WEI: Weibull distribution for fitting a GAMLSS In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

Description

The function `WEI` 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()`. [Note that the GAMLSS function `WEI2` uses a different parameterization for fitting the Weibull distribution.] The functions `dWEI`, `pWEI`, `qWEI` and `rWEI` define the density, distribution function, quantile function and random generation for the specific parameterization of the Weibul distribution.

Usage

 ```1 2 3 4 5``` ```WEI(mu.link = "log", sigma.link = "log") dWEI(x, mu = 1, sigma = 1, log = FALSE) pWEI(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE) qWEI(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE) rWEI(n, mu = 1, sigma = 1) ```

Arguments

 `mu.link` Defines the `mu.link`, with "log" link as the default for the mu parameter, other links are "inverse", "identity" and "own" `sigma.link` Defines the `sigma.link`, with "log" link as the default for the sigma parameter, other link is the "inverse", "identity" and "own" `x,q` vector of quantiles `mu` vector of the mu parameter `sigma` vector of sigma parameter `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

Details

The parameterization of the function `WEI` is given by

f(y|mu,sigma)= (sigma*y^(sigma-1)/mu^sigma)exp(-(y/mu)^sigma)

for y>0, μ>0 and σ>0. The GAMLSS functions `dWEI`, `pWEI`, `qWEI`, and `rWEI` 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 `WEI2` for a different parameterization of the Weibull.]

Value

`WEI()` returns a `gamlss.family` object which can be used to fit a Weibull distribution in the `gamlss()` function. `dWEI()` gives the density, `pWEI()` gives the distribution function, `qWEI()` gives the quantile function, and `rWEI()` generates random deviates. The latest functions are based on the equivalent `R` functions for Weibull distribution.

Note

The mean in `WEI` is given by mu*gamma((1/sigma)+1) and the variance (mu^2)*(gamma((2/sigma)+1)-gamma((1/sigma)+1)^2)

Author(s)

Mikis Stasinopoulos [email protected], Bob Rigby and Calliope Akantziliotou

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.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also http://www.gamlss.org/).

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, http://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.

`gamlss.family`, `WEI2`, `WEI3`
 ```1 2 3 4``` ```WEI() dat<-rWEI(100, mu=10, sigma=2) # library(gamlss) # gamlss(dat~1, family=WEI) ```