91Ó°ÊÓ

It is given that X₁,….., X_nis iid variables from a normal distribution with unknown mean µ and known variance σ²> 0. Therefore X₁….. X_n are iid 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 small variance or mean square error. Therefore, there is a slight deviation between the estimated and parameter values

So, to establish efficiency, we 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[(∂ / ∂θ logf(X|θ))²]

= E_θ°Ú(-∂² / ∂θ² ±ô´Ç²µ´Ú(³Ý´¥Î¸)±Õ

And

I_x(θ) = nl_x1 (θ)

Let X =X₁

From the definition

I_x(θ) = E_θ[(∂ / (∂σ²)² log f(X|θ²)]

= -3(x-µ)²/ σ⁴ + 1/σ²

= 1/2 σ²

And we know that,

I_x(σ²) = nI_X1(σ²)

= n/2 σ²

therefore fisher information isn/2σ²

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.³ÝÌ„_n is the most efficient estimator of µ.