GLRT for One-Sided Tests: PSTAT 120C, Assignments of Asian literature

Download GLRT for One-Sided Tests: PSTAT 120C and more Assignments Asian literature in PDF only on Docsity! PSTAT 120C: GLRT for 1-sided tests April 7, 2009 Announcements • HW #1 – hold question 2 until next week • Office hours: T 11-12, W 1:30-2:30 • Honors contract General Likelihood Ratio Tests • t test • Monotone likelihood ratios • One-sided tests • Log likelihood approximations • More parameters to worry about Finding k, or not? We find the likelihood ratio λ(x1, . . . , xn); we reject when the ratio is less than k. +++ When λ is a monotone function of a statistic t then the problem is easier. Statistic t is a function of the data x1, . . . , xn Increasing If t1 > t2 then λ(t1) ≥ λ(t2) =⇒ If λ(t1) < k then λ(t2) < k =⇒ There is a c such that if t < c then λ(t) < k. =⇒ Our critical region is of the form {T < c}. Decreasing If t1 > t2 then λ(t1) ≤ λ(t2) =⇒ If λ(t2) < k then λ(t1) < k =⇒ There is a c such that if t > c then λ(t) < k. =⇒ Our critical region is of the form {T > c}. We don’t have to calculate the k or the distribution of λ. All we need is to find c such that P{T < c} = α (or > c for a decreasing function) under the null hypothesis. 1 Example • Say that our likelihood ratio is λ(x1, . . . , xn) = exp [−x̄+ 2] • This is a decreasing function of x̄. • Therefore a critical region of the form {x̄ > c} is the GLRT. More parameters • t test is a GLRT • For n iid x1, . . . , xn H0 : µ = µ0 vs Ha : µ 6= µ0 • We have two parameters in our model µ and σ • MLE µ̂ = x̄ σ̂2 = 1 n ∑ (xi − x̄)2 = n− 1 n s2 • The tricky part max θ∈Θ0 L(θ) = max σ>0 n∏ i=1 f(xi | 10, σ) • We need to maximize the likelihood given µ = µ0 L(v) = 1 (2πv)n/2 exp [ − n∑ i=1 (xi − µ0)2 2v ] LL(v) = −n 2 log 2π − n 2 log v − n∑ i=1 (xi − µ0)2 2v ∂ ∂v LL(v) = − n 2v + 1 2v2 ∑ (xi − µ0)2 = 0 =⇒ v̂0 = 1 n n∑ i=1 (xi − µ0)2 • Likelihood ratio becomes λ = L(µ0, σ̂20) L(x̄, σ̂2) = ( σ̂2 σ̂20 )n/2 exp [ − 1 2σ̂20 ∑ (xi − µ0)2 + 1 2σ̂2 ∑ (xi − x̄)2 ] = ( σ̂2 σ̂20 )n/2 2