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Lecture 4

Lecture 4. The Fisherian Way or Likelihoodists. Pawitan 2001 page 15. Inference is possible directly from the likelihood function . For frequentists this is OK for large samples For Bayesians it is OK given a prior

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Lecture 4

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  1. Lecture 4 The Fisherian Way or Likelihoodists

  2. Pawitan 2001 page 15 • Inference is possibledirectly from the likelihoodfunction. • For frequentiststhis is OK for largesamples • For Bayesians it is OK given a prior • If probabilitystatementscan be constructedthenuse it, otherwiselilkelihoodbasedinference • Importantconcepts • Maximum likelihood • Fisher information • Likelihoodbasedconfidence interval • Likelihoodratio • Likelihood principle: Two data sets thatproduce the same (proportional) likelihoodcontain the same evidence (Birnbaum 1962)

  3. Likelihood-based intervals • Fisher (1973) proposed the useof the observedlikelihoodfunctiondirectlytocommunicate the uncertaintyof a parameter . • Whenexactprobability-basedinference is not available • Alsowhen the samplesize is too small toallowlarge-sampleresults Definition: Likelihood interval A set of parameter values where is a ”cut-off point” and is the normalizedlikelihood.

  4. Likelihood-based intervals Does a certaincut-off pointcorrespondto a specific P-value? Yes, butonlyif the likelihood is regular. Example on page 36 in Pawitan 2001.

  5. Let … be an iidsample from . Assume known. MLE: Normalized log-likelihood: (eq 1) => => (eq 2) So, if for somesignificancelevelwechoose a cut-off: Thenconfidence interval for For instance,

  6. Likeklihood-based CI vs. Wald CI Definition: 95% Wald CI • For a non-regularlikelihoodwecan try tofind a transformation so that it becomesmoreregular and use • For likelihood-based CI we do not needtofind a transformation!

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