# Little help with notation on estimated functions of random variables

I'm writing something involving random variables and I'm not sure about the notation being right or not, so I'd really appreciate it if someone could take a quick look and confirm.

We have a sample of $n$ iid discrete random variables $X_1, X_2, ..., X_n$, and then a function $f$ that maps them onto another sample of variables $f(X_1), f(X_2), ..., f(X_n)$. The true mean of this distribution is:

$\mu_{f(X)}=\sum{f(x)P(X=x)}$

Because we only have a sample, we use the sample mean as estimator. The expectation is:

$E[\hat\mu_{f(X)}]=\frac{1}{n}\sum{f(X_i)}$

However, the true mapping function $f$ is unknown, we only have an empirically estimated $\hat f$ (cubic) function. This means that we actually have $\hat f(X_1), \hat f(X_2), ..., \hat f(X_n)$, so the estimate above is:

$E[\hat\mu_{\hat f(X)}]=\frac{1}{n}\sum{\hat f(X_i)}$

$Var[\hat\mu_{\hat f(X)}]=\frac{sd_{\hat f(X)}^2}{n}$

I wonder if there are too many hats in there. Thanks in advance.

• Thanks for the answer. Is it OK to just write $\hat\mu_{\hat f(X)}=...$ rather than $E\left[\hat\mu_{\hat f(X)}\right]=...$ Jun 23, 2013 at 16:21
• That is a good point. You do not need E[ ], as you are defining your estimator to be the average. You actually have that $$E[\hat{μ}_{\hat{f}(X)}] = E[\hat{f}(X)]$$ and similarly $$E[\hat{μ}_{f(X)}] = E[f(X)]$$. Instead of saying "The expectation is:" I think you mean "The estimator is:", with the expectation E[ ] removed. Jun 23, 2013 at 17:10