Theory of Statistics - George Mason University

388 5 Unbiased Point Estimation
Example 5.4 an unbiased estimator with poor properties
Consider the problem of using a sample of size 1 for estimating g(θ) = e −3θ
where θ is the parameter in a Poisson distribution. An unbiased is
T(X) = (−2) X ,
as you are asked to show in Exercise 5.2.
The estimator is ridiculous. It can be negative, even though g(θ) > 0.
It is increasing in the positive integers, even though g(θ) decreases over the
positive integers.
Degree of a Statistical Function
If a statistical function is estimable, we may ask how many observations are
required to estimate it; that is, to estimate it unbiasedly. We refer to this
number as the degree of the statistical function. Obviously, this depends on
the distribution as well as the functional. A mean functional may not even
be exist, for example, in a Cauchy distribution, but if the mean functional
exists, it is estimable and its degree is 1. The variance functional in a normal
distribution is estimable and its degree is 2 (see page 401).
5.1 Uniformly Minimum Variance Unbiased Point
Estimation
An unbiased estimator may not be unique. If there are more than one unbiased
estimator, we will seek one that has certain optimal properties.
5.1.1 Unbiased Estimators of Zero
Unbiased estimators of 0 play a useful role in UMVUE problems.
If T(X) is unbiased for g(θ) then T(X) − U(X) is also unbiased for g(θ)
for any U such that E(U(X)) = 0; in fact, all unbiased estimators of g(θ)
belong to an equivalence class defined as
{T(X) − U(X)}, (5.4)
where Eθ(U(X)) = 0.
In Theorem 5.2 and its corollary we will see ways that unbiased estimators
of zero can be used to identify optimal unbiased estimators.
In some cases, there may be no such nontrivial U(X) that yields a different
unbiased estimator in (5.4). Consider for example, a single Bernoulli trial
with probability of success π, yielding the random variable X, and consider
T(X) = X as an estimator of π. We immediately see that T(X) is unbiased
for π. Now, let S be an estimator of π, and let S(0) = s0 and S(1) = s1. For
S to be unbiased, we must have
Theory of Statistics c○2000–2013 James E. Gentle