From Algorithms to Z-Scores - matloff - University of California, Davis

12.1. GENERAL METHODS OF PARAMETRIC ESTIMATION 245
and so set
Next,
So, the bias is 0.
12.1.6 What About Confidence Intervals?
ĉ = (3 − X)/3 (12.31)
Eĉ = E[(3 − X)/3) (12.32)
= 1
· (3 − EX)
3
(12.33)
= 1
[3 − EX]
3
(12.34)
= 1
[3 − (3 − 3c)]
3
(12.35)
= c (12.36)
Usually we are not satisfied with simply forming estimates (called point estimates). We also want
some indication of how accurate these estimates are, in the form of confidence intervals (interval
estimates).
In many special cases, finding confidence intervals can be done easily on an ad hoc basis. Look, for
instance, at the Method of Moments Estimator in Section 12.1.2. Our estimator (12.4) is a linear
function of X, so we easily obtain a confidence interval for c from one for EX.
Another example is (12.25). Taking the limit as n → ∞ the equation shows us (and we could
verify) that
c =
1
E[ln W ]
(12.37)
Defining Xi = ln Wi and X = (X1 + ... + Xn)/, we can obtain a confidence interval for EX in the
usual way. We then see from (12.37) that we can form a confidence interval for c by simply taking
the reciprocal of each endpoint of the interval, and swapping the left and right endpoints.
What about in general? For the Method of Moments case, our estimators are functions of the
sample moments, and since the latter are formed from sums and thus are asymptotically normal,