If we are attempting to fit a parametric distribution to data, we will obviously use the parametric Bootstrap. For each Bootstrap replication, we can calculate the necessary moments, and then use them to get estimates for the uncertainty about the parameters we are estimating. The Bootstrap/MoM method has the advantage of directly sampling from the joint uncertainty distribution of the parameters, so any correlation is automatically preserved. However, the MoM is not the most accurate method of estimating a parameter for low number of data points, so this method is preferable for its convenience rather than great accuracy. The primary use of MoM is to establish starting values for MLE techniques where one is using some optimizing algorithm to find the MLEs.
The model BootstrapMoM for Gamma performs a Bootstrap/MoM fit of data to a Gamma distribution, as follows:
A best fitting Gamma distribution is found by optimization (using Excel's Solver) to maximize a likelihood calculation to obtain MLEs. We did not use distribution fitting feature here, as we liked the "Location parameter" L in the Gamma to be equal to 0. Distribution fitting feature also varies L to find the best fitting distribution, but by maximizing the likelihood calculation to obtain MLEs in Excel, we can force L = 0.
This Gamma distribution is then used to draw Bootstrap replicates;
The mean and variance of the Bootstrap replicate is calculated, and translated into estimates of the Gamma distribution parameters a and b.
A scatter plot of the generated values for a and b show their correlation pattern:
The Bootstrap/MoM is a great way of understanding why such a pattern exists. The mean (=ab) is the moment that one most quickly reaches a narrow estimate because it does not require first estimating another parameter, and then calculating squared, cubed, etc deviations as the higher moments do. Thus the value of the product ab becomes quickly established. The Figure below shows that the Bootstrap estimate of the Gamma distribution mean gives a 90% confidence that the mean lies somewhere between 17.1 and 22.3. Thus in the scatter plot above, if a=2, then b » (8.6,11.1), but if a=4, then b » (4.3, 5.6). The joint distribution is also most concentrated around ab = 19.6, the sample mean.
The links to the BootstrapMoM for Gamma software specific models are provided here:
