Theory of Statistics - George Mason University

3.2 Statistical Inference: Approaches and Methods 253
We consider a generalization of the Fisher information (3.30) with sn(θ ; X) =
∂ logp(X; θ)/∂θ:
(sn(θ ; X)) (sn(θ ; X)) T
.
Eθ
Now we define efficiency of unbiased estimating functions in terms of this
quantity.
Definition 3.9 (efficiency of unbiased estimating functions)
Let sn(θ ; X) be an unbiased estimating function that is differentiable in θ.
The efficiency of sn is
T
(Eθ(∂sn(θ ; X)/∂θ)) Eθ
(sn(θ ; X))(sn(θ ; X)) T −1
(Eθ(∂sn(θ ; X)/∂θ)) .
The efficiency of unbiased estimating functions is sometimes called Godambe
efficiency, after V. P. Godambe. Compare this expression for the efficiency of
an unbiased estimating function with the CRLB, which is expressed in terms
of a score function.
Notice that for estimating functions, we define efficiency only for unbiased
functions. Just as in the case of point estimators, with estimating functions,
we use the word “efficient” in the sense of “most efficient”.
Definition 3.10 (efficient unbiased estimating functions)
Let s ∗ n(θ ; X) be an unbiased estimating function that is differentiable in θ. If
the efficiency of s∗ n is at least as great as the efficiency of any other unbiased
estimating function that is differentiable in θ, then we say s∗ n is efficient, or
(synonymously) Godambe efficient.
That is, while “efficiency” is a relative term, “efficient” is absolute. An efficient
estimating function is not necessarily unique, however.
Definition 3.11 (martingale estimating function)
Let {(Xt, Ft) : t ∈ T } be a forward martingale, and let {st(θ ; Xt) : t ∈ T }
be adapted to the filtration {Ft)}. Then {st(θ ; Xt) : t ∈ T } is called a
martingale estimating function iff
and
s0(θ ; X0) a.s.
= 0
E(st(θ ; Xt)|Ft−1) a.s.
= st−1(θ ; Xt−1).
Martingale estimating functions arise in applications of stochastic process
models, for example, in the analysis of financial data.
Our interest in estimating functions is due to their use in forming estimating
equations and subsequently in yielding estimators. We will consider some
asymptotic properties of solutions to estimating equations in Section 3.8.1
(consistency) and in Section 6.3.4 (asymptotic normality).
Theory of Statistics c○2000–2013 James E. Gentle