What is the maximum-likelihood estimate for each probability in a three-category model?
Conditions
- The model has three categories.
- The estimates are derived using maximum likelihood.
- The observed counts and sample size are known.
Reasoning, step by step
- Identify the observed count for each category.
- Identify the total sample size.
- Divide the observed count of each category by the total sample size.
- Verify that the resulting estimates sum to one.
- Confirm that the estimates respect the probability constraints.
Example
The script states: 'For the three-category model, maximum likelihood estimates each probability by its observed count divided by the sample size. These estimates sum to one and respect the probability constraints.'
Common misconceptions
- Assuming the estimates do not sum to one.
- Forgetting to divide by the total sample size.
- Believing that the estimates violate probability constraints.
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The maximum-likelihood estimate for the rate parameter is derived by maximizing the likelihood function . The resulting estimate is , which matches the moment estimate.
Conditions: The observations are independent.; The observations follow an exponential distribution.; The rate parameter is positive.
For a Uniform[0,θ] model, the maximum-likelihood estimate is the sample maximum, while the moment estimate is twice the sample mean. These two estimates need not agree because they are derived from different principles: the likelihood depends on the support covering every observation, whereas the moment estimate relies on the expected value.
Conditions: The model is Uniform[0,θ].; The parameter θ is positive.; Admissible parameter values must cover every observation.
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