Solution
A Normal Whose Mean Equals Its Spread
Show the problem again
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.
Worked solution
For a normal distribution N(μ, σ²) with both parameters functions of θ, the Fisher information is I(θ) = (μ′)²/σ² + 2(σ′)²/σ². (The mean and variance parts of the information are additive because the two score components are uncorrelated.) With μ = σ = θ, μ′ = σ′ = 1, so I(θ) = 1/θ² + 2/θ² = 3/θ². For θ = 2 this is 3/4 per observation, and 6 observations carry 6 · 3/4 = 9/2. The bound is 1/(9/2) = 2/9. (Using only the mean information would give the larger, looser value θ²/n = 2/3; the variance carries twice as much information as the mean.) The answer is 2/9.
Source: Standard example of a curved normal family with Fisher information computed by the chain rule; statement written for AxiomIQ.