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.

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*Yes, in any real world situation, if you know the value of the parameter, then you don't need to go about trying to estimate it.


*One scenario where you know the value of a parameter, and then may want to examine the behavior of a statistic (estimator), is when you are conducting a simulation to see how the statistic behaves in different circumstances.  For example, you can ask a computer to, say, start with a population of observations of known distribution and known parameters. You can then make many iterations of sampling this population, using a statistic to estimate some parameter, and see how that statistic behaves in certain situations. For example, what if the sample size is small compared with when it is large ?  What if the distribution is skewed compared with when it is symmetric ?  ... As a non-statistician, I find this approach useful in assessing how various statistics and tests behave in different circumstances.
