91Ó°ÊÓ

It is given that X_1,….,X_n is ii d variable from a normal distribution with known mean µ and unknown variance. Therefore X_1,….,X_n are ii d Normal (µ , σ)

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

The most efficient estimator among a group of unbiased estimators is the one with the minor variance.

An efficient estimator also fetches a slight variance or mean square error. Therefore, there is a slight deviation between the estimated and parameter values.

So, to establish efficiency, they have to compare the estimator's variance with the .

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

Then the fisher information is defined by

I_x(θ) = E_θ°Ú(∂ / ∂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°Ú(∂² / ∂ σ² ±ô´Ç²µ´Ú(³Ý´¥Ïƒ)]

= -3(x-µ)²/ σ⁴ + 1/σ²

= 2/ σ²

I_x(σ²) = nI_X1(σ²)

= 2n/ σ²

Therefore, the fisher information is 2n/ σ²

Therefore 1/l (σ) = σ²/ 2n

Now, by the Cramer Rao bound

V (σ)≥ I_x(σ²)^-1= n/2σ²

Since the lowest bound of variance is attained by CRLB equality, the M.L.E. X̄_n is the most efficient estimator of µ.

The expectations and variance of the estimator is E (xÌ„_n) = µ, Var (xÌ„_n) = σ² /n