# ParetoUC: The Pareto Distribution In VGAM: Vector Generalized Linear and Additive Models

## Description

Density, distribution function, quantile function and random generation for the Pareto(I) distribution with parameters `scale` and `shape`.

## Usage

 ```1 2 3 4``` ```dpareto(x, scale = 1, shape, log = FALSE) ppareto(q, scale = 1, shape, lower.tail = TRUE, log.p = FALSE) qpareto(p, scale = 1, shape, lower.tail = TRUE, log.p = FALSE) rpareto(n, scale = 1, shape) ```

## Arguments

 `x, q` vector of quantiles. `p` vector of probabilities. `n` number of observations. Same as in `runif`. `scale, shape` the alpha and k parameters. `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 `paretoff`, the VGAM family function for estimating the parameter k by maximum likelihood estimation, for the formula of the probability density function and the range restrictions imposed on the parameters.

## Value

`dpareto` gives the density, `ppareto` gives the distribution function, `qpareto` gives the quantile function, and `rpareto` generates random deviates.

## Author(s)

T. W. Yee and Kai Huang

## References

Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011). Statistical Distributions, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.

`paretoff`, `ParetoIV`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```alpha <- 3; k <- exp(1); x <- seq(2.8, 8, len = 300) ## Not run: plot(x, dpareto(x, scale = alpha, shape = k), type = "l", main = "Pareto density split into 10 equal areas") abline(h = 0, col = "blue", lty = 2) qvec <- qpareto(seq(0.1, 0.9, by = 0.1), scale = alpha, shape = k) lines(qvec, dpareto(qvec, scale = alpha, shape = k), col = "purple", lty = 3, type = "h") ## End(Not run) pvec <- seq(0.1, 0.9, by = 0.1) qvec <- qpareto(pvec, scale = alpha, shape = k) ppareto(qvec, scale = alpha, shape = k) qpareto(ppareto(qvec, scale = alpha, shape = k), scale = alpha, shape = k) - qvec # Should be 0 ```

### Example output ```Loading required package: stats4