VGAM (version 0.9-1)

gompertz: Gompertz Distribution Family Function

Description

Maximum likelihood estimation of the 2-parameter Gompertz distribution.

Usage

gompertz(lshape = "loge", lscale = "loge",
         ishape = NULL,   iscale = NULL,
         nsimEIM = 500, zero = NULL)

Arguments

lshape, lscale

Parameter link functions applied to the shape parameter a, scale parameter scale. All parameters are positive. See Links for more choices.

ishape, iscale

Optional initial values. A NULL means a value is computed internally.

nsimEIM, zero

See CommonVGAMffArguments.

Value

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

Warning

The same warnings in makeham apply here too.

Details

The Gompertz distribution has a cumulative distribution function $$F(x;\alpha, \beta) = 1 - \exp[-(\alpha/\beta) \times (\exp(\beta x) - 1) ]$$ which leads to a probability density function $$f(x; \alpha, \beta) = \alpha \exp(\beta x) \exp [-(\alpha/\beta) \times (\exp(\beta x) - 1) ]$$ for $\alpha > 0$, $\beta > 0$, $x > 0$. Here, $\beta$ is called the scale parameter scale, and $\alpha$ is called the shape parameter (one could refer to $\alpha$ as a location parameter and $\beta$ as a shape parameter---see Lenart (2012)). The mean is involves an exponential integral function. Simulated Fisher scoring is used and multiple responses are handled.

The Makeham distibution has an additional parameter compared to the Gompertz distribution. If $X$ is defined to be the result of sampling from a Gumbel distribution until a negative value $Z$ is produced, then $X = -Z$ has a Gompertz distribution.

References

Lenart, A. (2012) The moments of the Gompertz distribution and maximum likelihood estimation of its parameters. Scandinavian Actuarial Journal, in press.

See Also

dgompertz, makeham.

Examples

gdata <- data.frame(x2 = runif(nn <- 1000))
gdata <- transform(gdata, eta1  = -1,
                          eta2  = -1 + 0.2 * x2,
                          ceta1 =  1,
                          ceta2 = -1 + 0.2 * x2)
gdata <- transform(gdata, shape1 = exp(eta1),
                          shape2 = exp(eta2),
                          scale1 = exp(ceta1),
                          scale2 = exp(ceta2))
gdata <- transform(gdata, y1 = rgompertz(nn, shape = shape1, scale = scale1),
                          y2 = rgompertz(nn, shape = shape2, scale = scale2))

fit1 <- vglm(y1 ~ 1,  gompertz, data = gdata, trace = TRUE)
fit2 <- vglm(y2 ~ x2, gompertz, data = gdata, trace = TRUE)
coef(fit1, matrix = TRUE)
Coef(fit1)
summary(fit1)
coef(fit2, matrix = TRUE)
summary(fit2)