exponential: Exponential Distribution

In VGAM: Vector Generalized Linear and Additive Models

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)

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.

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.

Author(s)

T. W. Yee

References

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

See Also

amlexponential, gpd, laplace, expgeometric, explogff, poissonff, mix2exp, freund61, simulate.vlm, Exponential.

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))

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)

VGAM documentation built on Sept. 19, 2023, 9:06 a.m.