Generalised Extreme Value Distribution using MLE

For the Generalised Extreme Value distribution, when implementing maximum likelihood estimation for the parameters, a likelihood function is first required and Coles (2001) suggests:

Once this function is set up, a mathematical optimiser such as the ‘solver’ in excel, can be used to maximise the result of the function by iteratively changing the parameters; location, shape and scale. With the resulting parameters, the estimation of the flow for a given return period (QT) is then

Where:

ξ = location

k = shape

𝑎 = scale

n = sample size

T = Return Period

ln – the natural logarithm

To use this approach an initial starting point for the parameters is necessary. I tend to choose, the arithmetic mean for the location, the standard deviation for the scale and a value of 0.01 for the shape parameter.

L-moments

The L-moment process outlined by Hosking & Wallis (1997) begins by ordering the AMAX sample and calculating probability weighted moments as;

where j is the ordered rank. The first three sample L-moments are then defined by

L-moments and the Generalised Logistic Distribution

The three parameters for the generalised logistic distribution are then calculated as;

The QT estimate is then

As outlined in the FEH (1999), for the purposes of regional frequency analysis, the median of the annual maximum flows is used as the location parameter and is known as QMED (the index flood). In which case the parameters are slightly different. The shape parameter is as above,

and there is a standardised scale parameter,

The QT estimate is then

L-moments and the Generalised Extreme Value Distribution

With L-moments as above, the three parameters for the generalised extreme value distribution are calculated as

The QT estimate is then