VGAM (version 1.1-14)

expexpff: Exponentiated Exponential Distribution

Description

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)

Value

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

and vgam.

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.

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.

See Also

expexpff1, gammaR, weibullR, CommonVGAMffArguments.

Examples

if (FALSE) {
# 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
}