ed infine
E
(
z
2
)
{\displaystyle E(z^{2})}
=
σ
t
2
−
2
[
τ
′
(
θ
)
]
2
E
{
[
∂
(
ln
L
)
∂
θ
]
2
}
+
[
τ
′
(
θ
)
]
2
E
{
[
∂
(
ln
L
)
∂
θ
]
2
}
{\displaystyle =\;{\sigma _{t}}^{2}-2{\frac {\left[\tau '(\theta )\right]^{2}}{E\left\{\left[{\dfrac {\partial \left(\ln {\mathcal {L}}\right)}{\partial \theta }}\right]^{2}\right\}}}+{\frac {\left[\tau '(\theta )\right]^{2}}{E\left\{\left[{\dfrac {\partial \left(\ln {\mathcal {L}}\right)}{\partial \theta }}\right]^{2}\right\}}}}
=
σ
t
2
−
[
τ
′
(
θ
)
]
2
E
{
[
∂
(
ln
L
)
∂
θ
]
2
}
{\displaystyle =\;{\sigma _{t}}^{2}-{\frac {\left[\tau '(\theta )\right]^{2}}{E\left\{\left[{\dfrac {\partial \left(\ln {\mathcal {L}}\right)}{\partial \theta }}\right]^{2}\right\}}}}
Ma il valore medio del quadrato di una qualsiasi variabile casuale non può essere negativo, e dunque
0
≤
E
(
z
2
)
=
σ
t
2
−
[
τ
′
(
θ
)
]
2
E
{
[
∂
(
ln
L
)
∂
θ
]
2
}
{\displaystyle 0\;\leq \;E(z^{2})\;=\;{\sigma _{t}}^{2}-{\frac {\left[\tau '(\theta )\right]^{2}}{E\left\{\left[{\dfrac {\partial \left(\ln {\mathcal {L}}\right)}{\partial \theta }}\right]^{2}\right\}}}}
ed infine
σ
t
2
≥
[
τ
′
(
θ
)
]
2
E
{
[
∂
(
ln
L
)
∂
θ
]
2
}
=
[
τ
′
(
θ
)
]
2
R
(
θ
)
τ
′
(
θ
)
=
τ
′
(
θ
)
⋅
R
(
θ
)
{\displaystyle {\sigma _{t}}^{2}\geq {\frac {\left[\tau '(\theta )\right]^{2}}{E\left\{\left[{\dfrac {\partial (\ln {\mathcal {L}})}{\partial \theta }}\right]^{2}\right\}}}=\left[\tau '(\theta )\right]^{2}\,{\frac {R(\theta )}{\tau '(\theta )}}=\tau '(\theta )\cdot R(\theta )}
cioè:
Nessuna funzione dei valori osservati
t
(
x
1
,
x
2
,
…
,
x
N
)
{\displaystyle t(x_{1},x_{2},\ldots ,x_{N})}
τ
(
θ
)
{\displaystyle \tau (\theta )}
La varianza minima si raggiunge se e soltanto se
E
(
z
2
)
{\displaystyle E(z^{2})}
z
{\displaystyle z}
z
=
t
−
τ
(
θ
)
−
R
(
θ
)
∂
(
ln
L
)
∂
θ
≡
0
{\displaystyle z\;=\;t-\tau (\theta )-R(\theta )\,{\frac {\partial \left(\ln {\mathcal {L}}\right)}{\partial \theta }}\;\equiv \;0}
o, altrimenti detto, se la derivata logaritmica della verosimiglianza è proporzionale alla variabile casuale
t
−
τ
(
θ
)
{\displaystyle t-\tau (\theta )}
![{\displaystyle =\;{\sigma _{t}}^{2}-2{\frac {\left[\tau '(\theta )\right]^{2}}{E\left\{\left[{\dfrac {\partial \left(\ln {\mathcal {L}}\right)}{\partial \theta }}\right]^{2}\right\}}}+{\frac {\left[\tau '(\theta )\right]^{2}}{E\left\{\left[{\dfrac {\partial \left(\ln {\mathcal {L}}\right)}{\partial \theta }}\right]^{2}\right\}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12618939d17b2fcb5d98a305617ab46c0748ad97)
![{\displaystyle =\;{\sigma _{t}}^{2}-{\frac {\left[\tau '(\theta )\right]^{2}}{E\left\{\left[{\dfrac {\partial \left(\ln {\mathcal {L}}\right)}{\partial \theta }}\right]^{2}\right\}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2e7c54647a8e0f0ee98de9505a1be967d6bb76)
![{\displaystyle 0\;\leq \;E(z^{2})\;=\;{\sigma _{t}}^{2}-{\frac {\left[\tau '(\theta )\right]^{2}}{E\left\{\left[{\dfrac {\partial \left(\ln {\mathcal {L}}\right)}{\partial \theta }}\right]^{2}\right\}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d09e910bae532b7f27b7f6c9575168c1c1eb71f9)
![{\displaystyle {\sigma _{t}}^{2}\geq {\frac {\left[\tau '(\theta )\right]^{2}}{E\left\{\left[{\dfrac {\partial (\ln {\mathcal {L}})}{\partial \theta }}\right]^{2}\right\}}}=\left[\tau '(\theta )\right]^{2}\,{\frac {R(\theta )}{\tau '(\theta )}}=\tau '(\theta )\cdot R(\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb08f4e507e09fb1a9519955ca964d93fed14e4)
