R: Gompertz Distribution Family Function

gompertz {VGAM} | R Documentation

## Gompertz Distribution Family Function

### Description

Maximum likelihood estimation of the 2-parameter Gompertz distribution.

### Usage

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

### Arguments

`nowarning` | Logical. Suppress a warning? Ignored for VGAM 0.9-7 and higher.
`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`.

### Details

The Gompertz distribution has a cumulative distribution function

F(x;alpha, beta) = 1 - exp(-(alpha/beta) * (exp(beta * x) - 1) )

which leads to a probability density function

f(x; alpha, beta) = alpha * exp[-beta * x] * exp[-(alpha/beta) * (exp(beta * x) - 1) ]

for a > 0, b > 0, x > 0. Here, β is called the scale parameter `scale`, and α is called the shape parameter (one could refer to a as a location parameter and b 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.

### 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.

### Author(s)

T. W. Yee

### 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`, `simulate.vlm`.

### Examples

    ## Not run:
    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, scale = scale1, shape = shape1),
                              y2 = rgompertz(nn, scale = scale2, shape = shape2))

    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)

    ## End(Not run)

[Package VGAM version 1.0-3 Index]