pdfpe3: Probability Density Function of the Pearson Type III Distribution

Description

This function computes the probability density of the Pearson Type III distribution given parameters ($\mu$, $\sigma$, and $\gamma$) computed by parpe3. These parameters are equal to the product moments (pmoms): mean, standard deviation, and skew. The probability density function for $\gamma \ne 0$ is $$f(x) = \frac{Y^{\alpha -1} \exp({-Y/\beta})} {\beta^\alpha\, \Gamma(\alpha)} \mbox{,}$$ where $f(x)$ is the probability density for quantile $x$, $\Gamma$ is the complete gamma function in R as gamma, $\xi$ is a location parameter, $\beta$ is a scale parameter, $\alpha$ is a shape parameter, and $Y = x - \xi$ for $\gamma > 0$ and $Y = \xi - x$ for $\gamma < 0$. These three “new” parameters are related to the product moments ($\mu$, mean; $\sigma$, standard deviation; $\gamma$, skew) by $$\alpha = 4/\gamma^2 \mbox{,}$$ $$\beta = \frac{1}{2}\sigma |\gamma| \mbox{,\ and}$$ $$\xi = \mu - 2\sigma/\gamma \mbox{.}$$ If $\gamma = 0$, the distribution is symmetrical and simply is the probability density Normal distribution with mean and standard deviation of $\mu$ and $\sigma$, respectively. Internally, the $\gamma = 0$ condition is implemented by R function dnorm. The PearsonDS package supports the Pearson distribution system including the Type III (see Examples).

Usage

pdfpe3(x, para)

Value

Probability density ($f$) for $x$.

Arguments

x: A real value vector.
para: The parameters from parpe3 or vec2par.

Author

W.H. Asquith

References

Hosking, J.R.M., 1990, L-moments---Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, v. 52, pp. 105--124.