# Which Fisher information should I use for Cramer-Rao lower bound?

For $$X_1,\dots, X_n$$ iid sample from $$X\sim Bernoulli(p)$$. I try to verity that the estimator $$\hat{p}=\bar{X}$$ (sample mean) is the UMVUE for unknown parameter $$p$$.

I know that $$\hat{p}$$ is unbiased. I try to show that $$Var[\hat{p}]=CRLB$$ (Cramer-Rao lower bound), which is $$CRLB=\frac{[I'(\theta)]^2}{nI(\theta)},$$ and $$I$$ is the Fisher information.

My question is that what is the Fisher information here?

Note that the score function is $$U(p)=\frac{x-p}{p(1-p)}$$ for one sample.

Also, $$U(p)=\frac{\sum x_i-np}{p(1-p)}$$ for $$n$$ samples $$X_1,\dots, X_n$$.

I have two $$I(p)$$ here. I am not sure if I choose the Fisher information based on the $$U(p)$$ for one sample or $$n$$ sample...

$$I_1=E\left[\frac{(X-p)^2}{(p(1-p))^2}\right]=\frac{1}{p(1-p)}$$ and another one is $$I_2=E\left[\frac{(\sum X_i-np)^2}{(p(1-p))^2}\right]=\frac{n}{p(1-p)}$$

Cramér-Rao Lower Bound would be of the form $$\operatorname{Var}_\theta(T(\mathbf X) ) \geq \mathscr I(\theta)^{-1}.$$ For exponential family, $$\mathscr I(\theta) =\mathbb E_\theta\left[-\partial^2_\theta \ln f(\mathbf x;\theta)\right],$$ which for $$X_i\overset{\text{i.i.d.}}{\sim}\mathrm{Ber}(p)$$ is
\begin{align}\mathscr I(p) &=\mathbb E_p\left[-\partial^2_p\ln f(\mathbf x;p)\right]\\&=\mathbb E_p\left[-\sum_{i=1}^n\partial^2_p \ln f( x_i;p)\right]\\&=\mathbb E_p\left[-\sum_{i=1}^n\left\{-\frac{x_i}{p^2}+\frac{1-x_i}{(1-p)^2}\right\}\right]\\&=n[p(1-p)]^{-1}.\tag 1\label 1\end{align}
From $$\eqref 1,$$ it follows the CRLB is $$\frac{p(1-p)}n.$$
• So use the $n$ data sample? Not just use the one data point? Mar 7 at 17:14
• Yes @Hermi. You are working with a sample. You are inferring based on $n$ observations. It must get reflected in your calculation. Mar 7 at 17:23