How do you derive the maximum-likelihood estimate for the rate parameter of independent exponential observations?

Conditions

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

Reasoning, step by step

1. Write the likelihood function for independent exponential observations: $\lambda^n \exp(-\lambda \sum x_i)$.
2. Maximize this likelihood function with respect to $\lambda$.
3. Solve for the value of $\lambda$ that maximizes the function.
4. Obtain the maximum-likelihood estimate: $n / \sum x_i$.
5. Compare this estimate to the moment estimate, noting they are identical.

Example

The script states: 'For independent exponential observations with positive rate λ, the likelihood is λⁿexp(−λΣxᵢ). Its maximum gives $n/Σx$ᵢ, matching the moment estimate.'

Common misconceptions

• Assuming the maximum-likelihood estimate and the moment estimate are always different.
• Forgetting that the rate parameter $\lambda$ must be positive.
• Confusing the rate parameter $\lambda$ with the mean parameter $1/\lambda$.

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Connected concepts

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