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→Maximum Likelihood Estimator
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===Maximum Likelihood Estimator=== | ===Maximum Likelihood Estimator=== | ||
(MLE) | (MLE) | ||
Maximum Likelihood Estimator | |||
* Write out the likelihood function <math>L(\theta; \mathbf{x}) = f(\mathbf{x}; \theta)</math> | |||
* (Optional) Write out the log-likelihood function <math>l(\theta) = \log L(\theta; \mathbf{x})</math> | |||
* Take the derivative of the log-likelihood function w.r.t <math>\theta</math> | |||
* Find the maximum of the log-likelihood function by setting the first derivative to 0 | |||
* (Optional) Make sure it is the maximum by checking that the Hessian is positive definite | |||
* Your MLE <math>\hat{\theta}</math> is the value which maximizes <math>L(\theta)</math> | |||
* Note if the derivative is always 0, then any value is the MLE. If it is always positive, then take the largest possible value. | |||
;Notes | |||
* If <math>\hat{\theta}</math> is the MLE for <math>\theta</math> then the MLE for <math>g(\theta)</math> is <math>g(\hat{theta})</math> | |||
===Uniformly Minimum Variance Unbiased Estimator (UMVUE)=== | ===Uniformly Minimum Variance Unbiased Estimator (UMVUE)=== | ||
UMVUE, sometimes called MVUE or UMVU.<br> | UMVUE, sometimes called MVUE or UMVU.<br> | ||