
different parameterization for fitting the Weibull distribution.]
The functions `dWEI2`

, `pWEI2`

, `qWEI2`

and `rWEI2`

define the density, distribution function, quantile function and random
generation for the specific parameterization of the Weibull distribution.

```
WEI2(mu.link = "log", sigma.link = "log")
dWEI2(x, mu = 1, sigma = 1, log = FALSE)
pWEI2(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qWEI2(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rWEI2(n, mu = 1, sigma = 1)
```

`WEI2()`

returns a `gamlss.family`

object which can be used to fit a Weibull distribution in the `gamlss()`

function.

`dWEI2()`

gives the density, `pWEI2()`

gives the distribution
function, `qWEI2()`

gives the quantile function, and `rWEI2()`

generates random deviates. The latest functions are based on the equivalent `R`

functions for Weibull distribution.

- mu.link
Defines the

`mu.link`

, with "log" link as the default for the mu parameter, other links are "inverse" and "identity"- sigma.link
Defines the

`sigma.link`

, with "log" link as the default for the sigma parameter, other link is the "inverse" and "identity"- x,q
vector of quantiles

- mu
vector of the mu parameter values

- sigma
vector of sigma parameter values

- 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]

- p
vector of probabilities.

- n
number of observations. If

`length(n) > 1`

, the length is taken to be the number required

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

In `WEI2`

the estimated parameters `mu`

and `sigma`

can be highly correlated so it is advisable to use the
`CG()`

method for fitting [as the RS() method can be veru slow in this situation.]

The parameterization of the function `WEI2`

is given by
$$f(y|\mu,\sigma)= \sigma\mu y^{\sigma-1}e^{-\mu
y^{\sigma}}$$
for \(y>0\), \(\mu>0\) and \(\sigma>0\), see pp. 436-437 of Rigby et al. (2019).
The GAMLSS functions `dWEI2`

, `pWEI2`

, `qWEI2`

, and `rWEI2`

can be used to provide the pdf, the cdf, the quantiles and
random generated numbers for the Weibull distribution with argument `mu`

, and `sigma`

.
[See the GAMLSS function `WEI`

for a different parameterization of the Weibull.]

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, tools:::Rd_expr_doi("10.1201/9780429298547"). An older version can be found in 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, 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. tools:::Rd_expr_doi("10.1201/b21973")

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

`gamlss.family`

, `WEI`

,`WEI3`

,

```
WEI2()
dat<-rWEI(100, mu=.1, sigma=2)
hist(dat)
# library(gamlss)
# gamlss(dat~1, family=WEI2, method=CG())
```

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