R: The Topp-Leone Distribution

Topple {VGAM} | R Documentation

## 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`.

### 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.

### Value

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

### Note

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

### Author(s)

T. W. Yee

### 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

    ## End(Not run)

[Package VGAM version 1.1-3 Index]