MLE for Exponential Distribution

Overview

We derive the MLE for the Exponential distribution \(X \sim \text{Exp}(\lambda)\) with density \(f(x; \lambda) = \lambda e^{-\lambda x}\) for \(x > 0\).

Derivation

The log-likelihood is:

\[ \ell(\lambda) = n \log \lambda - \lambda \sum_{i=1}^n x_i \]

Setting the score to zero:

\[ \frac{d\ell}{d\lambda} = \frac{n}{\lambda} - \sum_{i=1}^n x_i = 0 \implies \hat{\lambda}_{MLE} = \frac{n}{\sum_{i=1}^n x_i} = \frac{1}{\bar{X}} \]

Properties

• \(\hat{\lambda}_{MLE} = 1/\bar{X}\) is biased but consistent
• Fisher information: \(I(\lambda) = 1/\lambda^2\)
• Asymptotic variance: \(\text{Var}(\hat{\lambda}) \approx \lambda^2/n\)