weibull {VGAM} | R Documentation

## Weibull Distribution Family Function

### Description

Maximum likelihood estimation of the 2-parameter Weibull distribution. No observations should be censored.

### Usage

    weibull(lshape = "loge", lscale = "loge",
            eshape = list(), escale = list(),
            ishape = NULL, iscale = NULL,
            nrfs = 1, imethod=1, zero = 2)

### Arguments

`lshape, lscale` | Parameter link functions applied to the (positive) shape parameter (called a below) and (positive) scale parameter (called b below). See `Links` for more choices.
`eshape, escale` | Extra argument for the respective links. See `earg` in `Links` for general information.
`ishape, iscale` | Optional initial values for the shape and scale parameters.
`nrfs` | Currently this argument is ignored. Numeric, of length one, with value in [0,1]. Weighting factor between Newton-Raphson and Fisher scoring. The value 0 means pure Newton-Raphson, while 1 means pure Fisher scoring. The default value uses a mixture of the two algorithms, and retaining positive-definite working weights.
`imethod` | Initialization method used if there are censored observations. Currently only the values 1 and 2 are allowed.
`zero` | An integer specifying which linear/additive predictor is to be modelled as an intercept only. The value must be from the set {1,2}, which correspond to the shape and scale parameters respectively. Setting `zero=NULL` means none of them.

### Details

The Weibull density for a response Y is

f(y;a,b) = a y^(a-1) * exp(-(y/b)^a) / [b^a]

for a > 0, b > 0, y > 0. The cumulative distribution function is

F(y;a,b) = 1 - exp(-(y/b)^a).

The mean of Y is b * gamma(1+ 1/a) (returned as the fitted values), and the mode is at b * (1- 1/a)^(1/a) when a>1. The density is unbounded for a<1. The kth moment about the origin is E(Y^k) = b^k * gamma(1+ k/a). The hazard function is a * t^(a-1) / b^a.

This VGAM family function currently does not handle censored data. Fisher scoring is used to estimate the two parameters. Although the Fisher information matrices used here are valid in all regions of the parameter space, the regularity conditions for maximum likelihood estimation are satisfied only if a>2 (according to Kleiber and Kotz (2003)). If this is violated then a warning message is issued. One can enforce a>2 by choosing `lshape = "logoff"` and `eshape=list(offset=-2)`.

### Value

An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`, and `vgam`.

### Warning

This function is under development to handle other censoring situations. The version of this function which will handle censored data will be called `cenweibull()`. It is currently being written and will use `Surv` as input. It should be released in later versions of VGAM.

If the shape parameter is less than two then misleading inference may result, e.g., in the `summary` and `vcov` of the object.

### Note

Successful convergence depends on having reasonably good initial values. If the initial values chosen by this function are not good, make use the two initial value arguments.

The Weibull distribution is often an alternative to the lognormal distribution. The inverse Weibull distribution, which is that of 1/Y where Y has a Weibull(a,b) distribution, is known as the log-Gompertz distribution.

### Author(s)

T. W. Yee

### References

Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ: Wiley-Interscience.

Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, 2nd edition, Volume 1, New York: Wiley.

Gupta, R. D. and Kundu, D. (2006) On the comparison of Fisher information of the Weibull and GE distributions, Journal of Statistical Planning and Inference, 136, 3130–3144.

### See Also

`dweibull`, `gev`, `lognormal`, `expexp`.

### Examples

    # Complete data
    x = runif(n <- 1000)
    y = rweibull(n, shape=exp(1+x), scale = exp(-0.5))
    fit = vglm(y ~ x, weibull, trace=TRUE)
    coef(fit, mat=TRUE)
    vcov(fit)
    summary(fit)

[Package VGAM version 0.7-9 Index]