R: Exponentiated Exponential Distribution

expexpff {VGAM} | R Documentation

## 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)

### Arguments

`lshape, lrate` | Parameter link functions for the shape and rate parameters. See `Links` for more choices. The defaults ensure both parameters are positive.
`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 `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;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.

### 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 `expexpff1`.

### 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

`expexpff1`, `gammaR`, `weibullR`, `CommonVGAMffArguments`.

### Examples

    # 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

[Package VGAM version 1.1-3 Index]