VGAM (version 1.1-13)

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)

Value

dtopple gives the density,

ptopple gives the distribution function,

qtopple gives the quantile function, and

rtopple generates random deviates.

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.

Author

T. W. Yee

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

if (FALSE)  shape <- 0.7; x <- seq(0.02, 0.999, length = 300)
plot(x, dtopple(x, shape = shape), type = "l", col = "blue",
     main = "Blue is density, orange is CDF", ylab = "", las = 1,
     sub = "Purple lines are the 10,20,...,90 percentiles")
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