Sampling issue: constructing weights to take into account sampling procedure?

[Isabel Martinez]

I have a question on how to deal with the following sampling problem:
I have administrative data from different regions of a country, so in principle I observe 100% of the population in each region.
However, for one large region X, I only have a random sample which covers 50% of the population with income below the 10th decile but 100% of the population belonging to the 10th decile of the income distribution.
For my analysis, I need to pool all the different regions.

My question: How do I take into account the sample construction in region X? Can I construct fweigths (observations for those below the 10th decile would get a weight of 2, everyone else gets a weigth of 1)? Or is there a better way to do this?

Help is highly appreciated!

[Phil Bromiley]

I would be tempted to look at the survey tools which explicitly allow for partial samples.

A big question about region X is whether the data are randomly sampled or not. It is clear that are not. You probably have a sample selection problem and need to look at Heckman estimators.