VGAM (version 1.1-4)

Topple: The Topp-Leone Distribution

Description

Density, distribution function, quantile function and random generation for the Topp-Leone distribution.

Usage

dtopple(x, shape, log = FALSE)
ptopple(q, shape, lower.tail = TRUE, log.p = FALSE)
qtopple(p, shape)
rtopple(n, shape)

Arguments

x, q, p, n

Same as Uniform.

shape

the (shape) parameter, which lies in \((0, 1)\).

log

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

lower.tail, log.p

Same meaning as in pnorm or qnorm.

Value

dtopple gives the density, ptopple gives the distribution function, qtopple gives the quantile function, and rtopple generates random deviates.

Details

See topple, the VGAM family function for estimating the (shape) parameter \(s\) by maximum likelihood estimation, for the formula of the probability density function.

References

Topp, C. W. and F. C. Leone (1955). A family of J-shaped frequency functions. Journal of the American Statistical Association, 50, 209--219.

See Also

topple, Triangle.

Examples

# NOT RUN {
 shape <- 0.7; x <- seq(0.02, 0.999, length = 300)
plot(x, dtopple(x, shape = shape), type = "l", col = "blue", las = 1,
     main = "Blue is density, orange is cumulative distribution function",
     sub = "Purple lines are the 10,20,...,90 percentiles", ylab = "")
abline(h = 0, col = "blue", lty = 2)
lines(x, ptopple(x, shape = shape), type = "l", col = "orange")
probs <- seq(0.1, 0.9, by = 0.1)
Q <- qtopple(probs, shape = shape)
lines(Q, dtopple(Q, shape), col = "purple", lty = 3, type = "h")
lines(Q, ptopple(Q, shape), col = "purple", lty = 3, type = "h")
abline(h = probs, col = "purple", lty = 3)
max(abs(ptopple(Q, shape) - probs))  # Should be zero
# }