expexp {VGAM}R Documentation

Exponentiated Exponential Distribution

Description

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

Usage

expexp(lshape = "loge", lscale = "loge",
       eshape=list(), escale=list(),
       ishape = 1.1, iscale = NULL, 
       tolerance = 1.0e-6, zero = NULL)

Arguments

lshape, lscale Parameter link functions for the shape and scale parameters. See Links for more choices. The defaults ensure both parameters are positive.
eshape, escale List. Extra argument for each of the links. See earg in Links for general information.
ishape Initial value for the shape parameter. If convergence fails try setting a different value for this argument.
iscale Initial value for the scale 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}.

Details

The exponentiated exponential distribution is an alternative to the Weibull and the gamma distributions. The formula for the density is

f(y;shape,scale) = shape scale (1-exp(-scale y))^(shape-1) exp(-scale y)

where y>0, shape>0 and scale>0. The mean of Y is (psi(shape+1)-psi(1))/scale (returned as the fitted values) where psi is the digamma function. The variance of Y is (psi'(1)-psi'(shape+1))/ scale^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.

Value

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

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.

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 expexp1.

Author(s)

T. W. Yee

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

expexp1, gamma2.ab, weibull.

Examples

# A special case: exponential data
y = rexp(n <- 1000)
fit = vglm(y ~ 1, fam=expexp, trace=TRUE, maxit=99)
coef(fit, matrix=TRUE)
Coef(fit)

# Ball bearings data (number of million revolutions before failure)
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=expexp(iscale=0.05, ish=5),
           trace=TRUE, maxit=300)
coef(fit, matrix=TRUE)
Coef(fit)   # Authors get c(shape=5.2589, scale=0.0314)
logLik(fit) # Authors get -112.9763

# Failure times of the airconditioning system of an airplane
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=expexp(ishape=0.8, isc=0.15),
           trace=TRUE, maxit=99)
coef(fit, matrix=TRUE)
Coef(fit)   # Authors get c(shape=0.8130, scale=0.0145)
logLik(fit) # Authors get log-lik -152.264

[Package VGAM version 0.7-7 Index]