IAI Actuarial Core Principles · Actuarial Statistics · Estimators and their properties
A random sample of n = 50 is drawn from an exponential distribution with mean θ (density f(x) = (1/θ)e^{-x/θ}). The Fisher information for one observation is 1/θ². If θ = 10, what is the Cramér-Rao lower bound for the variance of an unbiased estimator of θ?
The Cramér-Rao lower bound is 2. For n independent observations the bound is the reciprocal of n times the single-observation information, giving θ²/n = 100/50 = 2.
- A0.5
- B1
- C2Correct
- D5
- 10
Explanation
The bound is 1/(n I(θ)) = θ²/n = 100/50 = 2. Using n = 50 without squaring θ would give 0.2 and using θ/n would give 0.2. Using 100 without dividing by n gives 100. The value 2 matches the correct working.
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