# dpareto: Discrete Pareto Distribution In tolerance: Statistical Tolerance Intervals and Regions

## Description

Density (mass), distribution function, quantile function, and random generation for the discrete Pareto distribution.

## Usage

 ```1 2 3 4``` ```ddpareto(x, theta, log = FALSE) pdpareto(q, theta, lower.tail = TRUE, log.p = FALSE) qdpareto(p, theta, lower.tail = TRUE, log.p = FALSE) rdpareto(n, theta) ```

## Arguments

 `x, q` Vector of quantiles. `p` Vector of probabilities. `n` The number of observations. If `length>1`, then the length is taken to be the number required. `theta` The shape parameter, which must be greater than 0 and less than 1. `log, log.p` Logical vectors. If `TRUE`, then the probabilities are given as `log(p)`. `lower.tail` Logical vector. If `TRUE`, then probabilities are P[X≤ x], else P[X>x].

## Details

The discrete Pareto distribution has mass

p(x) = θ^(\log(1+x))-θ^(\log(2+x)),

where x=0,1,… and 0<θ<1 is the shape parameter.

## Value

`ddpareto` gives the density (mass), `pdpareto` gives the distribution function, `qdpareto` gives the quantile function, and `rdpareto` generates random deviates for the specified distribution.

## References

Krishna, H. and Pundir, P. S. (2009), Discrete Burr and Discrete Pareto Distributions, Statistical Methodology, 6, 177–188.

Young, D. S., Naghizadeh Qomi, M., and Kiapour, A. (2019), Approximate Discrete Pareto Tolerance Limits for Characterizing Extremes in Count Data, Statistica Neerlandica, 73, 4–21.

## See Also

`runif` and `.Random.seed` about random number generation.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```## Randomly generated data from the discrete Pareto ## distribution. set.seed(100) x <- rdpareto(n = 150, theta = 0.2) hist(x, main = "Randomly Generated Data", prob = TRUE) x.1 <- sort(x) y <- ddpareto(x = x.1, theta = 0.2) lines(x.1, y, col = 2, lwd = 2) plot(x.1, pdpareto(q = x.1, theta = 0.2), type = "l", xlab = "x", ylab = "Cumulative Probabilities") qdpareto(p = 0.80, theta = 0.2, lower.tail = FALSE) qdpareto(p = 0.95, theta = 0.2) ```

### Example output  ```Warning messages:
1: In rgl.init(initValue, onlyNULL) : RGL: unable to open X11 display
2: 'rgl.init' failed, running with 'rgl.useNULL = TRUE'.
 0
 4
```

tolerance documentation built on Feb. 6, 2020, 5:08 p.m.