Loss Models: From Data to Decisions, 4th Edition

10.3 Interval estimation

All of the estimators discussed to this point have been point estimators. That is, the estimation process produces a single value that represents our best attempt to determine the value of the unknown population quantity. While that value may be a good one, we do not expect it to exactly match the true value. A more useful statement is often provided by an interval estimator. Instead of a single value, the result of the estimation process is a range of possible numbers, any of which is likely to be the true value. A specific type of interval estimator is the confidence interval.

Definition 10.6 A 100(1 − α)% confidence interval for a parameter θ is a pair of random values, L and U, computed from a random sample such that Pr(L ≤ θ ≤ U) ≥ 1 − α for all θ.

Note that this definition does not uniquely specify the interval. Because the definition is a probability statement and must hold for all θ, it says nothing about whether or not a particular interval encloses the true value of θ from a particular population. Instead, the level of confidence, 1 − α, is a property of the method used to obtain L and U and not of the particular values obtained. The proper interpretation is that, if we use a particular interval estimator over and over on a variety of samples, at least 100(1 − α)% of the time our interval will enclose the true value. Keep in mind that it is the interval end points that are random.

Constructing confidence intervals is usually very difficult. For ...