The Topp-Leone Distribution

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

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.

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.

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.

Note

The Topp-Leone distribution is related to the triangle distribution.

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")
#> Error in h(simpleError(msg, call)): error in evaluating the argument 'y' in selecting a method for function 'plot': object 'shape' 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, ptopple(x, shape = shape), type = "l", col = "orange")
#> Error: object 'shape' not found
probs <- seq(0.1, 0.9, by = 0.1)
Q <- qtopple(probs, shape = shape)
#> Error: object 'shape' not found
lines(Q, dtopple(Q, shape), col = "purple", lty = 3, type = "h")
#> Error: object 'Q' not found
lines(Q, ptopple(Q, shape), col = "purple", lty = 3, type = "h")
#> Error: object 'Q' not found
abline(h = probs, col = "purple", lty = 3)
#> Error in int_abline(a = a, b = b, h = h, v = v, untf = untf, ...): plot.new has not been called yet
max(abs(ptopple(Q, shape) - probs))  # Should be zero
#> Error: object 'Q' not found
 # \dontrun{}