IAI Actuarial Core Principles · Actuarial Statistics · Estimators and their properties
Let X1,...,Xn be independent Poisson(λ) random variables. Which of the following is a sufficient statistic for λ?
The sample sum is sufficient for λ. The joint Poisson probability function factorises into a part depending on λ only through the sum of observations and a part free of λ, so the factorisation theorem applies.
- AThe sample maximum
- BThe sample sum X1+...+XnCorrect
- CThe sample range
- DThe first observation X1 alone
- The sample median
Explanation
The joint pmf is e^{-nλ} λ^{ΣxI} / ΠxI!, so by the factorisation theorem it depends on λ only through the sum. The maximum, range, median and X1 do not capture all the information about λ.
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