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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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definition of the likelihood
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.

notation cont
Notation (cont.)

Supposewehavecollectedn observations: …

The orderedvalues from smallesttolargestarethen given by: …

Hence, = maximum value

and = minimum value.

possible to combine different sources of information in a common likelihood
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, )

possible to combine different sources of information in a common likelihood1
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,

possible to combine different sources of information in a common likelihood2
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).