lecture 3
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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 .). - PowerPoint PPT PresentationTRANSCRIPT
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Lecture 3
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Notation
Likelihood
Parameter
Nuisance parameter
Probability being a function of the parameter stochastic variable
observed value
Density function
MLE Maximum likelihood estimate
Standard error
Score function, ie first derivative of Fisher Information, minus the second derivative of
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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 .
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Notation (cont.)Suppose we have collected n observations: …
The ordered values from smallest to largest are then given by: …
Hence, = maximum value and = minimum value.
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Possible to combine different sources of 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 average of 3 observations,
The likelihood for Sample 2 easiest to construct.We have So, )
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Possible to combine different sources of information in a common likelihood:
Example 2.7 (cont.)
The likelihood for Sample 1 is a little bit more tricky (see Example 2.4 for more details).Let be the cumulative distribution function for a standard normal distribution, and the probability density function.
We have The probability density function is the derivative of this function:So,
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Possible to combine different sources of information in a common likelihood:
Example 2.7 (cont.)
We have )
The two log-likelihoods can now simply be added:
The MLE, is computed by maximizing .But we also get the uncertainty in this estimated parameter.(See Figure 2.4 in Pawitan 2001).