5102-Lecture-11

5102-Lecture-11 - Lecture 11 Stat 5102-004 11 February 2011...

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Unformatted text preview: Lecture 11 Stat 5102-004 11 February 2011 Confidence Regions Let B be a set whose elements are subsets of the parameter space ϑ. B ∈ B =⇒ B ⊂ ϑ ˆ The region estimator Bγ : Y → B is a level γ confidence region if ˆ Pr(θ ∈ Bγ ) = γ The confidence region is approximate if ˆ Pr(θ ∈ Bγ ) ≈ γ for all θ ∈ ϑ. for all θ ∈ ϑ. STAT 5102 (Theory of Statistics) Lecture 11 1/9 Normal Mean, Variance Known Let Y1 , . . . , Yn ∼ N(µ, σ 2 ) with σ 2 known. σ σ ¯ ¯ Pr Y − √ z (γ ) ≤ µ ≤ Y + √ z (γ ) n n =γ iid point estimate ± standard error × critical value σ σ ˆ Bγ (y ) = y + − √ z (γ ), √ z (γ ) ¯ n n z (0.90) = 1.645 z (0.95) = 1.96 z (0.99) = 2.576 ˆ Pr(µ ∈ Bγ ) = γ for all µ STAT 5102 (Theory of Statistics) Lecture 11 2/9 Normal Mean, Variance Known, γ = 0.90 q qq q q qq q q q qq qq qq q qq q q q qq q qq q qq q q qq q q qq q q q q qq q q qq q q qq q q qq q q q qq q q qq qq q q qq qq qq q q q qq q qq q q qq q qq q q q qq q q q q qq q q q q qq q q qq q qq q q q q qq q q q qq q qq q q q q q q q qq q q q q q qqq qq q q q q q q q q qq q q qq q q q q q q q q qq qq q qq q q q q q qq q q qq q q q q q q q q q q qq q µ − 2σ n µ µ + 2σ n µ − 2σ n µ µ + 2σ n STAT 5102 (Theory of Statistics) Lecture 11 3/9 Normal Mean, Variance Known, γ = 0.95 q qq q q qq q q q qq qq qq q qq q q q qq q qq q qq q q qq q q qq q q q q qq q q qq q q qq q q qq q q q qq q q qq qq q q qq qq qq q q q qq q qq q q qq q qq q q q qq q q q q qq q q q q qq q q qq q qq q q q q qq q q q qq q qq q q q q q q q qq q q q q q qqq qq q q q q q q q q qq q q qq q q q q q q q q qq qq q qq q q q q q qq q q qq q q q q q q q q q q qq q µ − 2σ n µ µ + 2σ n µ − 2σ n µ µ + 2σ n STAT 5102 (Theory of Statistics) Lecture 11 4/9 Normal Mean, Variance Known, γ = 0.99 q qq q q qq q q q qq qq qq q qq q q q qq q qq q qq q q qq q q qq q q q q qq q q qq q q qq q q qq q q q qq q q qq qq q q qq qq qq q q q qq q qq q q qq q qq q q q qq q q q q qq q q q q qq q q qq q qq q q q q qq q q q qq q qq q q q q q q q qq q q q q q qqq qq q q q q q q q q qq q q qq q q q q q q q q qq qq q qq q q q q q qq q q qq q q q q q q q q q q qq q µ − 2σ n µ µ + 2σ n µ − 2σ n µ µ + 2σ n STAT 5102 (Theory of Statistics) Lecture 11 5/9 Normal Variance, Unknown Mean Let Y1 , . . . , Yn ∼ N(µ, σ 2 ) with µ and σ 2 both unknown. n σ2 ˆ ∼ χ2 (n − 1) σ2 Pr c1 (γ, n ) ≤ n σ2 ˆ ≤ c2 (γ, n ) σ2 =γ iid The shape of the reference distribution changes with the sample size. Pr n σ2 ˆ n σ2 ˆ ≤ σ2 ≤ c2 (γ, n ) c1 (γ, n ) =γ ˆ Bγ (y ) = σ 2 ˆ n n , c2 (γ, n ) c1 (γ, n ) STAT 5102 (Theory of Statistics) Lecture 11 6/9 Normal Variance, Unknown Mean γ = 0.90 n = 25 q q qq q qq q q qq q q q qq q q q qq q qq q q q qqq q qq qq qq q qq q qqq q qq q qq q qq q qq q q q q q qq qqq qq q q qq q qq q q q qq q q q qq q qq q q q qq q q qq qq q qq q qq q qq q qq qq q q qq q q qq q q q qq q q qq q qq q qq q q q qq q q q qq q q qq q qq q q q qq q q qq q q q qqq q qq q qq q q q qq qq qq q q qq qq qq q q q q qq q qq σ2 σ2 STAT 5102 (Theory of Statistics) Lecture 11 7/9 Normal Variance, Unknown Mean γ = 0.95 n = 25 q q qq q qq q q qq q q q qq q q q qq q qq q q q qqq q qq qq qq q qq q qqq q qq q qq q qq q qq q q q q q qq qqq qq q q qq q qq q q q qq q q q qq q qq q q q qq q q qq qq q qq q qq q qq q qq qq q q qq q q qq q q q qq q q qq q qq q qq q q q qq q q q qq q q qq q qq q q q qq q q qq q q q qqq q qq q qq q q q qq qq qq q q qq qq qq q q q q qq q qq σ2 σ2 STAT 5102 (Theory of Statistics) Lecture 11 8/9 Normal Variance, Unknown Mean γ = 0.99 n = 25 q q qq q qq q q qq q q q qq q q q qq q qq q q q qqq q qq qq qq q qq q qqq q qq q qq q qq q qq q q q q q qq qqq qq q q qq q qq q q q qq q q q qq q qq q q q qq q q qq qq q qq q qq q qq q qq qq q q qq q q qq q q q qq q q qq q qq q qq q q q qq q q q qq q q qq q qq q q q qq q q qq q q q qqq q qq q qq q q q qq qq qq q q qq qq qq q q q q qq q qq σ2 σ2 STAT 5102 (Theory of Statistics) Lecture 11 9/9 ...
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This note was uploaded on 02/24/2011 for the course STAT 5102 taught by Professor Staff during the Spring '03 term at Minnesota.

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