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Section 7.6

# Section 7.6 - Section 7.6 We had a brief...

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Section 7.6 We had a brief introduction/review to random sample (iid rvs), estimator, unbiased estimator, and consistent estimator. We now discuss a method of finding estimators which possess good properties. 1. Maximum Likelihood Estimation This method yields estimators that have many desirable properties; both finite as well as large sample properties. The basic idea to find an estimator ) ( ˆ x θ which is the most likely given the data ) ,..., ( 1 n X X X = . Example 1. Let ) 1 ( π , X ~ B , Bernoulli population with density 1 , 0 , ) 1 ( ) | ( ) | ( 1 x = - = = = - x x p x X P x π π π π where ) 1 ( = = X P π . We want to estimate population proportion π based on a random sample of size n . That is, n X X ,..., 1 are independent and identically distributed random variables. 1

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For } 1 , 0 { i x , we have ] ,..., [ 1 1 n n x X x X P = = ] [ . . . ] [ 1 1 n n x X P x X P = = = n n x x x x - - - - = 1 1 ) 1 ( ... ) 1 ( 1 1 π π π π = , ) 1 ( 1 1 - - n i n i x n x π π is called the joint density of n X X ,..., 1 . Write the above density as , ) 1 ( ) 1 ( ) | ( 1 1 s n s x n x n i n i x L - - - = - = π π π π π where = n i x s 1 .
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Section 7.6 - Section 7.6 We had a brief...

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