tolerance (version 3.0.0)

DiscretePareto: Discrete Pareto Distribution

Description

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

Usage

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)

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.

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 \(P[X\le x]\), else \(P[X>x]\).

Details

The discrete Pareto distribution has mass $$p(x) = \theta^{\log(1+x)}-\theta^{\log(2+x)},$$ where \(x=0,1,\ldots\) and \(0<\theta<1\) is the shape parameter.

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

## 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)