How is the maximum likelihood estimate defined from a likelihood curve?

The maximum likelihood estimator (MLE) is the value of the parameter that maximizes the likelihood function for the observed data. Graphically, it is the parameter value at the peak of the likelihood curve.

Conditions: A likelihood function $L(\theta \mid x_1, \dots, x_n)$ has been constructed.; One compares values of the parameter with the data fixed.

Calculus is used by taking the derivative of the log-likelihood function with respect to the parameter, setting it to zero to find critical points, and using the second derivative test to confirm that the critical point is a maximum.

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

We use the log-likelihood because the natural logarithm is a strictly increasing (monotonic) function, so maximizing the log-likelihood yields the exact same parameter value as maximizing the original likelihood. Additionally, the logarithm converts the product of independent probabilities into a sum, which makes differentiation much easier.

Conditions: The likelihood function $L(\theta)$ is strictly positive.; The base of the logarithm is greater than 1 (e.g., the natural logarithm).