2. Let X1, , Xn be iid exponential(9) random variables. Derive the LRT of Ho :...
Suppose that X1,X2, ,Xn are iid N(μ, σ2), where both parameters are unknown. Derive the likelihood ratio test (LRT) of Ho : σ2 < σ1 versus Ho : σ2 > σ.. (a) Argue that a LRT will reject Ho when w(x)S2 2 0 is large and find the critical value to confer a size α test. (b) Derive the power function of the LRT
Q6: Let X1, ..., Xn be a random sample of size n from an exponential distribution, Xi ~ EXP(1,n). A test of Ho : n = no versus Hain > no is desired, based on X1:n. (a) Find a critical region of size a of the form {X1:n > c}. (b) Derive the power function for the test of (a).
Suppose X1, X2, .., Xn is an iid sample from where >0. (a) Derive the size α likelihood ratio test (LRT) for Ho : θ-Bo versus H : θ θο. Derive the power function of the LRT (b) Suppose that n 10, Derive the most powerful (MP) level α-0.10 test of Ho : θ 1 versus Hi: 0-2. Calculate the power of your test
please see picture 5. Let X1, X2,..., Xn be Bin(2,0) random variables with Θ {.45, .65). For testing Ho : θ 45 versus HA : θ-66, determine the following: (a) the form of the Neyman-Pearson MP critical region for a size a test (b) the sampling distribution of 2iI X (c) the value of ho for α A.05 when n-20. (d) π(8) for α .05 when n-20. a random sample of lid 5. Let X1, X2,..., Xn be Bin(2,0) random...
Let Xi, .Χίο be a sample of iid Bin( 1, θ) random variables, and let e-{i : Σοί HA : θ 0.8. Determine a) the size of this critical region. b) the power of this critical region for 0 0.8. x,2 9} be a critical region for testing Ho:0 0.6 versus Let Xi, .Χίο be a sample of iid Bin( 1, θ) random variables, and let e-{i : Σοί HA : θ 0.8. Determine a) the size of this critical...
Suppose that X1, X2, ..., Xn are independent random variables (not iid) with densities ÍXi(z10,) -.2 e _ θ:/z1(z > 0), where θί 〉 0, for i = 1, 2, , n. (a) Derive the form of the likelihood ratio test (LRT) statistic for testing versuS H1: not Ho. You do not have to find the distribution of the likelihood ratio test (LRT) statistic under Ho- Just find the form of the statistic. (b) From your result in part (a),...
3. Suppose that we have two independent random samples: X1, Xn are exponential(), and Y1,... . Ym are μ. You do not need to find the critical value of exponential(μ). Find the LRT of H0 : θ the test. μ versus Ha : θ
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
3. Let X1, ..., Xn, ... be iid random variables from the shifted exponential distribution: Se-(2-0) f( x0) = л VI (a) Find the MLE for 0. (b) Find the MLE for ø= EX. (c) Find the MOM estimator for 0.
Suppose X1,X2, ,Xm are iid exponential with mean A. Suppose Yı,Yo, exponential with mean β2-Suppose the samples are independent. , Yn are iid (a) Derive the likelihood ratio test (LRT) statistic λ(x,y) for testing versus and show that it is a function of ti-ti (x)-Σ-iz; and t2-t2(y)-Σ1Uj. (b) Show how you could perform a size a test in part (a) using the F distribution Suppose X1,X2, ,Xm are iid exponential with mean A. Suppose Yı,Yo, exponential with mean β2-Suppose the...