maximum likelihood

• What Is the MLE for a Uniform(θ, 0) Distribution?
  For iid data from Uniform(θ, 0) with θ < 0, the likelihood is maximized at the sample minimum; if endpoints are reversed, it is the maximum.
• What Is the MLE of θ for a Uniform(0, θ) Distribution?
  For an i.i.d. sample from Uniform(0, θ), the maximum likelihood estimate is the largest observation, because the likelihood decreases for θ above that value.
• How to Find the MLEs of a Negative Binomial Distribution
  For negative-binomial data, the MLE of p has a closed form when the size is fixed; estimate unknown size by solving a digamma score equation numerically.
• How to Find the MLE of a Uniform Distribution
  For an i.i.d. sample from Uniform(0, θ), the maximum-likelihood estimate is the sample maximum; endpoint estimates differ when both bounds are unknown.