91Ó°ÊÓ

It is given that X is a random variable that follows Normal distribution with its mean of 0 and unknown standard deviation >0. Therefore X_1,….,X_n are ii d Normal (µ=0, σ > 0)

f(x| µ = 0, σ ) = 1/√ 2πσ²) exp(-x /2σ)²)

Assume X~ f (x| θ) (pdf or pmf) with θ ∈ ʘ ⊂ R

Then the fisher information is defined by

I_x(θ) = E_θ[(∂ / ∂θ logf(X|θ))²]

= E_θ°Ú(-∂² / ∂θ logf(X|θ)]

and

I_x(θ) = nl_x1 (θ)

Let X =X₁

From the definition

I_x(σ) = E_θ[(∂ / (∂σ²)² ±ô´Ç²µ´Ú(³Ý´¥Ïƒ²)]

= -3(x-µ)²/ σ⁴ + 1/σ²

= 2/ σ²

And we know that

I_x(σ²) = nI_X1(σ²)

= 2n/ σ²

Therefore, the fisher information is 2n/ σ²

Therefore 1/l (σ) = σ²/ 2n

By the CRLB theorem, the variance obtained by fisher information has the minor variance. Therefore, the variance of any other unbiased estimator will always be higher than the one obtained by CRLB bound.