Survey Statistics: Fat Bear Week 2026

Happy Fat Bear Week to all who celebrate. In 2024 I made a cartoon called Basu’s Bears, adapted from Basu’s (1971) elephants example, a lesson on the use of auxiliary information in survey statistics. For Fat Bear Week 2025, I wrote accompanying R code.

Gaurav’s post points out:

What makes the estimator wobble is not HT’s algebra but the sampling plan.
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If a proxy x can be measured …. use probability‑proportional‑to‑size (PPS)
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A complementary route…is to model y | x

As we saw in “equivalent models, equivalent weights (locally)”, Little 2004 linked these complementary routes, showing under which outcome model Horvitz-Thompson does well:

When is outcome model (8) correct ? When bear size y is proportional to probability we choose to measure them pi plus noise. If we follow Gaurav’s good advice and use PPS sampling, pi is proportional to auxiliary data x. So the outcome model is correct when bear size y is proportional to auxiliary data x plus noise. If x is bears before feasting, this means their expansion ratios thru salmon consumption are constant plus noise.

But what goes wrong in Basu’s (1971) example ? Instead of PPS, the compromise design is to set pi = 99/100 for Sambo and = 1/4900 for all other bears. So the outcome model is only correct when Sambo eats all other bears except for crumbs. If we can anticipate this, then HT even in this sampling plan will do just fine.