# Estimating parameters from quantiles, Pearson III distribution

The focus of this blog is similar to earlier posts (here and here); if we know the quantiles of a distribution, how do we determine the parameters? Here I’m interested in the Pearson III distribution which is commonly used in flood hydrology.

Usually, hydrologists use the frequency factor method of calculating quantiles as developed by Chow (1951).  When the logarithms of the floods are Pearson III distributed, the quantile function is:

$\log_e(Q_{aep}) = m + s \times K (g, aep)$

Where

• $Q_{aep}$ is the flood quantile for a particular Annual Exceedance Probability (AEP)
• $m$ is the mean parameter of the Pearson III distribution (mean of the logarithms of the flood magnitudes)
• $s$ is the standard deviation parameter of the Pearson III distribution (standard deviation of the logarithms of the flood magnitudes)
• $g$ is the skew parameter of the Pearson III distribution (skew of the logarithms of the flood magnitudes)
• $K$ is the frequency factor, which is a function of $g$ and AEP.

Calculation of frequency factors is discussed here.

Assume three quantiles at three Annual Exceedance Probabilities.

$\log_e(Q_1) = m + sK(g, aep_1)$       (1)
$\log_e(Q_2) = m + sK(g, aep_2)$       (2)
$\log_e(Q_3) = m + sK(g, aep_3)$       (3)

Subtracting (2) from (1) eliminates $m$.

$\log_e(Q_1) - \log_e(Q_2) = s \left( K(g, aep_1) - K(g, aep_2) \right)$       (4)

Developing a similar equation using $Q_2$ and $Q_3$ can then allow an expression based on $g$ alone.

$\frac{\log_e(Q_1) - \log_e(Q_2)}{\log_e(Q_2) - \log_e(Q_3)} = \frac{K(g, aep_1)-K(g, aep_2)}{K(g, aep_2) - K(g, aep_3)}$       (5)

Equation (5) can be solved numerically to obtain $g$ then substitute back into (4) to obtain $s$ and into any of (1), (2) or (3) to obtain $m$.

Here we will use a root finding approach to obtain $g$ from equation (5), which can be rearranged to give:

$f(g) = \frac{K(g, aep_1)-K(g, aep_2)}{K(g, aep_2) - K(g, aep_3)} - \frac{\log_e(Q_1) - \log_e(Q_2)}{\log_e(Q_2) - \log_e(Q_3)}$       (6)

When equation (6) is equal to zero, we have an estimate of $g$

Example

Three quantiles are available for Fifteen Mile Creek at Greta South (Gauge 403213). (Lat = -36.62, Long = 146.24).  These come from the regional flood frequency tool.

No. AEP Q
Discharge (cumec)
1 1% 356.59
2 10% 131.95
3 50% 38.21

A plot of equation (6), using these quantiles, suggests a root finding approach will work Ok.

Using the method outlined above, the parameter values are:

• g (skew) = -0.113
• s (standard deviation) = 0.991
• m (mean) = 3.624

A test using quantiles not include in the derivation suggest the method works well.

Details of calculations and R code available at this gist: