This question relates to fitting a distribution to the true severity of the operational risk loss we are
trying to model. The quality of the fit, or the precision of the fit, has two elements to the difference
between the severity as represented by our model and the true severity. To understand this, consider
the three data points below:
a. The true severity,
b. The best approximation of the true severity in the model space, and
c. The fit based on the dataset.
- True severity is what we are trying to model.
- The model space refers to the collection of analytical distributions (log-normal, burr etc) that we are
considering to arrive at the estimate of the severity.
- The 'best approximation of the true severity in the model space' is reached by estimating the
parameters of the distribution that optimizes the risk functional.
- The 'fit' is the actual parameter estimates we settle for with the distribution we have determined
best fits the true estimate of our severity. When estimating parameters, we have various methods
available for estimation - the least squares method, the maximum likelihood method, for example,
and we can get different estimates depending upon the method we choose to use.
Our severity model will be different from the true severity, and the total difference can be split into
two types of errors:
1. Fitting error, represented by 'c - b' above: The difference between the fit based on the dataset and
the best approximation of the true severity is called 'fitting error', ie, a measure of the extent to
which we could have estimated the parameters better.
2. Approximation error, represented by 'b - a' above: Approximation error is the difference between
the true severity, and the best approximation of the true severity that can be achieved within the
model space is called 'approximation error'.
One can reduce the approximation error by expanding the model space by adding more distributions.
This will reduce the approximation error, but generally has the effect of increasing the fitting error
because the complexity of the model space increases, and there are more ways to fit to the true
severity.
