Likelihood based theory for applicatoins

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Statistical theory has two primary methodologies for analyzing a statistical model with data, the default Bayesian and the frequentist; and these can give widely different answers with demonstrably serious risks. Both take likelihood as their primary ingredient but differ on other input. Discrete data represent a very large area of need for methodology that goes beyond the use of just likelihood. This research continues present work that demonstrates that likelihood gradient or mean likelihood gradient can give the additional substance needed to separate an interest parameter from nuisance parameters and then give highly accurate p-values for the interest parameter. The methodology covers the continous parameter context but its development for the discrete data and the contingency table context represents an urgent need for large areas of application.