Estimation - Common Errors in Statistics - page 42

ity of the method of maximum likelihood and other estimation procedures
that do not take these losses into consideration.
The majority of losses will be monotone nondecreasing in nature; that
is, the further apart the estimate q* and the true value q, the larger our
losses are likely to be. Typical forms of the loss function are the absolute
deviation |q* -q|, the square deviation (q* - q) 2 , and the jump—that is,
no loss if |q* - q|<d, and a big loss otherwise. Or the loss function may
resemble the square deviation but take the form of a step function increas-
ing in discrete increments.
Desirable estimators share the following properties: impartial, consistent,
efficient, robust, and minimum loss.
Impartiality
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 in location of the two populations involved.
Suppose further that the first sample includes the values a , b , c , d , and e ,
the second sample includes the values f , g , h , i , j , k , and our estimate of
the difference is q*. If the observations are completely reversed—that is,
if the first sample includes the values f , g , h , i , j , k and the second sample
the values a , b , c , d , and e —our estimation procedure should declare the
difference to be - q*.
The units we use in our observations should not affect the resulting
estimates. We should be able to take a set of measurements in feet,
convert to inches, make our estimate, convert back to feet, and get
absolutely the same result as if we'd worked in feet throughout. Similarly,
where we locate the zero point of our scale should not affect the
conclusions.
Finally, if our observations are independent of the time of day, the
season, and the day on which they were recorded (facts that ought to
be verified before proceeding further), then our estimators should be
independent of the order in which the observations were collected.
Consistency
Estimators should be consistent ; that is, the larger the sample, the greater
the probability the resultant estimate will be close to the true population
value.
Efficient
One consistent estimator certainly is to be preferred to another if the first
consistent estimator can provide the same degree of accuracy with fewer