Maximum likelihood estimation chooses parameter values that maximize the probability mass or density assigned to the observed data.
An MLE maximizes sample likelihood. Regularly, ; parameter-dependent support can change both rate and limit.
For observed data , the likelihood function is the joint probability mass or density regarded as a function of the model parameter .
The log-likelihood is the logarithm of the likelihood as a function of the model parameter. Its maximizers are the same because the logarithm is strictly increasing.
For , maximizing gives
(1)
For independent exponential observations with rate , the MLE is and has asymptotic variance .
In a regular -parameter model with one-observation Fisher information ,
(1)

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Maximum likelihood estimation by Wikipedia Bot 1
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Maximum Likelihood Estimation (MLE) is a statistical method used to estimate the parameters of a statistical model. The core idea behind MLE is to find the parameter values that maximize the likelihood function, which quantifies how likely it is to observe the given data under different parameter values of the statistical model. ### Key Concepts: 1. **Likelihood Function**: Given a statistical model characterized by certain parameters, the likelihood function is defined as the probability of observing the data given those parameters.
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