We seem to distinguish empirical estimates of parameters from 'true' values, and make comparisons between the two. I can understand what an empirical estimate is. What is a 'true' value?

For instance, my course notes have:

Definition 4.7 (Consistent estimator)An estimator $T(X_1,…,X_n)$ for a parameter $theta$ is consistent if, for any $epsilon>0$, we have

$$

lim_{nrightarrowinfty}P(|T(X_1,…,X_n)-theta|<epsilon)=1

$$

How do we determine $theta$ apart from by estimating it, in order to compare it to an estimator?

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#### Best Answer

In general the "true value" is a fiction, defined within a model that in reality won't fit perfectly, in which case consequently there is also no such thing as a "true value". *Assuming* that there is a true parameter value is a device for doing theory and developing methods. It allows us to theoretically show that this-or-that method to estimate it has this-or-that property and works better or worse, which is a motivation for these methods even though it doesn't correspond exactly to the real situation (but then no model does).

If we simulate artificial data, however, we can fix and control the true parameter values, in which case we can compare the estimate to the true value (ignoring here potential issues with random number generation).

There are also some real situations in which true values can be "controlled" or known to some extent, for example if we have a measurement instrument that is meant to measure a certain quantity with stochastic measurement error, and in some situations we may be able to control the quantity that is measured. This still cannot guarantee the truth of the measurement error model within which the model parameter is defined, but at least we can control the real quantity that is interpreted to correspond to the true parameter value. An example for this are indirect estimates from age determination methods applied to individuals of which we know the precise age.

Sometimes we estimate parameters that correspond to existing population quantities of a usually big but finite population (such as population means of something that is well defined for all population members such as age) from a sample, in which case the population quantity would correspond to the "true" parameter value.

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