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Unformatted text preview: Point Estimation population 2 , sample 2 , S X unknown parameters observed sample statistics estimation X = 2 2 S = estimators Point Estimation Parameter : Estimator : Exampl e ( 29 , ~ ,..., , 2 1 U X X X iid n Random sample from uniform ( 29 n X X X ,..., , max 2 1 = Reasonable to use maximum of sample to estimate the upper bound Also reasonable to use sample mean to estimate population mean X 2 = Which estimator is better? Bias ( 29 ( 29  = E bias ( 29 bias overestimate ( 29 < bias underestimate ( 29 = bias unbiased Example : Random sample ( 29 all for = X E of estimator unbiased is X ( 29 2 2 2 all for = S E 2 2 of estimator unbiased is S Bias Exampl e ( 29 , ~ ,..., , 2 1 U X X X iid n Random sample from uniform X 2 1 = ( 29 n X X X ,..., , max 2 1 2 = ( 29 = 1 E Unbiased ( 29 1 2 + = n n E Biased ( 29 1 1 1 2 + = + = n n n bias underestimate ( 29 n X X X n n ,..., , max 1 2 1 3 + = is unbiased Mean Square Error unbiased small variance unbiased large variance biased small variance biased large variance Mean Square Error target 1 unbiased is 1 2 biased is 2 1 2 n better tha is But ( 29 ( 29  = 2 E MSE ( 29 ( 29 2 bias Var + = ( 29 ( 29 1 2 MSE MSE < Efficiency 2 1 to compared of efficiency ( 29 ( 29 ( 29 1 2 2 1 , MSE MSE eff = ( 29 1 , if than efficient more is 2 1 2 1 eff ( 29 ( 29 ( 29 unbiased. are and both if , 2 1 1 2 2 1 Var Var eff = Efficiency Exampl e ( 29 , ~ ,..., , 2 1 U X X X iid n Random sample from uniform X 2 1 = ( 29 n X X X ,..., , max 2 1 2 = Unbiased Biased ( 29 1 1 2 + = n bias ( 29 ( 29 ( 29 n n X Var Var MSE 3 12 4 2 2 2 1 1 = = = = ( 29 ( 29 ( 29 2 1 2 2 2 + + = n n n Var ( 29 ( 29 ( 29 ( 29 ( 29 2 1 2 2 1 1 2 2 2 2 2 + + = + + + + = n n n n n n MSE ( 29 ( 29 ( 29 ( 29 ( 29 2 for 1 6 2 1 , 2 1 1 2 + + = = n n n MSE MSE eff efficient more is 2 Efficiency Exampl e ( 29 , ~ ,..., , 2 1 U X X X iid n Random sample from uniform ( 29 n X X X ,..., , max 2 1 2 = ( 29 ( 29 ( 29 2 1 2 2 2 + + = n n n Var ( 29 ( 29 ( 29 2 1 2 2 2 + + = n n MSE ( 29 ( 29 ( 29 1 for 1 1 2 , 3 2 2 3 + = = n n n MSE MSE eff efficient more is 3 ( 29 n X X X n n ,..., , max 1 2 1 3 + = Unbiased ( 29 ( 29 ( 29 ( 29 2 1 2 2 2 3 3 + = + = = n n Var n n Var MSE Consistency ( 29 1 lim 0, any for if of estimator consistent a is = < n n n P n Estimator of based on a sample of size...
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 Fall '10
 MrChung
 Statistics

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