Skip to content

IAI Actuarial Core Principles · Actuarial Statistics · Estimators and their properties

For a sample of size n from a Normal(μ, σ²) distribution with μ known, the estimator S0² = (1/n)Σ(Xi-μ)² is unbiased for σ². Its variance is 2σ⁴/n. What is the Cramér-Rao lower bound for unbiased estimators of σ² and the efficiency of S0²?

The Fisher information for σ² is n/(2σ⁴), so the Cramér-Rao bound is 2σ⁴/n. The estimator's variance equals this bound, so it is fully efficient with efficiency 1.

  1. ABound σ⁴/n; efficiency 0.5
  2. BBound 2σ⁴/n; efficiency 1Correct
  3. CBound 2σ⁴/n²; efficiency 1/n
  4. DBound σ²/n; efficiency 1
  5. Bound 2σ⁴/(n-1); efficiency (n-1)/n

Explanation

Fisher information for σ² with μ known is n/(2σ⁴), so the bound is 2σ⁴/n. S0² has variance equal to the bound, so efficiency is 1. Option with σ⁴/n confuses the bound.

Did you get it right without looking?

One question tells you little. A timed set on Estimators and their properties shows your real accuracy, how long you take and where you lose marks.

Practise Estimators and their propertiesFree diagnostic mock

More Estimators and their properties questions