Inverse Sampling
Also known as: Sequential Sampling
Inverse Sampling is a sequential sampling strategy where sampling continues until a fixed number of occurrences of a rare event or item of interest is observed. Introduced by J. B. S. Haldane in 1945, it is particularly efficient for estimating rare event probabilities or proportions when the target is sparse and costly to detect.
Key highlights
- Efficient for rare-event estimation: achieves target precision with fewer total samples on average
- Naturally concentrates effort on detecting the target, avoiding wasted sampling on non-occurrences
- Simple stopping rule (count to r successes) is easy to implement in the field
- Well-suited for adaptive and cost-constrained surveys
- Unbiased estimators of p exist and are well-established
Intuition
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How it works
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When to use it
Apply inverse sampling when the target event or item is rare (small p) and detection is expensive or time-consuming. Ideal in quality control (detecting defects in batches), disease surveillance (finding cases in low-prevalence populations), wildlife population estimation (trapping until a fixed number is captured), and any setting where direct fixed-size sampling would waste effort on non-occurrences. Less suitable for common events where standard fixed-size sampling is more efficient.
Strengths & limitations
- Efficient for rare-event estimation: achieves target precision with fewer total samples on average
- Naturally concentrates effort on detecting the target, avoiding wasted sampling on non-occurrences
- Simple stopping rule (count to r successes) is easy to implement in the field
- Well-suited for adaptive and cost-constrained surveys
- Unbiased estimators of p exist and are well-established
- Sample size n is random, complicating advance planning and budgeting
- Applicable only when the event or item can be clearly identified upon observation
- Requires sampling without replacement from a finite population for unbiased estimates
- Variance calculations are more complex than for fixed-size sampling
- Less efficient than fixed-size sampling for non-rare events (p not small)
Common pitfalls
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Applications
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Frequently asked
Why is inverse sampling more efficient for rare events?
With rare events, direct fixed-size sampling wastes effort on many non-events. Inverse sampling skips non-events and stops as soon as r successes accumulate, reducing wasted samples and total cost per unit precision for rare-event estimation.
How do I choose r?
Larger r improves precision but increases expected sample size. Choose r based on desired precision for estimating p. Theory provides formulas linking r, expected n, and variance; optimize r for your cost and precision trade-off.
What estimator should I use for p?
The most common unbiased estimator is p̂ = (r - 1) / (n - 1), though other forms exist. Variance calculations account for the random sample size. Consult specialized texts or software for your specific scenario.
Does inverse sampling work with replacement?
Yes, but the analysis differs. Sampling with replacement (Poisson-type) assumes an infinite population and has different unbiasedness and variance properties than sampling without replacement. Specify which scheme applies to your study.
Can I use inverse sampling if I do not know p in advance?
Yes. Inverse sampling is adaptive and does not require advance knowledge of p. It will automatically spend more effort if p is smaller than expected, making it robust to uncertainty about rarity.
Sources
- 1.Haldane, J. B. S. (1945). On a method of estimating frequencies. Biometrika, 33(3), 222–224.
- 2.Serfling, R. J. (1968). Contributions to central limit theory for dependent variables. Annals of Mathematical Statistics, 39(4), 1158–1175.
- 3.Lahiri, D. B. (1951). On the question of bias of some estimators and a suggestion for unbiased estimation. Journal of the Indian Statistical Association, 1(1), 25–42.
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Cite this page
ScholarGate. (2026, June 3). Inverse Sampling. ScholarGate. https://scholargate.app/sampling/inverse-sampling