R: Makeham Distribution Family Function

makeham {VGAM} | R Documentation

## Makeham Distribution Family Function

### Description

Maximum likelihood estimation of the 3-parameter Makeham distribution.

### Usage

    makeham(lscale = "loge", lshape = "loge", lepsilon = "loge",
            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

`nowarning` | Logical. Suppress a warning? Ignored for VGAM 0.9-7 and higher.
`lshape, lscale, lepsilon` | Parameter link functions applied to the shape parameter `shape`, scale parameter `scale`, and other parameter `epsilon`. All parameters are treated as positive here (cf. `dmakeham` allows `epsilon = 0`, etc.). See `Links` for more choices.

`ishape, iscale, iepsilon` | Optional initial values. A `NULL` means a value is computed internally. A value must be given for `iepsilon` currently, and this is a sensitive parameter!
`gshape, gscale, gepsilon` | See `CommonVGAMffArguments`.
`nsimEIM, zero` | See `CommonVGAMffArguments`. Argument `probs.y` is used only when `imethod = 2`.
`oim.mean` | To be currently ignored.

### Details

The Makeham distribution, which adds another parameter to the Gompertz distribution, has cumulative distribution function

F(y; alpha, beta, epsilon) = 1 - exp(-y * epsilon + (alpha / beta) * [1 - e^(beta * y)])

which leads to a probability density function

f(y; alpha, beta, epsilon) = (epsilon + alpha * e^(beta y) ) * exp(-y * epsilon + (alpha / beta) * [1 - e^(beta * y)])

for alpha > 0, beta > 0, epsilon >= 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.

### Value

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

### Warning

A lot of care is needed because this is a rather difficult distribution for parameter estimation, especially when the shape parameter is large relative to the scale parameter. If the self-starting initial values fail then try experimenting with the initial value arguments, especially `iepsilon`. Successful convergence depends on having very good initial values. More improvements could be made here. Also, monitor convergence by setting `trace = TRUE`.

A trick is to fit a `gompertz` distribution and use it for initial values; see below. However, this family function is currently numerically fraught.

### Author(s)

T. W. Yee

### See Also

`dmakeham`, `gompertz`, `simulate.vlm`.

### Examples

    ## Not run:  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)

    ## End(Not run)

[Package VGAM version 1.0-3 Index]