Let X1, . . . , Xn ∼ Exp(θ) and we wish to test H0 : θ = θ_0 vs H1 : θ not= θ_0. Find the asymptotic LRT for this scenario.
Let X1, . . . , Xn ∼ Exp(θ) and we wish to test H0 : θ = θ_0 vs H1 : θ not= θ_0. Find the asympto...
Let X1, . . . , Xn ∼ Exp(θ) and we wish to test H0 : θ = θ_0 vs H1 : θ not= θ_0. Find the asymptotic LRT for this scenario.
Let X1, . . . , Xn ∼ Exp(θ) and consider the test for H0 : θ ≥ θ_0 vs H1 : θ < θ_0. (a) Find the size-α LRT. With rejection region, R = {sample mean > c} where c will depend on a value from the χ ^2 df=2n distribution. (b) Find the appropriate value of c.
Let X1, . . . , Xn ∼ Geo(θ), f(x)= θ(1-θ)^x, and we wish to test H0 : θ ≤ 1/3 vs H1 : θ > 1/3. a) Using the full sample, X1....Xn, find the form of the UMP test for the hypotheses H0: θ=1/3 vs H1: θ=1/2. b)If n=15 and α = 0.1, what is the rejection region and the size of test in (a)?
Let X1, . . . , Xn ∼ Geo(θ), f(x)= θ(1-θ)^x, and we wish to test H0 : θ ≤ 1/3 vs H1 : θ > 1/3. a) Using the full sample, X1....Xn, find the form of the UMP test for the hypotheses H0: θ=1/3 vs H1: θ=1/2. b)If n=15 and α = 0.1, what is the rejection region and the size of test in (a)?
Suppose X1, . . . , Xn be a random sample from the Beta(θ, 1) distribution. Find the P-value for the LRT test of the hypotheses H0 : θ ≥ 1 vs H1 : θ < 1.
Let X1, . . . , Xn be independent Gamma(2, θ) random variables. The goal is to test H0 : θ = 2 versus H1 : θ not equal to 2. (1) Find the test statistic Λ. (2) Derive the rejection region of the corresponding LRT
Let X1, . . . , Xn ∼ Unif(0, θ). a) Is this family MLR in Y = X(n) (the sample maximum)? (b) Find the UMP size-α test for H0 : θ ≤ θ_0 vs H1 : θ > θ_0.
Let X1, . . . , Xn ∼ iid Exp(λ) and Y1, . . . , Ym ∼ iid Exp(τ ) be independent random samples. (a) Find the restricted MLEs under the null hypothesis H0 : λ = τ . (b) Write out a formula for the LRT statistic, and describe how you could perform this test asymptotically.
Let X1, . . . , Xn ∼ iid log Normal (µ, σ^2 ) for σ^ 2 known. Find the LRT for H0 : µ = µ_0 vs H1 : µ not= µ_0. f(x)=(2π)^(-1/2)(xσ)^(-1)*exp(-(ln x-µ)^2 /(2σ^2))
Let X1, . . . , Xn ∼ independent N(µ, τ ) for µ known. Find the Likelihood Ratio Test for H0 : τ ≥ τ_0 vs H1 : τ < τ_0. Use the fact that MLE of τ is (1/n)S^2.