Exponentiated Exponential Distribution

Estimates the two parameters of the exponentiated exponential distribution by maximum likelihood estimation.

Usage

expexpff(lrate = "loglink", lshape = "loglink",
         irate = NULL, ishape = 1.1, tolerance = 1.0e-6, zero = NULL)

Arguments

lshape, lrate

Parameter link functions for the $\alpha$ and $\lambda$ parameters. See Links for more choices. The defaults ensure both parameters are positive.

ishape

Initial value for the $\alpha$ parameter. If convergence fails try setting a different value for this argument.

irate

Initial value for the $\lambda$ parameter. By default, an initial value is chosen internally using ishape.

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}. See CommonVGAMffArguments for more information.

Details

The exponentiated exponential distribution is an alternative to the Weibull and the gamma distributions. The formula for the density is $$f(y;\lambda,\alpha) = \alpha \lambda (1-\exp(-\lambda y))^{\alpha-1} \exp(-\lambda y) $$ where $y>0$, $\lambda>0$ and $\alpha>0$. The mean of $Y$ is $(\psi(\alpha+1)-\psi(1))/\lambda$ (returned as the fitted values) where $\psi$ is the digamma function. The variance of $Y$ is $(\psi'(1)-\psi'(\alpha+1))/\lambda^2$ 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: $\alpha=1$ is the exponential distribution.

Value

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

References

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.

Author

T. W. Yee

Note

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.

Warning

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.

Examples

if (FALSE) { # \dontrun{
# 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 -152.264
} # }