One of the main focuses of study in inferential statistics is determining the "true" value of a parameter. It is interesting because generally the actual value for a parameter is never known. However, with statistical methods, it is possible to get an estimation for a parameter. Finding the population proportion is one of the oldest and most basic problems. Although the problem may seem easy at first, it soon becomes clear that it is not. Because of that, even after a lot of research, it is still a much discussed topic. Finding an interval estimate seems a reasonable method for estimating a proportion because especially under re-sampling, the point estimate can be very inaccurate.
We start by defining two classes of one-sided and a class of two-sided 1-alpha confidence intervals with certain monotonicity and symmetry on the confidence limits for the probability of success, the parameter in a binomial distribution. In the class of one-sided confidence intervals we search for the smallest interval, in the sense of set inclusion, with direct analysis of the coverage probability function. This method gives us the same interval as the traditional one-sided 1-alpha Clopper-Pearson interval, which is in fact, although rarely mentioned, the smallest interval in the specified class. We provide a simple sufficient and neccessary condition for the existence of the smallest interval in the class of two-sided intervals. A method is provided for deriving the smallest interval if the condition holds. The proposed confidence intervals are uniformly most accurate and have the uniformly minimum expected length.
|
