Exponential Distribution

Maximum likelihood estimation for the exponential distribution.

Usage

exponential(link = "loglink", location = 0, expected = TRUE,
            type.fitted = c("mean", "percentiles", "Qlink"),
            percentiles = 50,
            ishrinkage = 0.95, parallel = FALSE, zero = NULL)

Arguments

link

Parameter link function applied to the positive parameter $rate$. See Links for more choices.

location

Numeric of length 1, the known location parameter, $A$, say.

expected

Logical. If TRUE Fisher scoring is used, otherwise Newton-Raphson. The latter is usually faster.

ishrinkage, parallel, zero

See CommonVGAMffArguments for information.

type.fitted, percentiles

See CommonVGAMffArguments for information.

Details

The family function assumes the response $Y$ has density $$f(y) = \lambda \exp(-\lambda (y-A))$$ for $y > A$, where $A$ is the known location parameter. By default, $A=0$. Then $E(Y) = A + 1/ \lambda$ and $Var(Y) = 1/ \lambda^2$.

Value

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

References

Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011). Statistical Distributions, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.

Author

T. W. Yee

Note

Suppose $A = 0$. For a fixed time interval, the number of events is Poisson with mean $\lambda$ if the time between events has a geometric distribution with mean $\lambda^{-1}$. The argument rate in exponential is the same as rexp etc. The argument lambda in rpois is somewhat the same as rate here.

Examples

edata <- data.frame(x2 = runif(nn <- 100) - 0.5)
edata <- transform(edata, x3 = runif(nn) - 0.5)
edata <- transform(edata, eta = 0.2 - 0.7 * x2 + 1.9 * x3)
edata <- transform(edata, rate = exp(eta))
edata <- transform(edata, y = rexp(nn, rate = rate))
with(edata, stem(y))
#> 
#>   The decimal point is at the |
#> 
#>   0 | 00000000001111111111112222222233333333344444444455555666667888888899
#>   1 | 111122344566788
#>   2 | 0123466
#>   3 | 469
#>   4 | 02
#>   5 | 
#>   6 | 
#>   7 | 
#>   8 | 1
#> 

fit.slow <- vglm(y ~ x2 + x3, exponential, data = edata, trace = TRUE)
#> Iteration 1: deviance = 131.89819
#> Iteration 2: deviance = 128.14608
#> Iteration 3: deviance = 128.05522
#> Iteration 4: deviance = 128.05496
#> Iteration 5: deviance = 128.05496
fit.fast <- vglm(y ~ x2 + x3, exponential(exp = FALSE), data = edata,
                 trace = TRUE, crit = "coef")
#> Iteration 1: coefficients = 
#>  0.6460197, -1.1990895,  2.5411612
#> Iteration 2: coefficients = 
#>  0.40326186, -0.99750891,  1.98503817
#> Iteration 3: coefficients = 
#>  0.34045523, -0.91758530,  1.78525925
#> Iteration 4: coefficients = 
#>  0.33582307, -0.91157421,  1.77067002
#> Iteration 5: coefficients = 
#>  0.33580009, -0.91154499,  1.77059908
#> Iteration 6: coefficients = 
#>  0.33580009, -0.91154499,  1.77059907
coef(fit.slow, mat = TRUE)
#>             loglink(rate)
#> (Intercept)     0.3357996
#> x2             -0.9115381
#> x3              1.7705873
summary(fit.slow)
#> 
#> Call:
#> vglm(formula = y ~ x2 + x3, family = exponential, data = edata, 
#>     trace = TRUE)
#> 
#> Coefficients: 
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)   0.3358     0.1008   3.331 0.000865 ***
#> x2           -0.9115     0.3181  -2.866 0.004158 ** 
#> x3            1.7706     0.3353   5.280 1.29e-07 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Name of linear predictor: loglink(rate) 
#> 
#> Residual deviance: 128.055 on 97 degrees of freedom
#> 
#> Log-likelihood: -73.0496 on 97 degrees of freedom
#> 
#> Number of Fisher scoring iterations: 5 
#> 


# Compare results with a GPD. Has a threshold.
threshold <- 0.5
gdata <- data.frame(y1 = threshold + rexp(n = 3000, rate = exp(1.5)))

fit.exp <- vglm(y1 ~ 1, exponential(location = threshold), data = gdata)
coef(fit.exp, matrix = TRUE)
#>             loglink(rate)
#> (Intercept)      1.506818
Coef(fit.exp)
#>    rate 
#> 4.51235 
logLik(fit.exp)
#> [1] 1520.454

fit.gpd <- vglm(y1 ~ 1, gpd(threshold =  threshold), data = gdata)
coef(fit.gpd, matrix = TRUE)
#>             loglink(scale) logofflink(shape, offset = 0.5)
#> (Intercept)      -1.504118                      -0.6985687
Coef(fit.gpd)
#>        scale        shape 
#>  0.222213165 -0.002703437 
logLik(fit.gpd)
#> [1] 1520.465