Reference no: EM132254868
Assignment Questions -
Q1. Assume that X is a discrete random variable.
(a) Based on an observed value of X, derive the most powerful test of H0 : X ∼ GEO(0.05) vs H0 : X -∼ POI(0.95) with α = 0.0975.
(b) Find the power of this test under the alternative.
Q2. Let X1 . . . Xn have joint pdf f(x1, . . . , xn; θ) and S be a sufficient statistic for θ. Show that a most powerful test of H0 : θ = θ0 vs H0 : θ = θ1 can be expressed in terms of S.
Q3. Consider a random sample of size n from a normal distribution with mean zero, X ∼ N(0, σ2). It is desired to test H0 : σ2 = σ02 vs Ha : σ2 ≠ σ02 based on the test statistic S0 = ΣXi2/σ02. Consider a test with critical region of the form C = {(x1, . . . , xn)| s0 ≤ c1 or s0 ≥ c2} where ΣXi2/σ02, c1 and c2 are chosen to provide a test of size α. Show that the power function of such a test has the form π(σ2) = 1 - H(c2σ02/σ2; n) + H(c1σ02/σ2; n) where H(c; n) is the CDF of Χ2(n).
Q4. Let X1, . . . , Xn, be a random sample from a continuous distribution.
(a) Show that the GLR for testing H0 : X ∼ N(μ, σ2) against Ha : X ∼ EXP(θ, η) is a function of θ^/σ^.
(b) Is the distribution of this statistic free of unknown parameters under H0?
(c) Show that the GLR for testing H0 : X ∼ N(μ, σ2) vs Ha : X ∼ DE(θ, η) is a function of θ^/σ^.