I have a real-life problem similar to the following two subproblems, which are about maximizing samples' representativeness (and maybe getting smaller variance than with simple random sampling, but due to the small sample size, the stratification gains cannot be used):

Example 1:

Given is a population of 1000 people, their height is known. The sample size shall be n=5. Their average weight is to be estimated.

What is a reasonable way to draw a sample, taking into account the small sample size and the prior information? Intuitively, better than simple random sampling is drawing people equally distributed from small to large (small, semi-small, medium, semi-large, large).

Example 2:

Given is a population of 1000 people again, the sample size is n=5. Now 45% of the population belong to group 1, 25% to group 2, 15% to group 3, 10% to group 4 and 5 % to group 5. One can assume that members of the same group have similar weight.

What is a good way now to draw a sample of this population? Intuitively about 45% of the sample should be of group 1, 25% of group 2... (But how to calculate mean and variance then?)

The sampling procedures have to include a random component still, and –- in addition to the expectation estimates -- variance estimates are needed for constructing confidence intervals for the unknown average weight.

  • $\begingroup$ Actually, your intuition is incorrect in example 2. Suppose these were institutions of higher learning in the US, the classes are types of institution (e.g., state schools, liberal arts colleges, etc), and you want to estimate average enrollment. The optimum sample includes disproportionately more of the huge state schools, even though they comprise only 2% of the total count, because they contribute so much to the total student population. $\endgroup$ – whuber Aug 30 '11 at 18:04
  • $\begingroup$ Look up ratio estimators and their generalizations for the first example and use standard stratified estimators for the second. A Horvitz-Thompson estimator will give confidence intervals. $\endgroup$ – whuber Aug 30 '11 at 18:06
  • $\begingroup$ What is your prior information in Example 1? You cannot get something from nothing, though: if you deeply stratify, you still need to have at least $n\ge2$ for variances to be estimable, at least within the design-based paradigm. If you need to have at least one element in each stratum (from some other considerations), you can take the samples within each stratum, and then your (stratified) estimate of the mean will be $\bar y_s = \sum_{h=1}^5 W_h \bar y_h$ where $W_1 = 0.45$, $W_2=0.25$, etc. This may be the best sampling strategy if the strata are sufficiently homogeneous. $\endgroup$ – StasK Aug 31 '11 at 2:47

I would suggest Bayesian methods. Specify priors over these values and compute the posterior. This will take into account the amount of data you have, as requested, and it will incorporate a notion of uncertainty into the assignment of people to each of your discrete groups.

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    $\begingroup$ This is pointless. Bayesian methods are not a panacea. In survey statistics, the random elements are the indicators of sample inclusion, not the measured values, and the likelihood of any SRS is just $(C_N^n)^{-1}$, i.e., non-informative. $\endgroup$ – StasK Aug 31 '11 at 2:40
  • $\begingroup$ I don't agree. He asked how to sample from something given prior knowledge but after observing the data. Unless I'm missing something, that's what Bayes Theorem is for. Not to be an ideologue, but this seems like a pretty obvious place to apply Bayesian ideas. $\endgroup$ – William Aug 31 '11 at 14:33
  • $\begingroup$ No, I asked how to sample having prior information on the population's structure, of course not having observed data. My question is which individuals of [which group]/[what size] to include in the sample, and what estimators are to be used then. $\endgroup$ – RichardN Sep 1 '11 at 13:38

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