ABSTRACT
If we have in hand the exact or approximate distribution of a particular estimator, we may then be able to make assertions about the likelihood that the estimator will be within a certain distance of the target parameter. An interval estimate of a parameter θ is an interval (L,U) whose endpoints depend on an estimator of θ, but not on any unknown parameters, and for which the probability that the interval contains the unknown parameter θ can be explicitly derived or reliably approximated. When experimental data is available, the endpoints of the interval (L,U) can be computed, resulting in a numerical interval (l,u) called a confidence interval for the parameter θ. If the probability that the interval (L,U) contains the parameter θ is p, then the numerical interval (l,u) is referred to as a 100p% confidence interval for θ. It may seem tempting to use the phrase “probability interval” in describing the interval (l,u), but this phrase would be inappropriate due to the fact that a given numerical interval either contains the true value of the parameter or it doesn’t, so that the inequality l < θ< u has no immediate probabilistic meaning. Because the interval (l,u) is obtained as a realization of the interval (L,U), where P(L < θ<U) = p, it is quite reasonable to say that we have 100p% confidence that the numerical interval (l,u) will contain the true value of parameter θ.
