We let $X_1, . . . , X_n$ be independent identical normal distributed stochastic variables hvor $X_i$ ∼ N(θ, $θ^2$) with an unknown θ > 0. You can for example estimate θ using the mean: $θ_n$ = X = S/n, (theta tilde) and where $S=\sum_i X_i$.
Alternatively, θ can be estimated from the empirical variance:
$ˇθ_n$=$\sqrt(\sum_iX_i-X)/(n-1)=\sqrt(SS-S^2/n)/(n-1)$ where SS=$S=\sum_i X^2_i$.
The distribution of $ˇθ^2_n$ is $(θ^2\chi^2(n-1))/(n-1)$.
We can use without proof that $ˇθ^2_n$ is asymptotically normal distributed so N{$θ^2$,$2θ^4$/n}.
Now I have to find the score function and the information function. I know I have to find them if i know the likelihhood function. But can someone help me to find the density function so I can find the like likelihhood function?