Interval Estimates for Parameters of Discrete Sampling Distributions

Moustafa, Maha Wael Elbakri (2021). Interval Estimates for Parameters of Discrete Sampling Distributions. PhD thesis The Open University.



Well recommended methods of forming confidence intervals for discrete distributions parameter give interval estimates that do not actually meet the definition of a confidence interval, in that their coverage is sometimes lower than the nominal confidence level. The methods are favored because their intervals have a shorter average length than the exact method, whose intervals really are confidence intervals. Comparison of such methods is tricky as the best method should perhaps be the one that gives the shortest intervals (on average), but when is the coverage of a method so poor it should not be classed as a means of forming confidence interval.

As the definition of a confidence interval is being flouted, a better criterion for forming interval estimates for discrete distributions parameters is needed. The aim of this thesis is to suggest a new criterion: methods that meet the criterion are said to yield locally correct confidence intervals. We propose a method that yields such intervals and proves that its intervals have a shorter average length than those of any other method that meets the criterion. We refer to the new estimator as the optimal locally correct method or just the OLC method. The thesis begins by applying the new criterion and method to the binomial parameter. Then we extend the method so as to obtain locally correct confidence intervals for parameters of the Poisson distribution and the negative binomial distribution.

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