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2. STATISTICAL INFERENCE Figure 20: Relationships <strong>of</strong> the tests to the log-likelihood function Solve for θ 0 in W 2 ≤ z(α/2) 2 . Recall, W = √ ni(ˆθ)(ˆθ − θ 0 ) =⇒ ˆθ − √ z(α/2) ni(ˆθ) ≤ θ 0 ≤ ˆθ + √ z(α/2) ni(ˆθ) i.e., ˆθ ± z(α/2) √ . ni(ˆθ) Score Test Need to solve for θ 0 in V 2 = ( U(θ 0 ; x) √ ni(θ0 )) 2 ≤ z(α/2) 2 . LR Test Solve for θ 0 in 2(l(ˆθ; x) − l(θ 0 ; x)) ≤ χ 2 1,α = z(α/2) 2 . Example: X 1 , . . . , X n i.i.d. Po(λ). 109
2. STATISTICAL INFERENCE Recall Wald statistic is W = ˆλ − λ 0 √ˆλ/n =⇒ ˆλ ± √ z(α/2) ˆλ/n ⇐⇒ ¯x ± z(α/2) √¯x/n. Score test: Recall that the test statistic is V = ¯x − λ 0 √ . Hence to find a confidence interval, we λ0 /n need to solve for λ 0 in the equation V 2 ≤ z(α/2) 2 =⇒ (¯x − λ 0) 2 λ 0 /n ≤ z(α/2) 2 =⇒ (¯x − λ 0 ) 2 ≤ λ 0z(α/2) 2 ) =⇒ λ 2 0 − (2¯x + z(α/2)2 λ + ¯x 2 ≤ 0, n which is a quadratic in λ 0 and hence can be solved: −b ± √ b 2 − 4ac . 2a n For the LR test, we have { G 2 = 2n ¯x log ¯x } − (¯x − λ 0 ) λ 0 → can solve numerically for λ 0 in G 2 ≤ z(α/2) 2 . See Figure 21. Optimal Tests Consider simple null and alternative hypotheses: H 0 : θ = θ 0 and H a : θ = θ a Recall that the type I error rate is defined by α = P (reject H 0 |H 0 true). and the type II error rate by β = P (accept H 0 |H 0 false). The power is defined by 1 − β = P (reject H 0 |H 0 false) → so we want high power. 110
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MATHEMATICAL STATISTICS III Lecture
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1.9 Moments . . . . . . . . . . . .
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1. DISTRIBUTION THEORY ⎧∑ h(x)p
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n(1 − p) E(X) = , p n(1 − p) Va
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1. DISTRIBUTION THEORY ) ( N ) p(x)
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1. DISTRIBUTION THEORY that is, ⎧
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1. DISTRIBUTION THEORY 3. Gamma (K,
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1. DISTRIBUTION THEORY F Y (y) = P
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1. DISTRIBUTION THEORY Solution. Ob
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1. DISTRIBUTION THEORY 1.6 Moments
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1. DISTRIBUTION THEORY where φ(a)
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1. DISTRIBUTION THEORY Examples 1.
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1. DISTRIBUTION THEORY Possible val
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1. DISTRIBUTION THEORY Remarks 1. O
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1. DISTRIBUTION THEORY 2. Suppose Z
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