Page 378 - Probability and Statistical Inference

7. Point Estimation 355

There are situations where a parametric function of interest
has no unbiased estimator. In such situations, naturally we do not
proceed to seek out the best unbiased estimator.
Look at the Example 7.3.2.
Example 7.3.2 Suppose that X , ..., X are iid Bernoulli(p) where 0 < p <
1
n
1 is unknown and θ = p. We now wish to estimate the parametric function
T(θ) = 1/p. Of course is sufficient for p and we also know that T
is distributed as Binomial(n,p) for all fixed 0 < p < 1. In this case, there is no
unbiased estimator of T(θ). In order to prove this result, all we need to show
is that no function of T can be unbiased for T(θ). So let us assume that h(T) is
an unbiased estimator of T(θ) and write, for all 0 < p < 1,

which can be rewritten as

But the lhs in (7.3.3) is a polynomial of degree n+1 in the variable p and it
must be zero for all p ∈ (0, 1). That is the lhs in (7.3.3) must be identically
equal to zero and hence we must have −1 ≡ 0 which is a contradiction. That
is, there is no unbiased estimator of 1/p based on T. In this situation, there is
no point in trying to find the “best” unbiased estimator of 1/p. Look at a
closely related Exercise 7.3.7. !

In the Example 7.3.2, we could not find any unbiased estimator
-1
of p . But, one can appropriately modify the sampling
scheme itself and then find an unbiased estimator of p .
-1
Look at the Example 7.3.3.
Example 7.3.3 (Example 7.3.2 Continued) In many areas, for example,
ecology, entomology, genetics, forestry and wild life management, the
problem of estimating p is very important. In the previous example, the
-1
sample size n was held fixed and out of the n independent runs of the
Bernoulli experiment, we counted the number (T) of observed successes.
If the data was collected in that fashion, then we had shown the nonexist-
-1
ence of any unbiased estimator of p . But we do not suggest that p can
-1
not be estimated unbiasedly whatever be the process of data collection.