Theory of Statistics - George Mason University

398 5 Unbiased Point Estimation
The Bhattacharyya Lower Bound
We now consider a simple case in which θ is a scalar (and, hence the estimand
g(θ) and the estimator T(X) are scalars).
For the PDF f(x; θ) and the Borel scalar function g(θ) assume that each
is differentiable r times, and write
and
f (r) = ∂r f(x; θ)
∂ r
g (r) = ∂rg(θ) ∂r .
Let T be an unbiased estimator of g(θ).
Now, form the function
Ds = T − g(θ) −
s
arf (r) /f, (5.19)
r=1
where the ar are constants to be determined. Now, we have
E(f (r) /f) = 0 (5.20)
as before, and since T be an unbiased estimator for g(θ), we have
The variance of Ds is therefore,
E(Ds) = 0.
E(D 2
s) =
T − g(θ) −
s
arf (r) /f
r=1
2
fdx. (5.21)
We now seek to minimize this quantity in the ar. To do so, for p = 1, . . ., s,
we differentiate and set equal to zero:
s
T − g(θ) − arf (r)
/f (f (p) /f)fdx = 0, (5.22)
which yields
r=1
(T − g(θ))f (p) dx =
s
r=1
ar
f (r)
f
f (p)
f
fdx. (5.23)
Because of (5.20), the left-hand side of (5.23) is
Tf (p) dx = g (p) (θ). (5.24)
Theory of Statistics c○2000–2013 James E. Gentle