$W9_L8$: Parameter estimation: finding MME & ML estimators

The maximum-likelihood estimate for the rate parameter $\lambda$ is derived by maximizing the likelihood function $\lambda^n \exp(-\lambda \sum x_i)$. The resulting estimate is $n / \sum x_i$, which matches the moment estimate.

Conditions: The observations are independent.; The observations follow an exponential distribution.; The rate parameter $\lambda$ is positive.

In a three-category model, the maximum-likelihood estimate for each probability is its observed count divided by the sample size. These estimates sum to one and respect the probability constraints.

Conditions: The model has three categories.; The estimates are derived using maximum likelihood.; The observed counts and sample size are known.

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.