MLE_Lecture 1 - LECTURE 1 Some Examples of Maximum Likelihood Estimation Maximum Likelihood Estimation is a statistical estimation technique that

# MLE_Lecture 1 - LECTURE 1 Some Examples of Maximum...

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LECTURE 1 Some Examples of Maximum Likelihood Estimation Maximum Likelihood Estimation is a statistical estimation technique that chooses as estimates of the parameters of a statistical model the parameter values that, given the data, maximize the “likelihood” that the observed data was generated from a given model. Example 1: Consider a binomial experiment of n trials resulting in the observations n x x x ,..., , 2 1 , where 1 = i x if the trial was a success, 0 = i x otherwise. The likelihood of the observed sample n x x x ,..., , 2 1 is the product of the likelihoods of the individual observations i x , namely, ) ; ( ) : ( ) , ( ) ; , , , ( 2 1 2 1 p x L p x L p x L p x x x L n n = (1) p) (1 p ) 1 ( p) (1 p ... p) (1 p p) (1 p X n X x 1 x x 1 x x 1 x 1 1 n n 2 2 1 1 = = = = = n i i n i i x n x p p where = = n i i x X 1 and p is the unknown probability of success. We have assumed the independence of the observations n x x x ,..., , 2 1 and hence the ability to factor the likelihood of the sample ) ; ,..., , ( 2 1 p x x x L n into its respective parts ) ; ( ),..., ; ( ), ; ( 2 1 p x L p x L p x L n . We note that ln(L) is a monotonically increasing function of the likelihood L and thus both L and ln(L) are maximized at the same value of p.

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• Spring '16
• Econometrics, Maximum likelihood, Estimation theory, 2 second, log likelihood function

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