Is the variance of an estimator a random-variable?
If so, the mean of the variance and the variance of the variance exist.
An estimator of the variance of this variance of an estimator also exists.
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2Do you mean the empirical variance of an estimator, which itself is an estimator and hence a random variable? – Xi'an Sep 06 '22 at 20:06
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2An estimator is a random variable. The variance of a random variable, by definition, is a number. – whuber Sep 06 '22 at 20:06
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Is the variance of an estimator a random-variable?
The (population) variance of any random variable, including an estimator $T$ of a population parameter $\theta$, is not a random variable. From the definition,
$$\operatorname{Var}[T] \triangleq \mathbb{E}[(T-\mathbb{E}[T])^2] \triangleq\int_{\Omega} (T - \mathbb{E}[T])^2\ dF(T)$$
you are averaging across all possible instances of $T$. This leaves you with a scalar.
If so, the mean of the variance and the variance of the variance exist.
As above, it isn't so.
An estimator of the variance of this variance of an estimator also exists.
If the population variance $\operatorname{Var}[T]$ exists, then you could come up with an estimator $Q$ to estimate it.
But then you might think that you want $\operatorname{Var}[Q]$... But then how to estimate that? Welcome to Siphonaptera:
Great fleas have little fleas upon their backs to bite 'em, And little fleas have lesser fleas, and so ad infinitum. And the great fleas themselves, in turn, have greater fleas to go on; While these again have greater still, and greater still, and so on. -- Augustus De Morgan
Or perhaps even more apt is It's turtles all the way down.
Galen
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