NotesNov17 - The value of that maximizes the likelihood...

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The value of θ that maximizes the likeli- hood function is called the maximum like- lihood estimator (MLE) of the unknown parameter. Intuition : the joint pmf p X 1 ,...,X n ( x 1 , . . . , x n ; θ ) = n Y j =1 p X ( x j ; θ ) expresses the probability (likelihood) of ob- taining the observed values X 1 = x 1 , X 2 = x 2 , . . . , X n = x n if the true value of the parameter is θ . The maximum likelihood estimator of the unknown parameter is that value of θ that makes the observed values the most likely.
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In practice it is often easier to maximize the logarithm of the likelihood function, the so- called log-likelihood function L ( x 1 , . . . , x n ; θ ) = log p X 1 ,...,X n ( x 1 , . . . , x n ; θ ) = n X j =1 log p X ( x j ; θ ) .
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The unknown parameter θ may be multidi- mensional ( a vector). Example : Estimate both parameters n and p in the Binomial distribution. In this case θ = ( n, p ).
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Example : Let X 1 , . . . , X n be a sample from a Poisson distribution with an unknown parame- ter λ . Find the MLE of λ .
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Maximum likelihood estimation in the con- tinuous case Suppose the observations X 1 , X
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