How do we know the true value of a parameter, in order to check estimator properties? For example, we say that an estimator is unbiased if the expected value of the estimator is the true value of the parameter we're trying to estimate. However, if we already know the true value of the parameter, why are we trying to estimate it? I know there's something I'm missing here, like some theoretical true value that we can find, otherwise I don't see how we can check for unbiasedness, consistency etc.


How do we know the true value of a parameter, in order to check estimator properties?

We don't know the true value and we don't know the estimator properties.

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*One way to deal with it is that we estimate the true value of the parameter, and based on the estimate of the value, we estimate the estimator properties.
This approach is used for instance with the Wald test. This uses the estimated values of the parameters to describe the properties of the estimate.
This approach is an approximation and does not give an exact result.

we say that an estimator is unbiased if the expected value of the estimator is the true value of the parameter we're trying to estimate.

This doesn't require to know the true value. We only need to know that give any true value the expectation of the estimator is equal to that true value.
If for any true value of the parameter the bias is zero, then without knowing the value of the parameter we know that the bias is zero.
However, in cases when the bias is not always zero or a constant, then indeed we can not know the exact bias and we need make an estimate.
