A Heavy-Tailed Measurement
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.
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Source: Standard Pareto maximum-likelihood exercise. Statement written for AxiomIQ.