cdfpe3 {lmom}

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))

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.

Value

cdfpe3 gives the distribution function; quape3 gives the quantile function.

Note

The functions expect the distribution parameters in a vector, rather than as separate arguments as in the standard R distribution functions pnorm, qnorm, etc.

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.

See Also

cdfgam for the gamma distribution.

cdfnor for the normal distribution.

Examples

# 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)))