How is calculus used to find the maximum likelihood estimator?

Conditions

• The log-likelihood function is differentiable with respect to the parameter $\theta$.
• The parameter space allows for a maximum to exist.

Reasoning, step by step

1. Write down the likelihood function $L(\theta \mid x_1, \dots, x_n)$.
2. Take the natural logarithm to get the log-likelihood $\ell(\theta \mid x_1, \dots, x_n)$.
3. Compute the first derivative of the log-likelihood with respect to $\theta$: $\frac{d}{d\theta} \ell(\theta)$.
4. Set the first derivative equal to zero and solve for $\theta$ to find the critical points.
5. Compute the second derivative of the log-likelihood with respect to $\theta$.
6. Verify that the second derivative is negative at the critical point to confirm it is a maximum.

Example

The speaker outlines the steps: "write down likelihood, take log, take first derivative, set to zero, find maximizer, take second derivative." She notes, "we can just go ahead and use calculus to take derivatives and find the maximizer."

Common misconceptions

• Forgetting to check the second derivative to ensure the critical point is a maximum rather than a minimum.
• Assuming that setting the derivative to zero always yields a unique global maximum without checking boundary conditions.

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Connected concepts

Explore next

• What are the standard steps to find the Maximum Likelihood Estimator?
• Why do we use the log-likelihood instead of the raw likelihood function when finding the MLE?

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