Estimate Event Rates from Unequal-Probability Samples

# Estimate Event Rates from Unequal-Probability Samples You have a population table of units and their known probabilities of being selected for inspection. A second table records which units were inspected and whether an event occurred. Explain how to estimate the population event rate. A related logging system retains all nighttime observations but only a known random fraction of daytime observations. Explain how to estimate the overall weekly event rate and how you would choose sampling fractions across strata if you controlled the logging budget. Clarify the event-rate denominator rather than assuming that a unit rate and a per-mile rate are identical. ### What a Strong Answer Covers - Inverse-inclusion weighting with positive known probabilities and a defined target population. - A distinction between a total divided by a known denominator and a ratio of estimated totals. - Valid treatment of day/night retention and uncertainty under the sampling design. - Sampling allocation based on population size, variability, cost, and minimum coverage. ```hint Restore what was less likely to be seen An observed unit selected with probability one tenth represents a different amount of population information from a census unit. ``` ### Follow-up Questions - Why is the raw event fraction among inspected units generally biased for the population fraction? - What cannot be recovered if a relevant group has zero chance of inclusion?