Kari -

Are you talking about survey sampling where some groups have smaller sample selection-based weights than for other subsamples or strata, in randomized sampling plans, in order to obtain better (more accurate) results for certain segments of the population?  Note that if you call that stratified random sampling, it isn't exactly, in that an optimum allocation is geared only toward overall results, not good results for certain strata, better, I think, then called subpopulations.

Oversampling of this nature will increase the overall sample size in order to know more about certain subpopulations, but I don't see how that can cause any analysis problems.  Any time you have a stratified random sample, or unequal probability sampling, you are going to have various sampling weights.  In a cluster sample, especially multistage sampling, it gets more complicated, but that can happen, regardless.

I have not been involved in this, but you might want to research sampling from "rare populations," to see if there is anything of interest for you there. 

In general, there are a number of good sampling books, which may be of interest to you.  Here are a few:

Cochran, W.G(1977), Sampling Techniques, 3rd ed., John Wiley & Sons. 

Thompson, S.K.(2012), Sampling, 3rd ed, John Wiley & Sons.

Lohr, S.L(2010), Sampling: Design and Analysis, 2nd ed., Brooks/Cole.

Särndal, CE, Swensson, B. and Wretman, J. (1992), Model Assisted Survey Sampling, Springer-Verlang.    

If you are considering a model-assisted approach, then there are two kinds of weights to consider: probability-of-selection-based weights, and regression weights.  You may also want to research calibration weights which can use both of the above.  Here is a paper which notes the use of probability-of-selection weights and regression weights, in ratio estimators:

 https://www.researchgate.net/publication/319434665_Comparison_of_Model-Based_to_Design-Based_Ratio_Estimators

I hope something above is relevant to what you want to know. 

Cheers - Jim

Thanks Jim - this is very helpful!

Hi Kari,

A modeling/sampling experiment (with freedom to deduce guidance about what samples may be useful) just recently concluded that it should be possible to solve numerically a certain class of (errors-in-variables) linear regression problems (free access until mid-Oct at https://authors.elsevier.com/c/1Xgrv7qzSjUi5) at least in geophysics and possibly other (lucky) fields with data surplus.