weibull {VGAM} | R Documentation |
Maximum likelihood estimation of the 2-parameter Weibull distribution. Allows for Type-I right censored data.
weibull(lshape = "logoff", lscale = "loge", eshape=if(lshape == "logoff") list(offset=-2) else list(), escale=list(), ishape = NULL, iscale = NULL, imethod=1, zero = 2)
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. |
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.
|
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).
This VGAM family function handles Type-I right censored data as well as complete data. Fisher scoring is used to estimate the two parameters. The Fisher information matrices used here are only valid if a>2; these are where the regularity conditions for maximum likelihood estimation are satisfied. For this reason, the default link function for the shape parameter is a log-link with an offset value of -2.
An object of class "vglmff"
(see vglmff-class
).
The object is used by modelling functions such as vglm
,
and vgam
.
If the shape parameter is less than two then numerical problems may
occur during the fitting and/or misleading inference may result in
the summary
of the object.
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. For censored data, numerical integration is used to compute the expected working weight matrices; this may fail if the data is censored `too much' and/or may be quite slow. See the example below on how to input censored data.
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.
T. W. Yee
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.
dweibull
,
gev
,
lognormal
,
expexp
.
# Complete data x = runif(n <- 1000) y = rweibull(n, shape=2+exp(1+x), scale = exp(-0.5)) fit = vglm(y ~ x, weibull, tra=TRUE) coef(fit, mat=TRUE) Coef(fit) # Type-I right censored data cutpt = 0.6 # Making this too small results in numerical problems rcensored = y > cutpt cy = ifelse(rcensored, cutpt, y) table(rcensored) ## Not run: par(mfrow=1:2) hist(y, xlim=range(y)) hist(cy, xlim=range(y), main="Censored y") ## End(Not run) cfit = vglm(cy ~ x, weibull, trace=TRUE, crit="l", extra=list(rightcensored=rcensored)) coef(cfit, mat=TRUE) summary(cfit)