Marginal Distribution Estimators for the Gamma-Prior Parameters for a Group of Poisson Processes

Multiple data-sets of experimental times and failure counts from Poisson processes are used to estimate the parameters of the gamma distributions which are assumed appropriate for the failure rates. The experimental data are combined in two ways to estimate the failure rates; they are called unweighted and time weighted. These lead in turn to two different sets of gamma-parameter estimates. Marginal maximum likelihood estimates (MLE) are also considered. The concept of linkage is introduced, wherein some of the data sets are associated with priors which have common values of either scale parameters or shape parameters or both. A numerical example is presented with real data, showing the three sets of estimators for scale and shape parameters (unweighted, time-weighted, and MLE) for each type of linkage.