STATISTICS
is trivially extended to multiple integrals and shows that for two real functions,g(x)and
h(x),
(∫
g^2 (x)dNx
)(∫
h^2 (x)dNx)
≥
(∫
g(x)h(x)dNx) 2
. (31.22)
If we now letg=[ˆa−α(a)]
√
Pandh=(∂lnP/∂a)√
P, we find
{∫
[ˆa−α(a)]^2 PdNx}[∫(
∂lnP
∂a) 2
PdNx]
≥
(
1+
∂b
∂a) 2
.
On the LHS, the factor in braces represents the expected spread ofˆa-values around the
pointα(a). The minimum value that this integral may take occurs whenα(a)=E[aˆ].
Making this substitution, we recognise the integral as the varianceV[aˆ],andsoobtainthe
result
V[aˆ]≥(
1+
∂b
∂a) 2 [∫ (
∂lnP
∂a) 2
PdNx]− 1
. (31.23)
We note that the factor in brackets is the expectation value of (∂lnP/∂a)^2.
Fisher’s inequality is, in fact, often quoted in the form (31.23). We may recover the form
(31.18) by noting that on differentiating (31.20) with respect toawe obtain
∫ (
∂^2 lnP
∂a^2
P+
∂lnP
∂a∂P
∂a)
dNx=0.Writing∂P /∂aas (∂lnP/∂a)Pand rearranging we find that
∫(
∂lnP
∂a
) 2
PdNx=−∫
∂^2 lnP
∂a^2PdNx.Substituting this result in (31.23) gives
V[aˆ]≥−(
1+
∂b
∂a) 2 [∫
∂^2 lnP
∂a^2PdNx]− 1
.
Since the factor in brackets is the expectation value of∂^2 lnP/∂a^2 , we have recovered
result (31.18).
31.3.3 Standard errors on estimatorsFor a given samplex 1 ,x 2 ,...,xN, we may calculate the value of an estimatoraˆ(x)
for the quantitya. It is also necessary, however, to give some measure of the
statistical uncertainty in this estimate. One way of characterising this uncertainty
is with the standard deviation of the sampling distributionP(aˆ|a), which is given
simply by
σˆa=(V[ˆa])^1 /^2. (31.24)If the estimatoraˆ(x) were calculated for a large number of samples, each of size
N, then the standard deviation of the resultingaˆvalues would be given by (31.24).
Consequently,σˆais called thestandard erroron our estimate.
In general, however, the standard errorσaˆdepends on the true values of some