What is the maximum likelihood estimator according to this clip?
Conditions
- A likelihood function has been constructed.
- One compares values of the parameter with the data fixed.
Reasoning, step by step
- Construct the likelihood function from the observed data and the model.
- Plot or consider the likelihood as a function of the parameter .
- Identify the highest point (peak) on the likelihood curve.
- Determine the parameter value corresponding to this peak.
- Conclude that this maximizing parameter value is the maximum likelihood estimator.
Example
The instructor points to the peak of the sketched likelihood curve and asks, "what is this value here of theta that maximizes the likelihood function?" She then states, "that maximizer is called the maximum likelihood estimator."
Common misconceptions
- Assuming the MLE is the average of all possible parameter values.
- Thinking that the MLE guarantees the parameter is exactly the true value, rather than just the most plausible one given the data.
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Related questions
The maximum likelihood estimate is defined as the parameter value that corresponds to the peak (maximum) of the likelihood curve. It is the value of the parameter that gives the highest likelihood for the observed data.
Conditions: A likelihood function over candidate parameter values is available.; The data are 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 .; 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 is strictly positive.; The base of the logarithm is greater than 1 (e.g., the natural logarithm).
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