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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.

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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.

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  • $\begingroup$ Thanks for the answer. Is it OK to just write $\hat\mu_{\hat f(X)}=...$ rather than $E\left[\hat\mu_{\hat f(X)}\right]=...$ $\endgroup$ Jun 23, 2013 at 16:21
  • $\begingroup$ 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. $\endgroup$ Jun 23, 2013 at 17:10

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