Solution
Two Estimators for a Stretched Interval
Show the problem again
A sample from the Uniform(θ, 3θ) distribution (θ > 0 unknown) is 4, 5, 7, 10. Let θ̂_MoM be the method of moments estimate of θ and θ̂_ML the maximum likelihood estimate. Compute θ̂_ML − θ̂_MoM. Give an exact fraction.
Worked solution
The mean of Uniform(θ, 3θ) is 2θ, so the method of moments sets x̄ = 2θ. Here x̄ = 26/4 = 13/2, so θ̂_MoM = 13/4. For maximum likelihood, the density is 1/(2θ) on [θ, 3θ], so the likelihood (2θ)^(−4) is valid only if θ ≤ min = 4 and 3θ ≥ max = 10, i.e. 10/3 ≤ θ ≤ 4. Within that range the likelihood decreases in θ, so the best choice is the smallest legal θ: θ̂_ML = 10/3. The difference is 10/3 − 13/4 = 1/12.
Source: Original problem written for AxiomIQ.