VGAM (version 0.9-3)

perks: Perks Distribution Family Function

Description

Maximum likelihood estimation of the 2-parameter Perks distribution.

Usage

perks(lshape = "loge", lscale = "loge",
      ishape = NULL,   iscale = NULL,
      gshape = exp(-5:5), gscale = exp(-5:5),
      nsimEIM = 500, oim.mean = FALSE, zero = NULL)

Arguments

lshape, lscale

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

ishape, iscale

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

gshape, gscale

See CommonVGAMffArguments.

nsimEIM, zero

See CommonVGAMffArguments.

oim.mean

To be currently ignored.

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. If the self-starting initial values fail then try experimenting with the initial value arguments, especially iscale. Successful convergence depends on having very good initial values. Also, monitor convergence by setting trace = TRUE.

Details

The Perks distribution has cumulative distribution function $$F(x; \alpha, \beta) = 1 - \left{ \frac{1 + \alpha}{1 + \alpha e^{\beta y}} \right}^{1 / \beta}$$ which leads to a probability density function $$f(x; \alpha, \beta) = \left[ 1 + \alpha \right]^{1 / \beta} \alpha e^{\beta y} / (1 + \alpha e^{\beta y})^{1 + 1 / \beta}$$ for $\alpha > 0$, $\beta > 0$, $x > 0$. 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.

References

Perks, W. (1932) On some experiments in the graduation of mortality statistics. Journal of the Institute of Actuaries, 63, 12--40.

Richards, S. J. (2012) A handbook of parametric survival models for actuarial use. Scandinavian Actuarial Journal. 1--25.

See Also

dperks.

Examples

set.seed(123)
pdata <- data.frame(x2 = runif(nn <- 1000))  # x2 unused
pdata <- transform(pdata, eta1  = -1,
                          ceta1 =  1)
pdata <- transform(pdata, shape1 = exp(eta1),
                          scale1 = exp(ceta1))
pdata <- transform(pdata,
                   y1 = rperks(nn, shape = shape1, scale = scale1))

fit1 <- vglm(y1 ~ 1, perks, data = pdata, trace = TRUE)
coef(fit1, matrix = TRUE)
summary(fit1)