A Normal Whose Mean Equals Its Spread

Observations are independent and normally distributed with mean θ and standard deviation θ (so the variance is θ²), where θ > 0 is unknown. A sample of n = 6 is drawn. What is the Cramér-Rao lower bound on the variance of an unbiased estimator of θ, evaluated at θ = 2? Give a reduced fraction.

• bias
• distributions
• estimation
• variance

Source: Standard example of a curved normal family with Fisher information computed by the chain rule; statement written for AxiomIQ.