PDF of Lecture Notes - School of Mathematical Sciences
PDF of Lecture Notes - School of Mathematical Sciences
PDF of Lecture Notes - School of Mathematical Sciences
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2. STATISTICAL INFERENCE<br />
Finally observe that:<br />
⎛<br />
∂ 2 ⎞<br />
f(X; θ)<br />
⎜<br />
E ⎝<br />
∂θ2 ⎟<br />
⎠ =<br />
f(X; θ)<br />
∫ ∞<br />
−∞<br />
∫ ∞<br />
. . .<br />
−∞<br />
∂ 2<br />
f(x; θ)<br />
∂θ2 f(x; θ)dx 1 . . . dx n<br />
f(x; θ)<br />
=<br />
∫ ∞<br />
−∞<br />
∫ ∞<br />
. . .<br />
−∞<br />
= ∂2<br />
∂θ 2 ∫ ∞<br />
= ∂2<br />
∂θ 2 1<br />
= 0<br />
−∞<br />
Hence, we have proved Var{U(θ; X)} = I(θ).<br />
. . .<br />
∂ 2<br />
∂θ 2 f(x; θ)dx 1 . . . dx n<br />
∫ ∞<br />
−∞<br />
f(x; θ)dx 1 . . . dx n<br />
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