Solution
A Unit Window of Unknown Position
Show the problem again
A sample 2.3, 2.9, 2.6 is drawn from the Uniform(θ, θ+1) distribution, with θ unknown. The set of values of θ that maximize the likelihood is an interval. What is its length? Give an exact fraction or a decimal.
Worked solution
The density is 1 on [θ, θ+1], so the likelihood equals 1 when every observation lies in that window and 0 otherwise. All observations lie in the window exactly when θ ≤ min = 2.3 and θ + 1 ≥ max = 2.9, i.e. 1.9 ≤ θ ≤ 2.3. Every such θ gives the same maximal likelihood 1, so the MLE is not unique: the set of maximizers is [1.9, 2.3], of length 0.4 = 2/5.
Source: Standard example of non-unique maximum likelihood estimators. Statement written for AxiomIQ.