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stat1801_7 - PointEstimation population 2 unknown...

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Point Estimation population 2 , σ μ sample 2 , S X unknown  parameters observed sample statistics estimation X = μ 2 2 S = σ estimators
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Point Estimation Parameter :  θ Estimator :  θ Exampl e   ( 29 θ , 0 ~ ,..., , 2 1 U X X X iid n Random sample from uniform ( 29 n X X X ,..., , max ˆ 2 1 = θ Reasonable to use maximum of sample to estimate the upper bound Also reasonable to use sample mean to estimate population mean X 2 ˆ = θ Which estimator is better?
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Bias ( 29 ( 29 θ θ θ - = E bias ( 29 0 θ bias overestimate ( 29 0 < θ bias underestimate ( 29 0 = θ bias unbiased Example : Random sample   ( 29 μ μ all for = X E μ of estimator unbiased is X ( 29 2 2 2 all for σ σ = S E 2 2 of estimator unbiased is σ S
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Bias Exampl e   ( 29 θ , 0 ~ ,..., , 2 1 U X X X iid n Random sample from uniform X 2 ˆ 1 = θ ( 29 n X X X ,..., , max ˆ 2 1 2 = θ ( 29 θ θ = 1 ˆ E Unbiased ( 29 θ θ 1 ˆ 2 + = n n E Biased ( 29 θ θ θ θ 1 1 1 ˆ 2 + - = - + = n n n bias underestimate ( 29 n X X X n n ,..., , max 1 ˆ 2 1 3 + = θ is unbiased
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Mean Square Error unbiased small variance unbiased large variance biased small variance biased large variance
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Mean Square Error θ target 1 θ unbiased is 1 θ 2 θ biased is 2 θ 1 2 n better tha is But θ θ ( 29 ( 29 - = 2 ˆ θ θ θ E MSE ( 29 ( 29 2 θ θ bias Var + = ( 29 ( 29 1 2 θ θ MSE MSE <
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Efficiency 2 1 to compared of efficiency θ θ ( 29 ( 29 ( 29 1 2 2 1 , θ θ θ θ MSE MSE eff = ( 29 1 , if than efficient more is 2 1 2 1 θ θ θ θ eff ( 29 ( 29 ( 29 unbiased. are ˆ and ˆ both if , 2 1 1 2 2 1 θ θ θ θ θ θ Var Var eff =
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Efficiency Exampl e   ( 29 θ , 0 ~ ,..., , 2 1 U X X X iid n Random sample from uniform X 2 ˆ 1 = θ ( 29 n X X X ,..., , max ˆ 2 1 2 = θ Unbiased Biased ( 29 θ θ 1 1 ˆ 2 + - = n bias ( 29 ( 29 ( 29 n n X Var Var MSE 3 12 4 2 ˆ ˆ 2 2 1 1 θ θ θ θ = × = = = ( 29 ( 29 ( 29 2 1 ˆ 2 2 2 + + = n n n Var θ θ ( 29 ( 29 ( 29 ( 29 ( 29 2 1 2 2 1 1 ˆ 2 2 2 2 2 + + = + + + + = n n n n n n MSE θ θ θ θ ( 29 ( 29 ( 29 ( 29 ( 29 2 for 1 6 2 1 ˆ ˆ ˆ , ˆ 2 1 1 2 + + = = n n n MSE MSE eff θ θ θ θ efficient more is 2 θ
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Efficiency Exampl e   ( 29 θ , 0 ~ ,..., , 2 1 U X X X iid n Random sample from uniform ( 29 n X X X ,..., , max ˆ 2 1 2 = θ ( 29 ( 29 ( 29 2 1 ˆ 2 2 2 + + = n n n Var θ θ ( 29 ( 29 ( 29 2 1 2 ˆ 2 2 + + = n n MSE θ θ ( 29 ( 29 ( 29 1 for 1 1 2 ˆ ˆ ˆ , ˆ 3 2 2 3 + = = n n n MSE MSE eff θ θ θ θ efficient more is 3 θ ( 29 n X X X n n ,..., , max 1 ˆ 2 1 3 + = θ Unbiased ( 29 ( 29 ( 29 ( 29 2 ˆ 1 ˆ ˆ 2 2 2 3 3 + = + = = n n Var n n Var MSE θ θ θ θ
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Consistency ( 29 1 ˆ lim 0, any for if of estimator consistent a is = < - ε θ θ ε θ θ n n n P n θ ˆ Estimator of θ based on a sample of size n θ θ → p n ˆ ( 29 0 ˆ lim = n n MSE θ of estimator consistent a is θ θ n
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Consistency Exampl e   ( 29 θ , 0 ~ ,..., , 2 1 U X X X iid n Random sample from uniform ( 29 ( 29 ( 29 2 1 2 ˆ 2 2 + + = n n MSE θ θ X 2 ˆ 1 = θ ( 29 n MSE 3 ˆ 2 1 θ θ = ( 29 n X X X ,..., , max ˆ 2 1 2 = θ ( 29 n X X X n n ,..., , max 1 ˆ 2 1 3 + = θ ( 29 ( 29 2 ˆ 2 3 + = n n MSE θ θ All of these three estimators are consistent.
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Method of Moment Estimator Population moment ( 29 θ of function =
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