Solution
A Heavy-Tailed Measurement
Show the problem again
Observations are independent with density f(x) = θ x^(−(θ+1)) for x ≥ 1, where θ > 0 is unknown. Three observations are e, e² and e⁴ (e = 2.71828...). What is the maximum likelihood estimate of θ? Give an exact fraction.
Worked solution
The log-likelihood is n ln θ − (θ+1) Σ ln xᵢ. Setting the derivative to zero gives θ̂ = n / Σ ln xᵢ. Here Σ ln xᵢ = 1 + 2 + 4 = 7 and n = 3, so θ̂ = 3/7. The log-likelihood is strictly concave in θ, so this stationary point is the global maximum.
Source: Standard Pareto maximum-likelihood exercise. Statement written for AxiomIQ.