# toppleUC: The Topp-Leone Distribution In VGAM: Vector Generalized Linear and Additive Models

## Description

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

## Usage

 ```1 2 3 4``` ```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.

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.

`topple`, `Triangle`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```## 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) ```