# Cramér–Rao bound to multiple parameters

Let $\displaystyle {\boldsymbol {T}}(X)$ be an estimator of any vector function of parameters...

I could not understand what $\displaystyle {\boldsymbol {T}}(X)$ and $\displaystyle {\boldsymbol{\psi}}({\boldsymbol {\theta}})$ mean.

Can someone help?

Thanks

$\displaystyle {\boldsymbol {T}}(X)$ is a vector of estimators and $\displaystyle {\boldsymbol{\psi}}({\boldsymbol {\theta}})$ is its expectation. That is $$\mathbb{E}\left[\displaystyle {\boldsymbol {T}}(X)\right] = \displaystyle {\boldsymbol{\psi}}({\boldsymbol {\theta}}).$$
As an example, consider a random sample from a univariate Normal distribution. Then ${\boldsymbol {\theta}} = (\mu, \sigma^2)^T$, $\displaystyle {\boldsymbol {T}}(X) = \left(\sum_i X_i , \sum_i X_i^2\right)^T$ and $\displaystyle {\boldsymbol{\psi}}({\boldsymbol {\theta}}) = (n\mu, n\sigma^2 + n\mu^2)^T$.
• This may sound silly and I am sorry for that. I am interested in deciding a confidence interval for two parameters, then how do I get $\displaystyle {\boldsymbol {T}}(X)$ and ${\boldsymbol{\psi}}({\boldsymbol {\theta}})$ – pkj Mar 14 '18 at 18:07
• on the last page of this document stat.umn.edu/geyer/old03/5102/notes/fish.pdf , you have $\mathbf{t}$ and I do not know what is that – pkj Mar 14 '18 at 18:49
• Thank you @Taylor, you have been generous enough and spared time to answer my doubts. I thought the $t$s are same. I will give it another try, if I could not succeed, I will draft an another question. Thank you again. – pkj Mar 15 '18 at 2:45