Little help with notation on estimated functions of random variables

Question

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.

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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:

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

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

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

Answer 1

Your notation looks good :). Depending on how precise you feel like being - you could add subscripts and/or superscripts to your summations to indicate directly what you are summing over. Although this is clear from the context it would be nice I think. Also I would add an 'an' between the 'as' and 'estimator' in the sentence: "we use the sample mean as estimator". Hope this is helpful.