5102-Lecture-03

5102-Lecture-03 - Lecture 3 Stat 5102-004 24 January 2011...

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Lecture 3 Stat 5102-004 24 January 2011
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Likelihoods Under Random Sampling Suppose y = ( y 1 ,...,y n ) are a sample from a population. The iid random variables Y 1 ,...,Y n model random sampling. The joint density and likelihood are f ( y | θ ) = n Y i =1 f ( y i | θ ) and lik( θ | y ) = n Y i =1 lik( θ | y i ) . The log likelihood is ( θ | y ) = n X i =1 ( θ | y i ) . STAT 5102 (Theory of Statistics) Lecture 3 1 / 7
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Exponential Family Likelihoods Under Random Sampling Suppose the iid variables Y 1 ,...,Y n come from an exponential family. f ( y i | θ ) = h ( y i )exp ( y i θ - κ ( θ ) ) The likelihood given y = ( y 1 ,...,y n ) is lik( θ | y ) = " n Y i =1 h ( y i ) # exp ( n ¯ - ( θ ) ) . Exercise : Derive this. STAT 5102 (Theory of Statistics) Lecture 3 2 / 7
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Log Likelihoods and Exponential Families The log likelihood ( θ | y ) = n - κ ( θ )] + n i =1 log h ( y i ) has derivatives 0 ( θ | y ) = n y - κ 0 ( θ )] and 00 ( θ | y ) = -
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5102-Lecture-03 - Lecture 3 Stat 5102-004 24 January 2011...

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