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.
 A: *

*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.
A: Consider i.i.d. sample $X_1, \ldots, X_n \sim N(\mu, 1)$. We assume that all samples are from normal distribution with mean $\mu$, which is unknown, and known variance $1$.
Now, let $\bar X = \frac{1}{n} \sum_{i=1}^n X_i$.
Then $E(\bar X) = \mu$.
So no matter what $\mu$ is, $E(\bar X) = \mu$. It is not necessary to know what $\mu$ is.
A: Phil has given a very good answer regarding the unbiasedness of the estimator. But I think your question may actually stem from another confusion:
You seem to mixing up the estimate and the estimator. The estimator is a general rule how to calculate a specific value from a given sample. This sample-specific value is called the estimate.
If you use the estimator $\bar X = \frac{1}{n} \sum_{i=1}^n X_i$ to estimate the unknown population mean $\mu$, then you have an unbiased estimator because $E(\bar X) = \mu$. Applying this estimator to a given sample, you get a specific value $\hat X$ (the sample mean), which is called the estimate. However, this does not mean that $\hat X = \mu$, i.e. the actually estimated mean $\hat X$ for a given sample is not necessarily equal (or even close) to the true value. To get an idea how well we estimated the true value, therefore we usually express our uncertainty of the estimate, e.g. via confidence intervals.
A: In practice we don't know the true value of the parameter. However, if we are to use the estimator, we have to know its properties - to be sure that it is a good one (or at least suitable/acceptable under the circumstances.)
What we do usually assume to know is the underlying statistical distribution or, at least, some kind of a model structure (in case of non-parametric estimators.)
A: 
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.

*

*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.
