Exponentiated Exponential Distribution
Estimates the two parameters of the exponentiated exponential distribution by maximum likelihood estimation.
expexpff(lrate = "loglink", lshape = "loglink",
irate = NULL, ishape = 1.1, tolerance = 1.0e-6, zero = NULL)lshape, lrate |
Parameter link functions for the
shape and rate parameters.
See |
ishape |
Initial value for the shape parameter. If convergence fails try setting a different value for this argument. |
irate |
Initial value for the rate parameter.
By default, an initial value is chosen internally using
|
tolerance |
Numeric. Small positive value for testing whether values are close enough to 1 and 2. |
zero |
An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The default is none of them. If used, choose one value from the set {1,2}. |
The exponentiated exponential distribution is an alternative to the Weibull and the gamma distributions. The formula for the density is
f(y;rate,shape) = shape rate (1-\exp(-rate y))^(shape-1) \exp(-rate y)
where y>0, rate>0 and shape>0. The mean of Y is (psi(shape+1)-psi(1))/rate (returned as the fitted values) where psi is the digamma function. The variance of Y is (psi'(1)-psi'(shape+1))/ rate^2 where psi' is the trigamma function.
This distribution has been called the two-parameter generalized exponential distribution by Gupta and Kundu (2006). A special case of the exponentiated exponential distribution: shape=1 is the exponential distribution.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm
and vgam.
Practical experience shows that reasonably good initial values really
helps. In particular, try setting different values for the ishape
argument if numerical problems are encountered or failure to convergence
occurs. Even if convergence occurs try perturbing the initial value
to make sure the global solution is obtained and not a local solution.
The algorithm may fail if the estimate of the shape parameter is
too close to unity.
Fisher scoring is used, however, convergence is usually very slow. This is a good sign that there is a bug, but I have yet to check that the expected information is correct. Also, I have yet to implement Type-I right censored data using the results of Gupta and Kundu (2006).
Another algorithm for fitting this model is implemented in
expexpff1.
T. W. Yee
Gupta, R. D. and Kundu, D. (2001). Exponentiated exponential family: an alternative to gamma and Weibull distributions, Biometrical Journal, 43, 117–130.
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.
# A special case: exponential data
edata <- data.frame(y = rexp(n <- 1000))
fit <- vglm(y ~ 1, fam = expexpff, data = edata, trace = TRUE, maxit = 99)
coef(fit, matrix = TRUE)
Coef(fit)
# Ball bearings data (number of million revolutions before failure)
edata <- data.frame(bbearings = c(17.88, 28.92, 33.00, 41.52, 42.12, 45.60,
48.80, 51.84, 51.96, 54.12, 55.56, 67.80, 68.64, 68.64,
68.88, 84.12, 93.12, 98.64, 105.12, 105.84, 127.92,
128.04, 173.40))
fit <- vglm(bbearings ~ 1, fam = expexpff(irate = 0.05, ish = 5),
trace = TRUE, maxit = 300, data = edata)
coef(fit, matrix = TRUE)
Coef(fit) # Authors get c(rate=0.0314, shape=5.2589)
logLik(fit) # Authors get -112.9763
# Failure times of the airconditioning system of an airplane
eedata <- data.frame(acplane = c(23, 261, 87, 7, 120, 14, 62, 47,
225, 71, 246, 21, 42, 20, 5, 12, 120, 11, 3, 14,
71, 11, 14, 11, 16, 90, 1, 16, 52, 95))
fit <- vglm(acplane ~ 1, fam = expexpff(ishape = 0.8, irate = 0.15),
trace = TRUE, maxit = 99, data = eedata)
coef(fit, matrix = TRUE)
Coef(fit) # Authors get c(rate=0.0145, shape=0.8130)
logLik(fit) # Authors get log-lik -152.264Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.