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1. Is there any naming convention regarding the hat and the tilde symbol in stats? I found $\hat{\beta}$ is describing an estimator for $\beta$ ( Wikipedia ) But I also found $\tilde{\beta}$ is describing an estimator for $\beta$ (Wolfram ). Is there any difference in the meaning?
On the web I found kind of a difference but I am not sure about the meaning Reference for Stats Symbols. There it is distinguished between "estimates of parameters" and "estimates of variables". Could someone be so kind to explain in which case to use tilde and hat?

2. Regarding the expectation operator, is there any difference in $E(X)$ and $E[X]$ and $E\{X\}$ regarding the brackets? I got the advice to use the curly brackets. But I am not sure about the meaning. I used to use the brackets only for reading/visualization rather than pointing on some meaning. Any advice on that?

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Hats and tildes

The convention in (my end of) applied statistics is that $\hat{\beta}$ is an estimate of the true parameter value $\beta$ and that $\tilde{\beta}$ is another, possibly competing estimate.

Following the Wolfram example, these can both be distinguished from a statistic (function of the data) that also happens to be an estimate, e.g. the sample mean $\bar{x}$ could be an estimate of the population mean $\mu$ so it it could also be called $\hat{\mu}$.

Contra Wolfram, I'd call $\bar{X}$ the estimator (upper case roman letters denote random variables) and $\bar{x}$ the estimate (lower case roman letters denote observations of random variables), but only if I was feeling pedantic or it mattered to the argument.

Similarly, in the 'Reference for Stats Symbols' the thing that suggests to me that $\tilde{u}$ is a random variable rather than a parameter is the fact that it's a roman letter not a greek one. Again, this is why in the example above the sample mean involved the letter $x$ when it was a function of the data but $\mu$ when it was considered as an estimator. (And frankly, it's unclear to me what the tilde denotes on $u$. The mean? the mode? the actual but unobserved value? The surrounding text would have to say.)

Expectations

Re the expectation operator: I've never seen curly brackets used. Maybe that's a mathematical statistics thing, in which case someone around here should recognize it.

The empirical approach to notation

One simple situation where estimators, random variables, and expectations collide in the notation is in the discussion of EM algorithms. You might want to look at a few careful expositions to get a sense of the normal range of notational variation. This is the empirical approach to notation, which always beats theory provided you are looking at variation from the right population, i.e. your discipline or expected audience.

The bottom line

Stay within the normal range described above, and anyway say what you mean by the symbols once in the text before using them. It doesn't take much space and your readers will thank you.

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