Why do the maximum-likelihood and moment estimates not agree for a Uniform[0,θ] model?
Conditions
- The model is Uniform[0,θ].
- The parameter θ is positive.
- Admissible parameter values must cover every observation.
Reasoning, step by step
- Identify the maximum-likelihood estimate for θ, which is the sample maximum.
- Identify the moment estimate for θ, which is twice the sample mean.
- Compare the two estimates.
- Conclude that they need not agree because they are based on different statistical principles.
Example
The script states: 'For Uniform[0,θ] with positive θ, admissible parameter values must cover every observation. The MLE is the sample maximum, while the moment estimate is twice the sample mean; they need not agree.'
Common misconceptions
- Assuming the maximum-likelihood estimate and the moment estimate are always identical.
- Forgetting that the parameter θ must be positive.
- Believing that the moment estimate is the sample maximum.
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