Page 379 - Probability and Statistical Inference

356 7. Point Estimation

One may run the experiment a little differently as follows. Let N = the number
of runs needed to observe the first success so that the pmf of the random
y-1
variable N is given by P(N = y) = p(1 - p) , y = 1, 2, 3, ... which means that
N is a Geometric(p) random variable. One can verify that E [N] = p for all p
-1
p
-1
∈ (0, 1). That is the sample size N is an unbiased estimator of p . This method
of data collection, known as the inverse binomial sampling, is widely used in
applications mentioned before. In his landmark paper, Petersen (1896) gave
the foundation of capture-recapture sampling. The 1956 paper of the famous
geneticist, J. B. S. Haldane, is cited frequently. Look at the closely related
Exercise 7.3.8. !

How should we go about comparing performances of
any two unbiased estimators of T(θ)?

This is done by comparing the variances of the rival estimators. Since the
rival estimators are assumed unbiased for T(θ), it is clear that a smaller vari-
ance will indicate a smaller average (squared) error. So, if T , T are two
1
2
unbiased estimators of T(θθ θθ θ), then T is preferable to (or better than) T if
1
2
V (T ) ≤ V (T ) for all θθ θθ θ ∈ Θ but V (T ) < V (T ) for some θθ θθ θ ∈ Θ. Now, in the
2
1
θ
θ
2
θ
1
θ
class of unbiased estimators of T(θθ θθ θ), the one having the smallest variance is
called the best unbiased estimator of T(θθ θθ θ). A formal definition is given shortly.
Using such a general principle, by looking at (7.3.2), it becomes apparent
that among the unbiased estimators T , T , T and T for the unknown mean µ,
6
2
5
3
the estimator T is the best one to use because it has the smallest variance.
3
Definition 7.3.4 Assume that there is at least one unbiased estimator of
the unknown real valued parametric function T(θθ θθ θ). Consider the class C of all
unbiased estimators of T(θθ θθ θ). An estimator T ∈ C is called the best unbiased
estimator or the uniformly minimum variance unbiased estimator (UMVUE)
of T(θθ θθ θ) if and only if for all estimators T* ∈ C, we have
In Section 7.4 we will introduce several approaches to locate the UMVUE.
But first let us focus on a smaller subset of C for simplicity. Suppose that we
have located estimators T , ..., T which are all unbiased for T(θθ θθ θ) such that
1 k
2
V (T ) = δ and the T ’s are pairwise uncorrelated, 0 < δ < ∞, i = 1, ..., k.
?
i
i
Denote a new subclass of estimators