post3 - Point Estimation Likelihood function X1, X2, , Xn:...

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Point Estimation X 1 ,X 2 , ··· n : a random sample, considered as outcomes of a random experi- ment X f ( x ; 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 , n need a point estimate of θ . Want to define a statistic u ( X 1 , n )a sa good point estimator. 1 Likelihood function f ( x 1 , ,x n ; θ ) is the joint pdf of X 1 , n . Given that ( X 1 , n )=( x 1 , n ) is ob- served, the function of θ defined by L ( θ ; x 1 , n )= f ( x 1 , n ; θ ) is called the likelihood function. ? Likelihood is not a probability Example X N ( μ, 1), n =1 Example X 1 2 : iid N ( μ, 1), n =2 2 “observable” x population 01 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 1 4 ( c 1 16 L ( a ;1)= 8 25 ( b 2 4 ( c 6 16 L ( a ;2)= 1 25 ( b 1 4 ( c 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 , n f ( x ; θ ) Ω w/ joint pdf f ( x 1 ; θ ) f ( x 2 ; θ ) f ( x n ; θ ) L ( θ ; x 1 , n f ( x 1 ; θ ) f ( x 2 ; θ ) f ( x n ; θ ) Suppose u ( x 1 , n ) s.t.
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post3 - Point Estimation Likelihood function X1, X2, , Xn:...

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