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Lecture 3

Lecture 3. Notation. Definition of the likelihood. Pawitan (2001) page 22: Assuming a statistical model parameterized by a fixed and unknown , the likelihood is the probability of the observed data considered as a function of. Notation ( cont .).

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Lecture 3

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  1. Lecture 3

  2. Notation

  3. Definition of the likelihood Pawitan (2001) page 22: Assuming a statistical modelparameterized by a fixed and unknown, the likelihood is the probabilityof the observed data considered as a functionof.

  4. Notation (cont.) Supposewehavecollectedn observations: … The orderedvalues from smallesttolargestarethen given by: … Hence, = maximum value and = minimum value.

  5. Possibletocombine different sourcesof information in a common likelihood: Example 2.7 (page 28) Two independent samples from Sample 1: The maximum of 5 observations, , is reported. Sample 2: The averageof 3 observations, The likelihood for Sample 2 easiesttoconstruct. We have So, )

  6. Possibletocombine different sourcesof information in a common likelihood: Example 2.7 (cont.) The likelihood for Sample 1 is a little bit moretricky(seeExample 2.4 for moredetails). Let be the cumulative distribution function for a standard normal distribution, and the probabilitydensityfunction. We have The probabilitydensityfunction is the derivativeofthisfunction: So,

  7. Possibletocombine different sourcesof information in a common likelihood: Example 2.7 (cont.) Wehave ) The two log-likelihoodscannowsimply be added: The MLE, is computed by maximizing. Butwealso get the uncertainty in this estimated parameter. (SeeFigure 2.4 in Pawitan 2001).

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