Trading Bias for Variance
Six independent observations are drawn from a normal distribution with unknown mean and variance σ². Let S = Σ(Xᵢ − X̄)² be the sum of squared deviations about the sample mean, and consider estimators of σ² of the form S/d for a constant d > 0. Choose d to minimise the mean squared error E[(S/d − σ²)²] (the same d works for every σ²). Let σ = 1 for definiteness: what is the minimum mean squared error? Give a reduced fraction.
Related problems & prerequisites
Worth solving first
- A Heavy-Tailed Measurement 3.0/10
Source: Original problem written for AxiomIQ.