Let $X_1, . . . , X_n$ be i.i.d. with density function $f (x|θ)$. Let $T = t (X_1, . . . , X_n)$ be an unbiased estimate of $θ$. Then, under smoothness assumptions on $f (x|θ)$, $$Var(T) >= \frac{1}{I(θ)}$$
Can you please help me(explaining step by step) with understanding the derivation o this inequality? Because i undertand the idea of this inequality but i am completely lost with derivation.
Also i want to know why do we write Let $T = t (X_1, . . . , X_n)$ be an unbiased estimate of $θ$ ?Because usually i have seen only one symbol, smth like $\bar x$ is the estimator of $θ$. Appearance of $T = t (X_1, . . . , X_n)$ is confusing for me. And also in proof arises covariance and i don't understand why. Thanks