VGAM (version 1.1-12)

Makeham: The Makeham Distribution

Description

Density, cumulative distribution function, quantile function and random generation for the Makeham distribution.

Usage

dmakeham(x, scale = 1, shape, epsilon = 0, log = FALSE)
pmakeham(q, scale = 1, shape, epsilon = 0, lower.tail = TRUE,
         log.p = FALSE)
qmakeham(p, scale = 1, shape, epsilon = 0, lower.tail = TRUE,
         log.p = FALSE)
rmakeham(n, scale = 1, shape, epsilon = 0)

Value

dmakeham gives the density,

pmakeham gives the cumulative distribution function,

qmakeham gives the quantile function, and

rmakeham generates random deviates.

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. Same as in runif.

log

Logical. If log = TRUE then the logarithm of the density is returned.

lower.tail, log.p

Same meaning as in pnorm or qnorm.

scale, shape

positive scale and shape parameters.

epsilon

another parameter. Must be non-negative. See below.

Author

T. W. Yee and Kai Huang

Details

See makeham for details. The default value of epsilon = 0 corresponds to the Gompertz distribution. The function pmakeham uses lambertW.

References

Jodra, P. (2009). A closed-form expression for the quantile function of the Gompertz-Makeham distribution. Mathematics and Computers in Simulation, 79, 3069--3075.

See Also

makeham, lambertW.

Examples

probs <- seq(0.01, 0.99, by = 0.01)
Shape <- exp(-1); Scale <- exp(1); eps = Epsilon <- exp(-1)
max(abs(pmakeham(qmakeham(probs, sca = Scale, Shape, eps = Epsilon),
    sca = Scale, Shape, eps = Epsilon) - probs))  # Should be 0

if (FALSE)  x <- seq(-0.1, 2.0, by = 0.01);
plot(x, dmakeham(x, sca = Scale, Shape, eps = Epsilon), type = "l",
     main = "Blue is density, orange is the CDF",
     sub = "Purple lines are the 10,20,...,90 percentiles",
     col = "blue", las = 1, ylab = "")
abline(h = 0, col = "blue", lty = 2)
lines(x, pmakeham(x, sca = Scale, Shape, eps = Epsilon), col = "orange")
probs <- seq(0.1, 0.9, by = 0.1)
Q <- qmakeham(probs, sca = Scale, Shape, eps = Epsilon)
lines(Q, dmakeham(Q, sca = Scale, Shape, eps = Epsilon),
      col = "purple", lty = 3, type = "h")
pmakeham(Q, sca = Scale, Shape, eps = Epsilon) - probs # Should be all 0
abline(h = probs, col = "purple", lty = 3)