R: A specific parameterization of the Weibull distribution for...

WEI2 {gamlss.dist} | R Documentation

## A specific parameterization of the Weibull distribution for fitting a GAMLSS

### Description

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.

### Usage

    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)

### Arguments

`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

### Details

The parameterization of the function `WEI2` is given by

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

for y>0, μ>0 and σ>0. 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.]

### Value

`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.

### Warning

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.]

### Note

The mean in `WEI2` is given by mu^(-1/sigma)*gamma((1/sigma)+1) and the variance mu^{-2/sigma}*(gamma((2/sigma)+1)-gamma((1/sigma)+1)^2)

### Author(s)

Mikis Stasinopoulos, 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.

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. (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.

### See Also

`gamlss.family`, `WEI`,`WEI3`,

### Examples

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

[Package gamlss.dist version 5.3-2 Index]