DESIRABLE AND NOT-SO-DESIRABLE ESTIMATORS
The method of maximum likelihood is by far the most popular technique for deriving estimators.
—Casella and Berger [1990, p. 289].
The proper starting point for the selection of the best method of estimation is with the objectives of our study: What is the purpose of our estimate? If our estimate is θ* and the actual value of the unknown parameter is θ, what losses will we be subject to? It is difficult to understand the popularity of the method of maximum likelihood and other estimation procedures that do not take these losses into consideration.
The majority of losses will be monotonically nondecreasing in nature, that is, the further apart the estimate θ* and the true value θ, the larger our losses are likely to be. Typical forms of the loss function are the absolute deviation |θ* – θ|, the square deviation (θ* − θ)2, and the jump, that is, no loss if |θ* − θ| < i, and a big loss otherwise. Or the loss function may resemble the square deviation but take the form of a step function increasing in discrete increments.
Desirable estimators share the following properties: impartial, consistent, efficient, robust, and minimum loss.
Estimation methods should be impartial. Decisions should not depend on the accidental and quite irrelevant labeling of the samples. Nor should decisions depend on the units in which the measurements are made.
Suppose we have collected data from two samples with the object of estimating the difference ...