IMPROVED RESULTS
In many instances, we can obtain narrower interval estimates that have a greater probability of including the true value of the parameter by focusing on sufficient statistics, pivotal statistics, and admissible statistics.
A statistic T is sufficient for a parameter if the conditional distribution of the observations given this statistic T is independent of the parameter. If the observations in a sample are exchangeable, then the order statistics of the sample are sufficient; that is, if we know the order statistics x(1) ≤ x(2) ≤ … ≤ x(n ), then we know as much about the unknown population distribution as we would if we had the original sample in hand. If the observations are on successive independent binomial trials that result in either success or failure, then the number of successes is sufficient to estimate the probability of success. The minimal sufficient statistic that reduces the observations to the fewest number of discrete values is always preferred.
A pivotal quantity is any function of the observations and the unknown parameter that has a probability distribution that does not depend on the parameter. The classic example is the Student’s t, whose distribution does not depend on the population mean or variance when the observations come from a normal distribution.
A decision procedure d based on a statistic T is admissible with respect to a given loss function L, providing there does not exist a second procedure d* whose use would result in smaller ...
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