EPdist function

The Extended Pareto Distribution

Density, distribution function, quantile function and random generation for the Extended Pareto Distribution (EPD).

depd(x, gamma, kappa, tau = -1, log = FALSE)
pepd(x, gamma, kappa, tau = -1, lower.tail = TRUE, log.p = FALSE)
qepd(p, gamma, kappa, tau = -1, lower.tail = TRUE, log.p = FALSE)
repd(n, gamma, kappa, tau = -1)

Arguments

• x: Vector of quantiles.
• p: Vector of probabilities.
• n: Number of observations.
• gamma: The $\gamma$ parameter of the EPD, a strictly positive number.
• kappa: The $\kappa$ parameter of the EPD. It should be larger than $\max\{-1,1/\tau\}$.
• tau: The $\tau$ parameter of the EPD, a strictly negative number. Default is -1.
• log: Logical indicating if the densities are given as $\log(f)$, default is FALSE.
• lower.tail: Logical indicating if the probabilities are of the form $P(X\le x)$ (TRUE) or $P(X>x)$ (FALSE). Default is TRUE.
• log.p: Logical indicating if the probabilities are given as $\log(p)$, default is FALSE.

Details

The Cumulative Distribution Function (CDF) of the EPD is equal to $F(x) = 1-(x(1+\kappa-\kappa x^{\tau}))^{-1/\gamma}$ for all $x > 1$ and $F(x)=0$ otherwise.

Note that an EPD random variable with $\tau=-1$ and $\kappa=\gamma/\sigma-1$ is GPD distributed with $\mu=1$, $\gamma$ and $\sigma$.

Returns

depd gives the density function evaluated in $x$, pepd the CDF evaluated in $x$ and qepd the quantile function evaluated in $p$. The length of the result is equal to the length of $x$ or $p$.

repd returns a random sample of length $n$.

References

Beirlant, J., Joossens, E. and Segers, J. (2009). "Second-Order Refined Peaks-Over-Threshold Modelling for Heavy-Tailed Distributions." Journal of Statistical Planning and Inference, 139, 2800--2815.

Author(s)

Tom Reynkens.

See Also

Pareto, GPD, Distributions

Examples

# Plot of the PDF
x <- seq(0, 10, 0.01)
plot(x, depd(x, gamma=1/2, kappa=1, tau=-1), xlab="x", ylab="PDF", type="l")

# Plot of the CDF
x <- seq(0, 10, 0.01)
plot(x, pepd(x, gamma=1/2, kappa=1, tau=-1), xlab="x", ylab="CDF", type="l")