Theory of Statistics - George Mason University

3.2 Statistical Inference: Approaches and Methods 251
We consider an independent sample X1, . . ., Xn of random vectors with
orders d1, . . ., dn, with supdi < ∞. We assume the distributions of the Xi are
defined with respect to a common parameter θ ∈ Θ ⊆ IR k . We now define
Borel functions ψi(Xi, t) and let
sn(t ; X) =
If Eθ((sn(θ ; X)) 2 ) < ∞, we call
n
ψi(Xi, t) t ∈ Θ. (3.77)
i=1
sn(t ; X) (3.78)
an estimating function. We often write the estimating function simply as sn(t).
(Also, note that I am writing “t” instead of “θ” to emphasize that it is a
variable in place of the unknown parameter.)
Two prototypic estimating functions are the score function, equation (3.57)
and the function on the left side of the normal equations (3.70).
We call
sn(t) = 0 (3.79)
a generalized estimating equation (GEE) and we call a root of the generalized
estimating equation a GEE estimator. If we take
ψi(Xi, t) = ∂ρ(Xi, t)/∂t,
we note the similarity of the GEE to equation (3.68).
Unbiased Estimating Functions
The estimating function is usually chosen so that
Eθ(sn(θ ; X)) = 0, (3.80)
or else so that the asymptotic expectation of {sn} is zero.
If sn(θ ; X) = T(X) − g(θ), the condition (3.80) is equivalent to the estimator
T(X) being unbiased for the estimand g(θ). This leads to a more
general definition of unbiasedness for a function.
Definition 3.7 (unbiased estimating function)
The estimating function sn(θ ; X) is unbiased if
Eθ(sn(θ ; X)) = 0 ∀θ ∈ Θ. (3.81)
An unbiased estimating function does not necessarily lead to an unbiased
estimator of g(θ), unless, of course, sn(θ ; X) = T(X) − g(θ).
Theory of Statistics c○2000–2013 James E. Gentle