exponential: Exponential Distribution

Description

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)

Value

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

Arguments

link: Parameter link function applied to the positive parameter $r a t e$ . 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.

Author

T. W. Yee

Details

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

References

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

Examples

Run this code

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))

fit.slow <- vglm(y ~ x2 + x3, exponential, data = edata, trace = TRUE)
fit.fast <- vglm(y ~ x2 + x3, exponential(exp = FALSE), data = edata,
                 trace = TRUE, crit = "coef")
coef(fit.slow, mat = TRUE)
summary(fit.slow)


# 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)
Coef(fit.exp)
logLik(fit.exp)

fit.gpd <- vglm(y1 ~ 1, gpd(threshold =  threshold), data = gdata)
coef(fit.gpd, matrix = TRUE)
Coef(fit.gpd)
logLik(fit.gpd)

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