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 ∼ 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 asymptotic LRT for this scenario.
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)?
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
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 ∼ 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, X2, . . . , Xn be IID N(0, σ2 ) variables. Find the rejection region for the likelihood ratio test at level α = 0.1 for testing H0 : σ2 = 1 vs H1 : σ2 = 2.
Let X1, X2, . . . , Xn be a random sample of size n from a normal population with mean µX and variance σ ^2 . Let Y1, Y2, . . . , Ym be a random sample of size m from a normal population with mean µY and variance σ ^2 . Also, assume that these two random samples are independent. It is desired to test the following hypotheses H0 : σX = σY versus H1 : σX...