# A reliance on repeated sampling ideas can lead to logical paradoxes that appear in common rather than esoteric procedures?

I am currently studying the textbook In All Likelihood -- Statistical Modelling and Inference Using Likelihood by Yudi Pawitan. Section Repeated sampling principle: the frequentists of chapter 1 says the following:

True frequentism states that measures of uncertainties are to be interpreted only in a repeated sampling sense. In areas of statistical application, such as medical laboratory science or industrial quality control, where procedures are naturally repeated many times, the frequentist measures are very relevant. The problem arises as the requirement of repeat experimentation is allowed to be hypothetical. There are many areas of science where experiments are unlikely to be repeated, for example in archaeology, economics, geology, astronomy, medicine, etc. A reliance on repeated sampling ideas can lead to logical paradoxes that appear in common rather than esoteric procedures.

It is this last part that I am curious about:

A reliance on repeated sampling ideas can lead to logical paradoxes that appear in common rather than esoteric procedures.

How do such paradoxes come about, and what are some specific examples?

I would greatly appreciate it if people would please take the time to elaborate on this.

Say you have a measurement of real value interfered with noise. The measurement is given by $$y(n) = x + \omega_n$$ where $$x$$ is assumed to be a constant value. An unbiased estimator of $$x$$ can be the mean value: $$\hat{x}=\frac{\sum_n y(n)}{n}$$ and the variance of the estimation should be:
$$var(\hat{x})=\frac{\sigma_\omega^2}{n}$$
$$\hat{\sigma}_\hat{x}^2=\frac{\sum_n (y(n)-\frac{\sum_n y(n)}{n})}{n-1}$$
For repeatable tests, with a constant $$x$$, $$\hat{\sigma}_\hat{x}^2$$ should decrease as the number of the tests grow. However, if $$x$$ varies with time, say $$x= x(n)$$, then estimated variance will increase, rather than decrease.