IAI Actuarial Core Principles · Actuarial Statistics · Estimators and their properties
X1,...,Xn are independent Uniform(0, θ) variables. Which statement is correct?
The sample maximum is sufficient for θ because the joint density depends on the data only through it. The Cramér-Rao bound is not valid here, since the range of the variable depends on θ and regularity conditions fail.
- AThe sample mean is sufficient for θ and the Cramér-Rao bound applies directly
- BThe sample maximum is sufficient for θ, and the regularity conditions for the Cramér-Rao bound fail because the support depends on θCorrect
- CThe sample minimum is sufficient for θ
- DNo sufficient statistic of lower dimension than n exists
- The sample sum is sufficient and complete for θ
Explanation
The joint density is θ^{-n} on 0<x(n)<θ, so it factorises through the maximum. Since the support depends on θ, differentiation under the integral is invalid and the Cramér-Rao bound does not apply in the usual form.
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