Exponentiated Exponential Distribution

Estimates the two parameters of the exponentiated exponential distribution by maximizing a profile (concentrated) likelihood.

expexpff1(lrate = "loglink", irate = NULL, ishape = 1)

Arguments

lrate

Parameter link function for the (positive) \(\lambda\) parameter. See Links for more choices.

irate

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

ishape

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

Details

See expexpff for details about the exponentiated exponential distribution. This family function uses a different algorithm for fitting the model. Given \(\lambda\), the MLE of \(\alpha\) can easily be solved in terms of \(\lambda\). This family function maximizes a profile (concentrated) likelihood with respect to \(\lambda\). Newton-Raphson is used, which compares with Fisher scoring with expexpff.

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.

Author

T. W. Yee

Note

This family function works only for intercept-only models, i.e., y ~ 1 where y is the response.

The estimate of \(\alpha\) is attached to the misc slot of the object, which is a list and contains the component shape.

As Newton-Raphson is used, the working weights are sometimes negative, and some adjustment is made to these to make them positive.

Like expexpff, good initial values are needed. Convergence may be slow.

Warning

The standard errors produced by a summary of the model may be wrong.

See also

expexpff, CommonVGAMffArguments.

Examples

# 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, expexpff1(ishape = 4), trace = TRUE,
            maxit = 250, checkwz = FALSE, data = edata)
#> Applying Greenstadt modification to 23 matrices
#> Iteration 1: loglikelihood = -142.86097
#> Applying Greenstadt modification to 19 matrices
#> Iteration 2: loglikelihood = -138.55752
#> Applying Greenstadt modification to 18 matrices
#> Iteration 3: loglikelihood = -134.61146
#> Applying Greenstadt modification to 18 matrices
#> Iteration 4: loglikelihood = -131.02887
#> Applying Greenstadt modification to 18 matrices
#> Iteration 5: loglikelihood = -127.81272
#> Applying Greenstadt modification to 18 matrices
#> Iteration 6: loglikelihood = -124.96291
#> Applying Greenstadt modification to 18 matrices
#> Iteration 7: loglikelihood = -122.47572
#> Applying Greenstadt modification to 17 matrices
#> Iteration 8: loglikelihood = -120.34414
#> Applying Greenstadt modification to 17 matrices
#> Iteration 9: loglikelihood = -118.55463
#> Applying Greenstadt modification to 18 matrices
#> Iteration 10: loglikelihood = -117.08958
#> Applying Greenstadt modification to 18 matrices
#> Iteration 11: loglikelihood = -115.9242
#> Applying Greenstadt modification to 18 matrices
#> Iteration 12: loglikelihood = -115.02504
#> Applying Greenstadt modification to 18 matrices
#> Iteration 13: loglikelihood = -114.35458
#> Applying Greenstadt modification to 19 matrices
#> Iteration 14: loglikelihood = -113.87332
#> Applying Greenstadt modification to 20 matrices
#> Iteration 15: loglikelihood = -113.5415
#> Applying Greenstadt modification to 23 matrices
#> Iteration 16: loglikelihood = -113.32227
#> Applying Greenstadt modification to 23 matrices
#> Iteration 17: loglikelihood = -113.18287
#> Applying Greenstadt modification to 23 matrices
#> Iteration 18: loglikelihood = -113.09699
#> Applying Greenstadt modification to 23 matrices
#> Iteration 19: loglikelihood = -113.04553
#> Applying Greenstadt modification to 23 matrices
#> Iteration 20: loglikelihood = -113.01541
#> Applying Greenstadt modification to 23 matrices
#> Iteration 21: loglikelihood = -112.99811
#> Applying Greenstadt modification to 23 matrices
#> Iteration 22: loglikelihood = -112.98834
#> Applying Greenstadt modification to 23 matrices
#> Iteration 23: loglikelihood = -112.98288
#> Applying Greenstadt modification to 23 matrices
#> Iteration 24: loglikelihood = -112.97986
#> Applying Greenstadt modification to 23 matrices
#> Iteration 25: loglikelihood = -112.9782
#> Applying Greenstadt modification to 23 matrices
#> Iteration 26: loglikelihood = -112.9773
#> Applying Greenstadt modification to 23 matrices
#> Iteration 27: loglikelihood = -112.9768
#> Applying Greenstadt modification to 23 matrices
#> Iteration 28: loglikelihood = -112.97654
#> Applying Greenstadt modification to 23 matrices
#> Iteration 29: loglikelihood = -112.97639
#> Applying Greenstadt modification to 23 matrices
#> Iteration 30: loglikelihood = -112.97631
#> Applying Greenstadt modification to 23 matrices
#> Iteration 31: loglikelihood = -112.97627
#> Applying Greenstadt modification to 23 matrices
#> Iteration 32: loglikelihood = -112.97625
#> Applying Greenstadt modification to 23 matrices
#> Iteration 33: loglikelihood = -112.97624
#> Applying Greenstadt modification to 23 matrices
#> Iteration 34: loglikelihood = -112.97623
#> Applying Greenstadt modification to 23 matrices
#> Iteration 35: loglikelihood = -112.97623
#> Applying Greenstadt modification to 23 matrices
#> Iteration 36: loglikelihood = -112.97622
coef(fit, matrix = TRUE)
#>             loglink(rate)
#> (Intercept)      -3.43239
Coef(fit)  # Authors get c(0.0314, 5.2589) with log-lik -112.9763
#>       rate 
#> 0.03230964 
logLik(fit)
#> [1] -112.9762
fit@misc$shape  # Estimate of shape
#> [1] 5.288181


