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MathidiaMathematics Academy
Statistical inference
Lesson 37 of 75
Easy practice

Maximum likelihood estimation

Objective. Derive and assess maximum likelihood estimators.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Maximum likelihood estimation chooses the parameter value θ̂ that maximises the likelihood L(θ), typically found by solving d/dθ ℓ(θ) = 0 for the log-likelihood ℓ(θ). The resulting estimator is then checked using the second derivative to confirm a maximum, and its properties such as consistency and asymptotic normality can be assessed via the Fisher information. For n independent Exponential(λ) observations x1,...,xn, maximising ℓ(λ) = n ln λ − λΣxᵢ gives λ̂ = n/Σxᵢ, the reciprocal of the sample mean.

Worked example

Find the maximum likelihood estimator for p given n independent Bernoulli(p) observations with k successes.

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Illustrated card showing ?
  1. 1Log-likelihood is ℓ(p) = k ln p + (n−k) ln(1−p)
  2. 2Differentiate: ℓ′(p) = k/p − (n−k)/(1−p)
  3. 3Set ℓ′(p) = 0 and solve for p
  4. 4Simplify to the estimator in terms of k and n

Answer: p̂ = k/n

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Common mistake. A frequent error is setting the derivative of the likelihood itself to zero rather than the log-likelihood, or forgetting to verify the stationary point is a maximum rather than a minimum. Always differentiate ℓ(θ) and check the second derivative's sign.

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