makeham: Makeham Regression Family Function

Description

Maximum likelihood estimation of the 3-parameter Makeham distribution.

Usage

makeham(lscale = "loglink", lshape = "loglink", lepsilon = "loglink",
        iscale = NULL,   ishape = NULL,   iepsilon = NULL,
        gscale = exp(-5:5),gshape = exp(-5:5), gepsilon = exp(-4:1),
        nsimEIM = 500, oim.mean = TRUE, zero = NULL, nowarning = FALSE)

Arguments

Value

An object of class "vglmff"

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

Details

The Makeham distribution, which adds another parameter to the Gompertz distribution, has cumulative distribution function $$F(y; \alpha, \beta, \varepsilon) = 1 - \exp \left\{ -y \varepsilon + \frac {\alpha}{\beta} \left[ 1 - e^{\beta y} \right] \right\} $$ which leads to a probability density function $$f(y; \alpha, \beta, \varepsilon) = \left[ \varepsilon + \alpha e^{\beta y} \right] \; \exp \left\{ -y \varepsilon + \frac {\alpha}{\beta} \left[ 1 - e^{\beta y} \right] \right\}, $$ for $\alpha > 0$, $\beta > 0$, $\varepsilon \geq 0$, $y > 0$. Here, $\beta$ is called the scale parameter scale, and $\alpha$ is called a shape parameter. The moments for this distribution do not appear to be available in closed form.

Simulated Fisher scoring is used and multiple responses are handled.

Examples

Run this code

if (FALSE)  set.seed(123)
mdata <- data.frame(x2 = runif(nn <- 1000))
mdata <- transform(mdata, eta1  = -1,
                          ceta1 =  1,
                          eeta1 = -2)
mdata <- transform(mdata, shape1 = exp(eta1),
                          scale1 = exp(ceta1),
                          epsil1 = exp(eeta1))
mdata <- transform(mdata,
         y1 = rmakeham(nn, shape = shape1, scale = scale1, eps = epsil1))

# A trick is to fit a Gompertz distribution first
fit0 <- vglm(y1 ~ 1, gompertz, data = mdata, trace = TRUE)
fit1 <- vglm(y1 ~ 1, makeham, data = mdata,
             etastart = cbind(predict(fit0), log(0.1)), trace = TRUE)

coef(fit1, matrix = TRUE)
summary(fit1)