Pearson type III distribution
Distribution function and quantile function of the Pearson type III distribution
cdfpe3(x, para = c(0, 1, 0)) quape3(f, para = c(0, 1, 0))
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). |
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 != 0, let alpha=4/gamma^2, (1/2) sigma |gamma|, xi = mu - 2 sigma / gamma. The probability density function is
f(x) = |x-xi|^(α-1) exp(-|x-ξ|/β) / ( 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 -> infinity. 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.
cdfpe3 gives the distribution function;
quape3 gives the quantile function.
The functions expect the distribution parameters in a vector,
rather than as separate arguments as in the standard R
distribution functions pnorm, qnorm, etc.
Hosking, J. R. M. and Wallis, J. R. (1997). Regional frequency analysis: an approach based on L-moments, Cambridge University Press, Appendix A.10.
cdfgam for the gamma distribution.
cdfnor for the normal distribution.
# 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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