# 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, expexpff1(ishape = 0.8), trace = TRUE,
            maxit = 50, checkwz = FALSE, data = eedata)
#> Applying Greenstadt modification to 30 matrices
#> Iteration 1: loglikelihood = -229.62305
#> Applying Greenstadt modification to 29 matrices
#> Iteration 2: loglikelihood = -223.12728
#> Applying Greenstadt modification to 29 matrices
#> Iteration 3: loglikelihood = -216.82414
#> Applying Greenstadt modification to 29 matrices
#> Iteration 4: loglikelihood = -210.72536
#> Applying Greenstadt modification to 29 matrices
#> Iteration 5: loglikelihood = -204.84383
#> Applying Greenstadt modification to 29 matrices
#> Iteration 6: loglikelihood = -199.19363
#> Applying Greenstadt modification to 30 matrices
#> Iteration 7: loglikelihood = -193.79011
#> Applying Greenstadt modification to 30 matrices
#> Iteration 8: loglikelihood = -188.64997
#> Applying Greenstadt modification to 30 matrices
#> Iteration 9: loglikelihood = -183.79108
#> Applying Greenstadt modification to 30 matrices
#> Iteration 10: loglikelihood = -179.2324
#> Applying Greenstadt modification to 30 matrices
#> Iteration 11: loglikelihood = -174.99364
#> Applying Greenstadt modification to 30 matrices
#> Iteration 12: loglikelihood = -171.0948
#> Applying Greenstadt modification to 30 matrices
#> Iteration 13: loglikelihood = -167.55531
#> Applying Greenstadt modification to 30 matrices
#> Iteration 14: loglikelihood = -164.39285
#> Applying Greenstadt modification to 30 matrices
#> Iteration 15: loglikelihood = -161.62175
#> Applying Greenstadt modification to 30 matrices
#> Iteration 16: loglikelihood = -159.25086
#> Applying Greenstadt modification to 30 matrices
#> Iteration 17: loglikelihood = -157.28105
#> Applying Greenstadt modification to 30 matrices
#> Iteration 18: loglikelihood = -155.70247
#> Applying Greenstadt modification to 30 matrices
#> Iteration 19: loglikelihood = -154.4922
#> Applying Greenstadt modification to 30 matrices
#> Iteration 20: loglikelihood = -153.61294
#> Applying Greenstadt modification to 30 matrices
#> Iteration 21: loglikelihood = -153.01404
#> Applying Greenstadt modification to 30 matrices
#> Iteration 22: loglikelihood = -152.6357
#> Applying Greenstadt modification to 30 matrices
#> Iteration 23: loglikelihood = -152.41607
#> Applying Greenstadt modification to 27 matrices
#> Iteration 24: loglikelihood = -152.29971
#> Applying Greenstadt modification to 20 matrices
#> Iteration 25: loglikelihood = -152.24343
#> Applying Greenstadt modification to 18 matrices
#> Iteration 26: loglikelihood = -152.21827
#> Applying Greenstadt modification to 18 matrices
#> Iteration 27: loglikelihood = -152.20765
#> Applying Greenstadt modification to 18 matrices
#> Iteration 28: loglikelihood = -152.20337
#> Applying Greenstadt modification to 17 matrices
#> Iteration 29: loglikelihood = -152.20168
#> Applying Greenstadt modification to 17 matrices
#> Iteration 30: loglikelihood = -152.20104
#> Applying Greenstadt modification to 17 matrices
#> Iteration 31: loglikelihood = -152.20079
#> Applying Greenstadt modification to 17 matrices
#> Iteration 32: loglikelihood = -152.2007
#> Applying Greenstadt modification to 16 matrices
#> Iteration 33: loglikelihood = -152.20066
#> Applying Greenstadt modification to 16 matrices
#> Iteration 34: loglikelihood = -152.20065
#> Applying Greenstadt modification to 16 matrices
#> Iteration 35: loglikelihood = -152.20065
#> Applying Greenstadt modification to 16 matrices
#> Iteration 36: loglikelihood = -152.20064
coef(fit, matrix = TRUE)
#>             loglink(rate)
#> (Intercept)     -4.230272
Coef(fit)  # Authors get c(0.0145, 0.8130) with log-lik -152.264
#>       rate 
#> 0.01454843 
logLik(fit)
#> [1] -152.2006
fit@misc$shape  # Estimate of shape
#> [1] 0.8095687