Methods Based on Likelihood and Their Asymptotic properties
Asymptotic properties of maximum likelihood estimators (MLEs), likelihood ratio tests (LRTs) and frequency χ2 tests for multinomial probabilities involving unknown parameters estimated by their MLEs are topics covered in this chapter. Under regularity conditions, the MLEs are not only consistent, but converges almost surely to the true value of the parameter and are asymptotically normal. Most important is the property of efficiency (ie, attaining the Cramér-Rao lower bound for the variance in an appropriate sense). The LRT generalizes the most powerful test in the simple-against-simple hypothesis testing problem, using the unrestricted MLE and the MLE under the null hypothesis ...
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