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# post3 - Point Estimation Likelihood function X1 X2 Xn a...

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Point Estimation X 1 , X 2 , · · · , X n : a random sample, considered as outcomes of a random experi- ment X f ( x ; θ ) , θ Ω Family of pdfs= { f ( x ; θ ) : θ Ω } ex) { N ( θ, 1) : θ Ω = ( -∞ , ) } Need to select precisely one member of the family as the pdf of X 1 , · · · , X n need a point estimate of θ . Want to define a statistic u ( X 1 , · · · , X n ) as a good point estimator. 1 Likelihood function f ( x 1 , · · · , x n ; θ ) is the joint pdf of X 1 , · · · , X n . Given that ( X 1 , · · · , X n ) = ( x 1 , · · · , x n ) is ob- served, the function of θ defined by L ( θ ; x 1 , · · · , x n ) = f ( x 1 , · · · , x n ; θ ) is called the likelihood function. ? Likelihood is not a probability Example X N ( μ, 1), n = 1 Example X 1 , X 2 : iid N ( μ, 1), n = 2 2 “observable” x population 0 1 2 a 16/25 8/25 1/25 b 1/4 2/4 1/4 c 1/16 6/16 9/16 Suppose we need to determine the population from which the observed value was selected, based on an observation x . L ( a ; 0) = 16 25 , L ( b ; 0) = 1 4 , L ( c ; 0) = 1 16 L ( a ; 1) = 8 25 , L ( b ; 1) = 2 4 , L ( c ; 1) = 6 16 L ( a ; 2) = 1 25 , L ( b ; 2) = 1 4 , L ( c ; 2) = 9 16 For any given x , reasonable to find the popu- lation which maximizes L ( θ ; x ). MLE 3 Method of Maximum Likelihood X 1 , · · · , X n f ( x ; θ ) , θ

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