Maximum likelihood estimation, Advanced Statistics

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Maximum likelihood estimation is an estimation procedure involving maximization of the likelihood or the log-likelihood with respect to the parameters. Such type of estimators is particularly important because of their many desirable statistical properties such as consistency, and asymptotic efficiency. As an example considers the number of successes, X, in a sequence of random variables from a Bernoulli distribution with success probability p. The likelihood can be given by


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differentiating the log-likelihood, L, with respect to p gives the following

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