cdfpe3 | R Documentation |

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 |

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`

.

`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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