(A) study of bernoulli trials

Abstract

In application of the properties of Bernoulli trials the parameters are usually unknown and have to be estimated from the sample. This thesis discusses the probability distributions and the problems of parameter estimation under different dependence relations among the Bernoulli trials. Specifically, an independent trials model and a model with dependence relations between successive trials are considered. For each model three sampling plans or stopping rules are discussed. Sampling plan $S_1$ : a preassigned number of observations are taken. Sampling plan $S_2$ : observations are continued until a preass igned number m of successes are obtained. Sampling plan $S_3$ : observations are continued until at least $m_1$ success and $m_2$ failures are obtained where $m_1$ and $m_2$ are preassigned numbers. For each combination of dependent structures and sampling plans, relevant probability distributions and moments, are studied, and sufficient statistics and simple estimators for the parameters are presented. Especially this thesis synthesizes and complements the results of the previous studies done by many authors and studies in detail properties of the new sampling plan $S_3$.