The LLN refers to the convergence of the sample average to the population mean when the sample is iid. This is, for a random variable $X$ with $E(X)=\mu$, we know that in a sample of size N, and with iid observations:

$$ \frac{\sum_i x_i}{N} \rightarrow \mu $$

Now, I have mixture of census and sample data. In particular, there is a variable $Y$, for which information is available of all the population. For a predefined threshold $\eta$, the statistical office decided to produce a census for the $y_i>\eta$ group, and to produce a sample for the $y_i<\eta$ group.

I am puzzled about the LLN in this context.

  1. I think it makes no sense to talk about LLN for population data. Provided no measurement error or other issue exist, the census subgroup average is the average of that particular subgroup. Is this the case?
  2. LLN is relevant for the sampled subgroup. The sample is however not iid, at least not with respect to the whole population. Would the LLN still hold at the subsample level? The convergence would not be to $\mu$, unless $Y$ is in itself independently distributed, i.e. unrelated to $X$?
  3. I am ultimately interested in formalising $E(X)$. For instance, it is clear that given my dataset, $ \frac{\sum_i x_i}{N} \rightarrow \mu $ is not true anymore (from point 1 and 2 above). Is there another mathematical expression for this when I have population data too? I want to make transformations using covariance formulas, so I need to make sure I have a correct sample equivalent of $E(X)$.
  4. Maybe a way to overcome the whole problem is to simply produce a sample from my census data, even if I would be losing information. But how to ensure the resulting one is iid? Something perhaps related to $\eta$?

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