The Makeham Distribution

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

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)

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.

Value

dmakeham gives the density, pmakeham gives the cumulative distribution function, qmakeham gives the quantile function, and rmakeham generates random deviates.

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
#> [1] 5.551115e-16

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 = "")
#> Error in h(simpleError(msg, call)): error in evaluating the argument 'x' in selecting a method for function 'plot': object 'x' not found
abline(h = 0, col = "blue", lty = 2)
#> Error in int_abline(a = a, b = b, h = h, v = v, untf = untf, ...): plot.new has not been called yet
lines(x, pmakeham(x, sca = Scale, Shape, eps = Epsilon), col = "orange")
#> Error: object 'x' not found
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")
#> Error in plot.xy(xy.coords(x, y), type = type, ...): plot.new has not been called yet
pmakeham(Q, sca = Scale, Shape, eps = Epsilon) - probs # Should be all 0
#> [1] -4.163336e-17  5.551115e-17  5.551115e-17 -1.110223e-16  0.000000e+00
#> [6]  1.110223e-16  4.440892e-16 -5.551115e-16 -1.110223e-16
abline(h = probs, col = "purple", lty = 3)  # \dontrun{}
#> Error in int_abline(a = a, b = b, h = h, v = v, untf = untf, ...): plot.new has not been called yet