Multidimensional quantiles I have 1000 observations with 2 continuous variables :

Observation ID | X | Y

 A: The goal is to get good estimates and decent standard errors.
First, sort by x; then systematic sampling approaches will effectively stratify by x and by y.


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*Take a single systematic sample of n observations, where n is a multiple of 2. To estimate standard errors, "stratify" the sample, with each stratum consisting of two neighboring observations: So stratum 1 is observations 1 & 2; stratum 2 is observations 3 & 4; and so on. There will be H = n/2 strata, and will provide H-1 d.f. for standard errors. This is estimator v3 on page 300 in Wolter (2007). For examples of taking systematic samples see Kish (1965, p. 113). If N/n is not an integer, the last stratum might have 3 members.


I don't recommend this approach if you expect much non-response, because survey programs like Stata will complain if you have strata with < 2 observations. To correct this, you will have to merge neighboring strata by hand. Also this commits you to a specific sample size in advance, which might turn out to be something of a burden.


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*Take, say, m = 4 or 5 independent systematic samples in x order, each of size n/m. In the analysis, treat each sample as a cluster. This is the method of replicated subsamples. See Deming, 1960, for many examples.


This design has two advantages: First, it provides decent estimates of standard errors, although albeit with lower degrees of freedom. Second, it can adapt to your situation: Consider the first sample as a pilot test. You'll get some preliminary data, test different approaches, and get an idea of the effort needed to get completed interviews. Then do as many additional subsamples as you can afford, but complete the ones you start and devote effort to reducing non-response in those subsamples.
References:
Deming, W. Edwards. 1960. Sample design in business research. New York: Wiley.Wolter, Kirk M. 2007. Introduction to variance estimation. New York: Springer.
Kish, Leslie. 1965. Survey sampling. New York: Wiley.
Wolter, Kirk M. 2007. Introduction to variance estimation. New York: Springer.
