Solution
Information in a Cauchy Sample
Show the problem again
Ten independent observations come from the Cauchy distribution with density f(x; θ) = 1/(π(1 + (x − θ)²)), where the location θ is unknown. Using the Fisher information of the sample, what is the Cramér-Rao lower bound on the variance of any unbiased estimator of θ? Give a reduced fraction.
Worked solution
The score for one observation is ∂/∂θ log f = 2(x − θ)/(1 + (x − θ)²). The Fisher information is I = ∫ [2u/(1 + u²)]² / (π(1 + u²)) du = (4/π) ∫ u²/(1 + u²)³ du. Using ∫ u²/(1 + u²)³ du = π/8 over the real line, I = (4/π)(π/8) = 1/2. Information adds over independent observations, so ten of them carry 10 · 1/2 = 5, and the Cramér-Rao bound is 1/5. (The sample mean is useless here, having the same Cauchy distribution as a single observation; the bound is approached by the maximum-likelihood estimator only asymptotically.) The answer is 1/5.
Source: Standard Fisher-information computation for the Cauchy location family; statement written for AxiomIQ.