cdfpe3: Pearson type III distribution

Description

Distribution function and quantile function of the Pearson type III distribution

Usage

cdfpe3(x, para = c(0, 1, 0))
quape3(f, para = c(0, 1, 0))

Value

cdfpe3 gives the distribution function;

quape3 gives the quantile function.

Arguments

x: Vector of quantiles.
f: Vector of probabilities.
para: Numeric vector containing the parameters of the distribution, in the order $\mu, \sigma, \gamma$ (location, scale, shape).

Details

The Pearson type III distribution contains as special cases the usual three-parameter gamma distribution (a shifted version of the gamma distribution) with a finite lower bound and positive skewness; the normal distribution, and the reverse three-parameter gamma distribution, with a finite upper bound and negative skewness. The distribution's parameters are the first three (ordinary) moment ratios: $\mu$ (the mean, a location parameter), $\sigma$ (the standard deviation, a scale parameter) and $\gamma$ (the skewness, a shape parameter).

If $\gamma\ne0$, let $\alpha=4/\gamma^2$, $\beta={\scriptstyle 1 \over \scriptstyle 2}\sigma|\gamma|$, $\xi=\mu-2\sigma/\gamma$. The probability density function is $$f(x)={|x-\xi|^{\alpha-1}\exp(-|x-\xi|/\beta) \over \beta^\alpha\Gamma(\alpha)}$$ with $x$ bounded by $\xi$ from below if $\gamma>0$ and from above if $\gamma<0$. If $\gamma=0$, the distribution is a normal distribution with mean $\mu$ and standard deviation $\sigma$.

The Pearson type III distribution is usually regarded as consisting of just the case $\gamma>0$ given above, and is usually parametrized by $\alpha$, $\beta$ and $\xi$. Our parametrization extends the distribution to include the usual Pearson type III distributions, with positive skewness and lower bound $\xi$, reverse Pearson type III distributions, with negative skewness and upper bound $\xi$, and the Normal distribution, which is included as a special case of the distribution rather than as the unattainable limit $\alpha\rightarrow\infty$. This enables the Pearson type III distribution to be used when the skewness of the observed data may be negative. The parameters $\mu$, $\sigma$ and $\gamma$ are the conventional moments of the distribution.

The gamma distribution is obtained when $\gamma>0$ and $\mu=2\sigma/\gamma$. The normal distribution is the special case $\gamma=0$. The exponential distribution is the special case $\gamma=2$.

References

Hosking, J. R. M. and Wallis, J. R. (1997). Regional frequency analysis: an approach based on L-moments, Cambridge University Press, Appendix A.10.

Examples

Run this code

# Random sample from the Pearson type III distribution
# with parameters mu=1, alpha=2, gamma=3.
quape3(runif(100), c(1,2,3))

# The Pearson type III distribution with parameters
# mu=12, sigma=6, gamma=1, is the gamma distribution
# with parameters alpha=4, beta=3.  An illustration:
fval<-seq(0.1,0.9,by=0.1)
cbind(fval, qgamma(fval, shape=4, scale=3), quape3(fval, c(12,6,1)))

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