# gompertz: Gompertz Regression Family Function In VGAM: Vector Generalized Linear and Additive Models

## Description

Maximum likelihood estimation of the 2-parameter Gompertz distribution.

## Usage

 ```1 2 3``` ```gompertz(lscale = "loglink", lshape = "loglink", 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.

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.

`dgompertz`, `makeham`, `simulate.vlm`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22``` ```## 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) ```

### Example output

```Loading required package: stats4
VGLM    linear loop  1 :  loglikelihood = -228.41701
VGLM    linear loop  2 :  loglikelihood = -227.64391
VGLM    linear loop  3 :  loglikelihood = -227.64213
VGLM    linear loop  4 :  loglikelihood = -227.64213
VGLM    linear loop  1 :  loglikelihood = -1293.0365
VGLM    linear loop  2 :  loglikelihood = -1290.0635
VGLM    linear loop  3 :  loglikelihood = -1289.9114
VGLM    linear loop  4 :  loglikelihood = -1289.9105
VGLM    linear loop  5 :  loglikelihood = -1289.9105
VGLM    linear loop  6 :  loglikelihood = -1289.9105
loge(scale) loge(shape)
(Intercept)    1.012078   -1.064231
scale     shape
2.7513114 0.3449929

Call:
vglm(formula = y1 ~ 1, family = gompertz, data = gdata, trace = TRUE)

Pearson residuals:
Min      1Q  Median     3Q   Max
loge(scale) -10.968 -0.1549  0.2934 0.5350 0.616
loge(shape)  -1.325 -0.8906 -0.1406 0.7675 2.175

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept):1  1.01208    0.03624   27.93   <2e-16 ***
(Intercept):2 -1.06423    0.07375  -14.43   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Number of linear predictors:  2

Names of linear predictors: loge(scale), loge(shape)

Log-likelihood: -227.6421 on 1998 degrees of freedom

Number of iterations: 4
loge(scale) loge(shape)
(Intercept) -0.82934894  -1.0713993
x2          -0.09195182   0.3595059

Call:
vglm(formula = y2 ~ x2, family = gompertz, data = gdata, trace = TRUE)

Pearson residuals:
Min      1Q Median     3Q    Max
loge(scale) -10.542 -0.3881 0.2011 0.6227 0.9884
loge(shape)  -2.525 -0.7816 0.1353 0.8732 1.5797

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept):1 -0.82935    0.13756  -6.029 1.65e-09 ***
(Intercept):2 -1.07140    0.10850  -9.874  < 2e-16 ***
x2:1          -0.09195    0.25631  -0.359    0.720
x2:2           0.35951    0.18809   1.911    0.056 .
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Number of linear predictors:  2

Names of linear predictors: loge(scale), loge(shape)

Log-likelihood: -1289.91 on 1996 degrees of freedom

Number of iterations: 6
```

VGAM documentation built on Jan. 16, 2021, 5:21 p.m.